ARITHMETIC OVER FUNCTION FIELDS

JASON MICHAEL STARR

Abstract. These notes accompany lectures presented at the Clay Mathematics Institute 2006 Summer School on Arithmetic Geometry. The lectures summarize some recent progress on existence of rational points of projective varieties defined over a function field over an algebraically closed field.

1. Introduction

These notes accompany lectures presented at the Clay Mathematics Institute 2006 Summer School on Arithmetic Geometry. They are more complete than the lectures themselves. Exercises assigned during the lectures are proved as lemmas or propositions in these notes. Hopefully this makes the notes useful to a wider audience than the original participants of the summer school.

This report describes some recent progress on questions in the interface between arithmetic geometry and algebraic geometry. In fact the questions come from arithmetic geometry: what is known about existence and “abundance” of points on algebraic varieties defined over a non-algebraically closed field K. But the answers are in algebraic geometry, i.e., they apply only when the field K is the function field of an algebraic variety over an algebraically closed field. For workers in number theory, such answer are of limited interest. But hopefully the techniques will be of interest, perhaps as simple analogues for more advanced techniques in arithmetic. With regards to this hope, the reader is encouraged to look at two articles on the arithmetic side, [GHMS04a] and [GHMS04b]. Also, of course, the answers have interesting consequences within algebraic geometry itself.

There are three sections corresponding to the three lectures I delivered in the summer school. The first lecture proves the classical theorems of Chevalley-Warning and Tsen-Lang: complete intersections in projective space of sufficiently low degree defined over finite fields or over function fields always have rational points. These theorems imply corollaries about the Brauer group and Galois cohomology of these fields, which are also described.

The second section introduces rationally connected varieties and presents the proof of Tom Graber, Joe Harris and myself of a conjecture of Kollár, Miyaoka and Mori: every rationally connected fibration over a curve over an algebraically closed field of characteristic 0 has a section. The proof presented here incorporates simplifications due to A. J. de Jong. Some effort is made to indicate the changes necessary to prove A. J. de Jong’s generalization to separably rationally connected fibrations over curves over fields of arbitrary characteristic. In the course of the proof, we give a thorough introduction to the “smoothing combs” technique of Kollár, Miyaoka and Mori and its application to weak approximation for “generic jets” in smooth fibers of rationally connected fibrations. This has been significantly generalized to weak approximation for all jets in smooth fibers by Hassett and Tschinkel, cf. [HT06]. Some corollaries of the Kollár-Miyaoka-Mori conjecture to Mumford’s conjecture, fixed point theorems, and fundamental groups are also described (these were known to follow before the conjecture was proved).

Date: June 28, 2007.

Finally, the last section hints at the beginnings of a generalization of the Kollár-Miyaoka-Mori conjecture to higher-dimensional function fields (not just function fields of curves). A rigorous result in this area is a second proof of A. J. de Jong’s Period-Index Theorem: for a division algebra D whose center is the function field K of a surface, the index of D equals the order of [D] in the Brauer group of K. This also ties together the first and second sections. Historically the primary motivation for the theorems of Chevalley, Tsen and Lang had to do with Brauer groups and Galois cohomology. The subject has grown beyond these first steps. But the newer results do have consequences for Brauer groups and Galois cohomology in much the same vein as the original results in this subject.

2. The Tsen-Lang theorem

A motivating problem in both arithmetic and geometry is the following.

Problem 2.1. Given a field K and a K-variety X find sufficient, resp. necessary, conditions for existence of a K-point of X.

The problem depends dramatically on the type of K: number field, finite field, p-adic field, function field over a finite field, or function field over an algebraically closed field. In arithmetic the number field case is most exciting. However the geometric case, i.e., the case of a function field over an algebraically closed field, is typically easier and may suggest approaches and conjectures in the arithmetic case.

Two results, the Chevalley-Warning theorem and Tsen’s theorem, deduce a sufficient condition for existence of K-points by “counting”. More generally, counting leads to a relative result: the Tsen-Lang theorem that a strong property about existence of k-points for a field k propagates to a weaker property about K-points for certain field extensions K/k. The prototype result, both historically and logically, is a theorem of Chevalley and its generalization by Warning. The counting result at the heart of the proof is Lagrange’s theorem together with the observation that a nonzero single-variable polynomial of degree ≤ q − 1 cannot have q distinct zeroes.

Lemma 2.2. For a finite field K with q elements, the polynomial 1−x^{q−1} vanishes on K^{∗} and x^{q} − x vanishes on all of K. For every integer n ≥ 0, for the K-algebra homomorphism

ev_{n} : K[X_{0}, …, X_{n}] → Hom_{Sets}(K^{n+1}, K),
ev_{n}(p(X_{0}, …, X_{n})) = ((a_{0}, …, a_{n}) ↦ p(a_{0}, …, a_{n})),

the kernel equals the ideal

I_{n} = ⟨X_{0}^{q} − X_{0}, …, X_{n}^{q} − X_{n}⟩.

Finally, the collection (X_{i}^{q} − X_{i})_{i=0,...,n} is a Gröbner basis with respect to every monomial order refining the grading of monomials by total order. In particular, for every p in I_{n} some term of p of highest degree is in the ideal ⟨X_{0}^{q}, …, X_{n}^{q}⟩.

Proof. Because K^{∗} is a group of order q − 1, Lagrange’s theorem implies a^{q−1} = 1 for every element a of K^{∗}, i.e., 1 − x^{q−1} vanishes on K^{∗}. Multiplying by x shows that x^{q} − x vanishes on K. Thus the ideal I_{n} is at least contained in the kernel of ev_{n}.

Modulo X_{n}^{q} − X_{n}, every element of K[X_{0}, …, X_{n}] is congruent to one of the form

p(X_{0}, …, X_{n}) = p_{q−1} · X_{n}^{q} + ⋯ + p_{0}X_{n}^{0}, p_{0}, …, p_{q−1} ∈ K[X_{0}, …, X_{n−1}].

(Of course K^{n} is defined to be {0} and K[X_{0}, …, X_{n−1}] is defined to be K if n equals 0.) Since K has q elements and since a nonzero polynomial of degree ≤ q −1 can have at most q−1 distinct zeroes, for every (a_{0}, …, a_{n−1}) ∈ K^{n} the polynomial p(a_{0}, …, a_{n−1}, X_{n}) is zero on K if and only if

p_{0}(a_{0}, …, a_{n−1}) = ⋯ = p_{q−1}(a_{0}, …, a_{n−1}).

Thus ev_{n}(p) equals 0 if and only if each ev_{n−1}(p_{i}) equals 0. In that case, by the induction hypothesis, each p_{i} is in I_{n−1} (in case n = 0, each p_{i} equals 0). Then, since I_{n−1}K[X_{0}, …, X_{n}] is in I_{n}, p is in I_{n}. Therefore, by induction on n, the kernel of ev_{n} is precisely I_{n}.

Finally, Buchberger’s algorithms applied to the set (X_{0}^{q}−X_{0}, …, X_{n}^{q}−X_{n}) produces S-polynomials

S_{i,j} = X_{j}^{q}(X_{i}^{q} − X_{i}) − X_{i}^{q}(X_{j}^{q} − X_{j}) = X_{j}(X_{i}^{q} − X_{i}) − X_{i}(X_{j}^{q} − X_{j})

which have remainder 0. Therefore this set is a Gröbner basis by Buchberger’s criterion. □

Theorem 2.3. [Che35],[War35] Let K be a finite field. Let n and r be positive integers. Let F_{1}, …, F_{r} be nonconstant, homogeneous polynomials in K[X_{0}, …, X_{n}]. If

deg(F_{1}) + ⋯ + deg(F_{r}) ≤ n

then there exists (a_{0}, …, a_{n}) ∈ K^{n+1} − {0} such that for every i = 1, …, r, F_{i}(a_{0}, …, a_{n}) equals 0. Stated differently, the projective scheme 𝕍(F_{1}, …, F_{r}) ⊂ ℙ_{K}^{n} has a K-point.

Proof. Denote by q the number of elements in K. The polynomial

G(X_{0}, …, X_{n}) = 1 − ∏_{i=0}^{n}(1 − X_{i}^{q−1})

equals 0 on {0} and equals 1 on K^{n+1} − {0}. For the same reason, the polynomial

H(X_{0}, …, X_{n}) = 1 − ∏_{j=1}^{r}(1 − F_{j}(X_{0}, …, X_{n})^{q−1})

equals 0 on

{(a_{0}, …, a_{n}) ∈ K^{n+1}|F_{1}(a_{0}, …, a_{n}) = ⋯ = F_{r}(a_{0}, …, a_{n}) = 0}

and equals 1 on the complement of this set in K^{n+1}. Since each F_{i} is homogeneous, 0 is a common zero of F_{1}, …, F_{r}. Thus the difference G − H equals 1 on

{(a_{0}, …, a_{n}) ∈ K^{n+1} − {0}|F_{1}(a_{0}, …, a_{n}) = ⋯ = F_{r}(a_{0}, …, a_{n}) = 0}

and equals 0 on the complement of this set in K^{n+1}. Thus, to prove that F_{1}, …, F_{r} have a nontrivial common zero, it suffices to prove the polynomial G − H does not lie in the ideal I_{n}.

Since

deg(F_{1}) + ⋯ + deg(F_{r}) ≤ n,

H has strictly smaller degree than G. Thus the leading term of G − H equals the leading term of G. There is only one term of G of degree deg(G). Thus, for every monomial ordering refining the grading by total degree, the leading term of G equals

(−1)^{n+1}X_{0}^{q−1}X_{1}^{q−1} ⋯ X_{n}^{q−1}.

This is clearly divisible by none of X_{i}^{q} for i = 0, …, n, i.e., the leading term of G − H is not in the ideal ⟨X_{0}^{q}, …, X_{n}^{q}⟩. Because (X_{0}^{q} − X_{0}, …, X_{n}^{q} − X_{n}) is a Gröbner basis for I_{n} with respect to the monomial order, G − H is not in I_{n}. □

On the geometric side, an analogue of Chevalley’s theorem was proved by Tsen, cf. [Tse33]. This was later generalized independently by Tsen and Lang, cf. [Tse36], [Lan52]. Lang introduced a definition which simplifies the argument.

Definition 2.4. [Lan52] Let m be a nonnegative integer. A field K is called C_{m}, or said to have property C_{m}, if it satisfies the following. For every positive integer n and every sequence of positive integers (d_{1}, …, d_{r}) satisfying

d_{1}^{m} + ⋯ + d_{r}^{m} ≤ n,

every sequence (F_{1}, …, F_{r}) of homogeneous polynomials F_{i} ∈ K[X_{0}, …, X_{n}] with deg(F_{i}) = d_{i} has a common zero in K^{n+1} − {0}.

Remark 2.5. In fact the definition in [Lan52] is a little bit different than this. For fields having normic forms, Lang proves the definition above is equivalent to his definition. And the definition above works best with the following results.

With this definition, the statement of the Chevalley-Warning theorem is quite simple: every finite field has property C_{1}. The next result proves that property C_{m} is preserved by algebraic extension.

Lemma 2.6. For every nonnegative integer m, every algebraic extension of a field with property C_{m} has property C_{m}.

Proof. Let K be a field with property C_{m} and let L′/K be an algebraic extension. For every sequence of polynomials (F_{1}, …, F_{r}) as in the definition, the coefficients generate a finitely generated subextension L/K of L′/K. Thus clearly it suffices to prove the lemma for finitely generated, algebraic extensions L/K.

Denote by e the finite dimension dim_{K}(L). Because multiplication on L is Kbilinear, each homogeneous, degree d_{i}, polynomial map of L-vector spaces,

F_{i} : L^{⊕(n+1)} → L,

is also a homogeneous, degree d_{i}, polynomial map of K-vector spaces. Choosing a K-basis for L and decomposing F_{i} accordingly, F_{i} is equivalent to e distinct homogeneous, degree d_{i}, polynomial maps of K-vector spaces,

F_{i,j} : L^{⊕(n+1)} → K, j = 1, …, e.

The set of common zeroes of the collection of homogeneous polynomial maps (F_{i}|i = 1, …, r) equals the set of common zeroes of the collection of homogeneous polynomial functions (F_{i,j}|i = 1, …, r, j = 1, …, e). Thus it suffices to prove there is a nontrivial common zero of all the functions F_{i,j}.

By hypothesis,

∑_{i=1}^{r} deg(F_{i})^{m} is no greater than n.

Thus, also

∑_{i=1}^{r} ∑_{j=1}^{e} deg(F_{i,j})^{m} = e ∑_{i=1}^{r} deg(F_{i})^{m} is no greater than en.

Since K has property C_{m} and since

dim_{K}(L^{⊕(n+1)}), i.e., (n + 1)dim_{K}(L) = e(n + 1),

is larger than en, the collection of homogeneous polynomials F_{i,j} has a common zero in L^{⊕(n+1)} − {0}. □

The heart of the Tsen-Lang theorem is the following proposition.

Proposition 2.7. Let K/k be a function field of a curve, i.e., a finitely generated, separable field extension of transcendence degree 1. If k has property C_{m} then K has property C_{m+1}.

This is proved in a series of steps. Let n, r and d_{1}, …, d_{r} be positive integers such that

d_{1}^{m+1} + ⋯ + d_{r}^{m+1} ≤ n.

For every collection of homogeneous polynomials

F_{1}, …, F_{r} ∈ K[X_{0}, …, X_{n}], deg(F_{i}) = d_{i},

the goal is to prove that the collection of homogeneous, degree d_{i}, polynomial maps of K-vector spaces

F_{1}, …, F_{r} : K^{⊕(n+1)} → K

has a common zero. Of course, as in the proof of Lemma 2.6, this is also a collection of homogeneous polynomial maps of k-vector spaces. Unfortunately both of these k-vector spaces are infinite dimensional. However, using geometry, these polynomial maps can be realized as the colimits of polynomial maps of finite dimensional k-vector spaces. For these maps there is an analogue of the Chevalley-Warning argument replacing the counting argument by a parameter counting argument which ultimately follows from the Riemann-Roch theorem for curves. The first step is to give a projective model of K/k.

Lemma 2.8. For every separable, finitely generated field extension K/k of transcendence degree 1, there exists a smooth, projective, connected curve C over k and an isomorphism of k-extensions K ≅ k(C). Moreover the pair (C, K ≅ k(C)) is unique up to unique isomorphism.

Proof. This is essentially the Zariski-Riemann surface of the extension K/k. For a proof in the case that k is algebraically closed, see [Har77, Theorem I.6.9]. The proof in the general case is similar. □

The isomorphism K ≅ k(C) is useful because the infinite dimensional k-vector space k(C) has a plethora of naturally-defined finite dimensional subspaces. For every Cartier divisor D on C, denote by V_{D} the subspace

V_{D} := H^{0}(C, O_{C}(D)) = {f ∈ k(C)|div(f) + D ≥ 0}.

The collection of all Cartier divisors D on C is a partially ordered set where

D′ ≥ D if and only if D′ − D is effective.

The system of subspaces V_{D} of k(C) is compatible for this partial order, i.e., if D′ ≥ D then V_{D′} ⊃ V_{D}. And K is the union of all the subspaces V_{D}, i.e., it is the colimit of this compatible system of finite dimensional k-vector spaces. Thus for all k-multilinear algebra operations which commute with colimits, the operation on k(C) can be understood in terms of its restrictions to the finite dimensional subspaces k(C). The next lemma makes this more concrete for the polynomial map F.

Lemma 2.9. Let C be a smooth, projective, connected curve over a field k and let

F_{i} ∈ k(C)[X_{0}, …, X_{n}]_{d_{i}}, i = 1, …, r

be a collection of polynomials in the spaces k(C)[X_{0}, …, X_{n}]_{d_{i}} of homogeneous, degree d_{i} polynomials. There exists an effective, Cartier divisor P on C and for every i = 1, …, r there exists a global section F_{C,i} of the coherent sheaf O_{C}(P)[X_{0}, …, X_{n}]_{d_{i}} such that for every i = 1, …, r the germ of F_{C,i} at the generic point of C equals F_{i}.

Remark 2.10. In particular, for every Cartier divisor D on C and for every i = 1, …, r there is a homogeneous, degree d, polynomial map of k-vector spaces

F_{C,D,i} : V_{D}^{⊕(n+1)} → W_{d_{i},P,D}, W_{d_{i},P,D} := V_{d_{i}D+P} ,

such that for every i = 1, …, r the restriction of F_{i} to V_{D}^{⊕(n+1)} equals F_{C,D,i} considered as a map with target K (rather than the subspace V_{dD+P}).

Proof. The coefficients of each F_{i} are rational functions on C. Each such function has a polar divisor. Since there are only finitely many coefficients of the finitely many polynomials F_{1}, …, F_{r}, there exists a single effective, Cartier divisor P on C such that every coefficient is a global section of O_{C}(P). □

Because of Lemma 2.9, the original polynomial maps F_{1}, …, F_{r} can be understood in terms of their restrictions to the subspaces V_{D}. The dimensions of these subspaces are determined by the Riemann-Roch theorem.

Theorem 2.11 (Riemann-Roch for smooth, projective curves). Let k be a field. Let C be a smooth, projective, connected curve over k. Denote by ω_{C/k} the sheaf of relative differentials of C over k and denote by g(C) = genus(C) the unique integer such that deg(ω_{C/k}) = 2g(C) − 2. For every invertible sheaf L on C,

h^{0}(C, L) − h^{0}(C, ω_{C} ⊗_{O_{C}} L^{∨}) = deg(L) + 1 − g(C).

Remark 2.12. In particular, if deg(L) > deg(ω_{C}) = 2g(C) − 2 so that ω_{C} ⊗_{O_{C}} L^{∨} has negative degree, then h^{0}(C, ω_{C} ⊗_{O_{C}} L^{∨}) equals zero. And then

h^{0}(C, L) = deg(L) + 1 − g(C).

For a Cartier divisor D satisfying

deg(D) > 2g(C) − 2 and for each i = 1, …, r, d_{i}deg(D) + deg(P) > 2g(C) − 2, the Riemann-Roch theorem gives that V_{D}^{⊕(n+1)} and W_{d_{i},P,D} are finite dimensional k-vector spaces of respective dimensions,

dim_{k}(V_{D}^{⊕(n+1)}) = (n + 1)h^{0}(C, O_{C}(D)) = (n + 1)(deg(D) + 1 − g)

and

dim_{k}(W_{d_{i},P,D}) = dim(V_{d_{i}D+P}) = d_{i}deg(D) + deg(P) + 1 − g.

In this case, choosing a basis for W_{d_{i},P,D} and decomposing

F_{C,D,i} : V_{D}^{⊕(n+1)} → W_{d_{i},P,D}

into its associated components, there exist dim_{k}(W_{d_{i},P,D}) homogeneous, degree d, polynomial functions

(F_{C,D,i})_{j} : V_{D}^{⊕(n+1)} → k, j = 1, …, dim_{k}(W_{d_{i},P,D})

such that a zero of F_{C,D,i} is precisely the same as a common zero of all the functions (F_{C,D,i})_{j}.

Proof of Proposition 2.7. By hypothesis, each d_{i} and n+1−∑_{i=1}^{r} d_{i}^{m+1} are nonzero so that the fractions

\frac{2g(C) − 2 − deg(P)}{d_{i}} for each i = 1, …, r, \frac{(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)}{n + 1 − ∑_{i=1}^{r} d_{i}^{m+1}}

are all defined. Let D be an effective, Cartier divisor on C such that

deg(D) > 2g(C) − 2, deg(D) > \frac{2g(C) − 2 − deg(P)}{d_{i}}, i = 1, …, r, and

deg(D) > \frac{(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)}{n + 1 − ∑_{i=1}^{r} d_{i}^{m+1}}.

Because deg(D) > 2g(C) − 2, the Riemann-Roch theorem states that

dim_{k}(V_{D}^{⊕(n+1)}) = (n + 1)dim_{k}(V_{D}) = (n + 1)(deg(D) + 1 − g).

For every i = 1, …, r, because d_{i} is positive and because deg(D) > (2g(C) − 2 − deg(P))/d_{i}, also

deg(d_{i}D + P) = d_{i}deg(D) + deg(P) is greater than 2g(C) − 2.

Thus the Riemann-Roch theorem states that

dim_{k}(W_{d_{i},P,D}) = dim_{k}(V_{d_{i}D+P}) = d_{i}deg(D) + deg(P) + 1 − g(C).

Thus for the collection of polynomial functions (F_{C,D,i})_{j},

dim_{k}(V_{D}^{⊕(n+1)}) − ∑_{i=1}^{r} ∑_{j} deg((F_{C,D,i})_{j})^{m}

equals

(n + 1)(deg(D) + 1 − g) − ∑_{i=1}^{r}(d_{i}deg(D) + deg(P) + 1 − g(C))d_{i}^{m} =

(n + 1 − ∑_{i=1}^{r} d_{i}^{m+1})deg(D) − [(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)].

Because

deg(D) > \frac{(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)}{n + 1 − ∑_{i=1}^{r} d_{i}^{m+1}}

and because n + 1 − ∑_{i=1}^{r} d_{i}^{m+1} is positive, also

(n + 1 − ∑_{i=1}^{r} d_{i}^{m+1})deg(D) > [(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)].

Therefore

dim_{k}(V_{D}^{⊕(n+1)}) is greater than ∑_{i=1}^{r} ∑_{j} deg((F_{i,C,D})_{j})^{m}.

Because of the inequality above, and because k has property C_{m}, there is a nontrivial common zero of the collection of homogeneous polynomial functions (F_{C,D,i})_{j}, i = 1, …, r, j = 1, …, dim_{k}(W_{d_{i},P_{i},D}). Therefore there is a nontrivial common zero of the collection of homogeneous polynomial maps F_{C,D,i}, i = 1, …, r. By Lemma 2.9, the image of this nonzero element in K^{⊕(n+1)} is a nonzero element which is a common zero of the polynomials F_{1}, …, F_{r}. □

Proposition 2.7 is the main step in the proof of the Tsen-Lang theorem.

Theorem 2.13 (The Tsen-Lang Theorem). [Lan52] Let K/k be a field extension with finite transcendence degree, tr.deg.(K/k) = t. If k has property C_{m} then K has property C_{m+t}.

Proof. The proof of the theorem is by induction on t. When t = 0, i.e., when K/k is algebraic, the result follows from Lemma 2.6. Thus assume t > 0 and the result is known for t − 1. Let (b_{1}, …, b_{t}) be a transcendence basis for K/k. Let E_{t}, resp. E_{t−1}, denote the subfield of K generated by k and b_{1}, …, b_{t}, resp. generated by k and b_{1}, …, b_{t−1}. Since E_{t−1}/k has transcendence degree t − 1, by the induction hypothesis E_{t−1} has property C_{m+t−1}. Now E_{t}/E_{t−1} is a purely transcendental extension of transcendence degree 1. In particular, it is finitely generated and separable. Since E_{t−1} has property C_{m+t−1}, by Proposition 2.7 E_{t} has property C_{m+t}. Finally by Lemma 2.6 again, since K/E_{t} is algebraic and E_{t} has property C_{m+t}, also K has property C_{m+t}. □

The homogeneous version of the Nullstellensatz implies a field k has property C_{0} if and only if k is algebraically closed. Thus one corollary of Theorem 2.13 is the following.

Corollary 2.14. Let k be an algebraically closed field and let K/k be a field extension of finite transcendence degree t. The field K has property C_{t}.

In particular, the case t = 1 is the historically the first result in this direction.

Corollary 2.15 (Tsen’s theorem). [Tse36] The function field of a curve over an algebraically closed field has property C_{1}.

Chevalley and Tsen recognized that property C_{1}, which they called quasi-algebraic closure, has an important consequence for division algebras. Lang recognized that property C_{2} also has an important consequence for division algebras, cf. [Lan52, Theorem 13]. Let K be a field. A division algebra with center K is a K-algebra D with center K such that every nonzero element of D has a (left-right) inverse. Although this is not always the case, we will also demand that dim_{K}(D) is finite.

Denote by \overline{K} the separable closure of K. Every division algebra with center K is an example of a central simple algebra over K, i.e., a K-algebra A with center K and dim_{K}(A) finite such that A ⊗_{K} \overline{K} is isomorphic as a \overline{K}-algebra to the algebra Mat_{n×n}(\overline{K}) of n × n matrices with entries in \overline{K} for some integer n. In particular, dim_{K}(A) = n^{2} for a unique integer n. For a division algebra D with center K, the unique integer n is called the index of D.

Let ϕ : A ⊗_{K} \overline{K} → Mat_{n×n}(\overline{K}) be an isomorphism of \overline{K}-algebras. There is an induced homogeneous, degree n, polynomial map of \overline{K}-vector spaces

det ∘ ϕ : A → Mat_{n×n}(\overline{K}) → \overline{K}.

By the Skolem-Noether theorem, every other isomorphism ϕ′ : A⊗_{K} \overline{K} → Mat_{n×n}(\overline{K}) is of the form conj_{a} ∘ ϕ where a ∈ Mat_{n×n}(\overline{K}) is an invertible element and

conj_{a} : Mat_{n×n}(\overline{K}) → Mat_{n×n}(\overline{K}), conj_{a}(b) = aba^{−1}

is conjugation by a. But det∘conj_{a} equals det. Thus the map det∘ϕ is independent of the particular choice of ϕ. Since the Galois group of \overline{K}/K acts on the polynomial map through its action on ϕ, the polynomial map is also Galois invariant. Therefore there exists a unique homogeneous, degree n, polynomial map of K-vector spaces

Nrm_{A/K} : A → K

such that for every isomorphism of \overline{K}-algebras ϕ, det ∘ ϕ equals Nrm_{A/K} ⊗ 1.

The homogeneous, polynomial map of K-vectors spaces Nrm_{A/K} is the reduced norm of A. It is multiplicative, i.e.,

∀a, b ∈ A, Nrm_{A/K}(ab) = Nrm_{A/K}(a)Nrm_{A/K}(b).

And the restriction to the center K is the polynomial map λ ↦ λ^{n}. These properties characterize the reduced norm. By the same type of Galois invariance argument as above, and using Cramer’s rule, an element a of A has a (left and right) inverse if and only if Nrm_{A/K}(a) is nonzero. In particular, if D is a division algebra the only zero of Nrm_{A/K} is a = 0.

Proposition 2.16. Let K be a field

(i) If K has property C_{1}, then the only division algebra with center K is K itself.

(ii) If K has property C_{2} then for every division algebra D with center K the

reduced norm map

Nrm_{D/K} : D → K

is surjective.

Proof. Let D be a division algebra with center K. Denote by n the index of D. Because Mat_{n×n}(\overline{K}) has dimension n^{2} as a \overline{K}-vector space, also D has dimension n^{2} as a K-vector space. If K has property C_{1}, then since the homogeneous polynomial map Nrm_{D/K} has only the trivial zero,

n = deg(Nrm_{D/K}) ≥ dim_{K}(D) = n^{2},

i.e., n = 1. Thus for a field K with property C_{1}, the only finite dimensional, division algebra with center K has dimension 1, i.e., D equals K.

Next suppose that K has property C_{2}. Clearly Nrm_{D/k}(0) equals 0. Thus to prove that

Nrm_{D/K} : D → K

is surjective, it suffices to prove that for every nonzero c ∈ K there exists b in D with Nrm_{D/K}(b) = c. Consider the homogeneous, degree n, polynomial map

F_{c} : D ⊕ K → K, (a, λ) ↦ Nrm_{D/K}(a) − cλ^{n}.

Since

dim_{K}(D ⊕ K) = n^{2} + 1 > deg(F_{c})^{2},

by property C_{2} the map F_{c} has a zero (a, λ) ≠ (0, 0), i.e., Nrm_{D/k}(a) = cλ^{n}. In particular, λ must be nonzero since otherwise a is a nonzero element of D with Nrm_{D/K}(a) = 0. But then b = (1/λ)a is an element of D with Nrm_{D/k}(b) = c. □

It was later recognized, particularly through the work of Merkurjev and Suslin, that these properties of division algebras are equivalent to properties of Galois cohomology. The cohomological dimension of a field K is the smallest integer cd(K) such that for every Abelian, discrete, torsion Galois module A and for every integer m > cd(K),

H^{m}(\overline{K}/K, A) = \{0\}.

Theorem 2.17. [Ser02, Proposition 5, §I.3.1], [Sus84, Corollary 24.9] Let K be a field.

(i) The cohomological dimension of K is ≤ 1 if and only if for every finite extension L/K, the only division algebra with center L is L itself. (ii) If K is perfect, the cohomological dimension of K is ≤ 2 if and only if for every finite extension L/K, for every division algebra D with center L, the reduced norm map Nrm_{D/L} is surjective.

3. Rationally connected varieties

The theorems of Chevalley-Warning and Tsen-Lang are positive answers to Problem 2.1 for a certain class of fields. It is natural to ask whether these theorems can be generalized for such fields.

Problem 3.1. Let r be a nonnegative integer. Give sufficient geometric conditions on a variety such that for every C_{r} field K (or perhaps every C_{r} field satisfying some additional hypotheses) and for every K-variety satisfying the conditions, X has a K-point.

As with Problem 2.1, this problem is quite vague. Nonetheless there are important partial answers. One such answer, whose proof was sketched in the lectures of Hassett in this same Clay Summer School, is the following.

Theorem 3.2. [Man86] [CT87] Let K be a C_{1} field and let X be a projective K-variety. If X ⊗_{K} \overline{K} is birational to ℙ2 ; \overline{K} then X has a K-point.

10

This begs the question: What (if anything) is the common feature of rational surfaces and of the varieties occurring in the Chevalley-Warning and Tsen-Lang theorems, i.e., complete intersections in ℙ^{n} of hypersurfaces of degrees d_{1}, …, d_{r} with d_{1} + ⋯ + d_{r} ≤ n? One answer is rational connectedness. This is a property that was studied by Kollár-Miyaoka-Mori and Campana, cf. [Kol96].

Definition 3.3. Let k be an algebraically closed field. An integral (thus nonempty), separated, finite type, k-scheme X is rationally connected, resp. separably rationally connected, if there exists an integral, finite type k-scheme M and a morphism of k-schemes

u : M ×_{k} ℙ_{k}^{1} → X, (m, t) ↦ u(m, t)

such that the induced morphism of k-schemes

u^{(2)} : M ×_{k} ℙ_{k}^{1} ×_{k} ℙ_{k}^{1} → X ×_{k} X, (m, t_{1}, t_{1}) ↦ (u(m, t_{1}), u(m, t_{2}))

is surjective, resp. surjective and generically smooth.

In a similar way, X is rationally chain connected, resp. separably rationally chain connected, if for some integer n ≥ 1, the analogous property holds after replacing ℙ_{k}^{1} by the proper, connected, nodal, reducible curve C_{n} which is a chain of n smooth rational curves.

The definition of rational connectedness, resp. rational chain connectedness, mentions a particular parameter space M. However, using the general theory of Hilbert schemes, it suffices to check that every pair (x_{1}, x_{2}) of K-points of X ⊗_{k} K is contained in some rational K-curve, resp. a chain of rational K-curves, (not necessarily from a fixed parameter space) for one sufficiently large, algebraically closed, field extension K/k, i.e., for an algebraically closed extension K/k such that for every countable collection of proper closed subvarieties Y_{i} ⊊ X, there exists a K-point of X contained in none of the sets Y_{i}. For instance, K/k is sufficiently large if K is uncountable or if K/k contains the fraction field k(X)/k as a subextension.

A very closely related property is the existence of a very free rational curve, i.e., a morphism

f : ℙ_{k}^{1} → X_{smooth}

into the smooth locus of X such that f^{∗}T_{X} is ample, i.e.,

f^{∗}T_{X} ≅ O_{ℙ_{k}^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ_{k}^{1}}(a_{n}), a_{1}, …, a_{n} > 0.

Definition 3.4. The very free locus X_{v.f.} of X is the union of the images in X of all very free rational curves to X ⊗_{k} K as K/k varies over all algebraically closed extensions.

The next theorem explains the relation of these different properties.

Theorem 3.5. [Kol96, §IV.3], [HT06], Unless stated otherwise, all varieties below are quasi-projective over an algebraically closed field k.

(0) In characteristic 0, every rationally connected variety is separably rationally connected.

(i) For every flat, proper morphism π : X → B (not necessarily of quasiprojective varieties over a field), the subset of B parameterizing points whose geometric fiber is rationally chain connected is stable under specialization. (If one bounds the degree of the chains with respect to a relatively ample invertible O_{X}-module, then it is a closed subset.)

(ii) The very free locus of a quasi-projective variety is open. More generally, for every flat, quasi-projective morphism π : X → B (not necessarily of quasiprojective varieties over a field) having irreducible geometric fibers, there exists an open subset X_{π,v.f.} of X such that for every geometric fiber X_{b} of π, (X_{b})_{v.f.} equals X_{π,v.f.} ×_{B} X_{b}. In particular, the subset of B parameterizing points whose geometric fiber contains a very free rational curve is an open subset, π(X_{π,v.f.}).

(iii) The very free locus X_{v.f.} of a quasi-projective variety is (separably) rationally connected in the following strong sense. For every positive integer N, for every positive integer m, and for every positive integer a, for every collection of distinct closed points t_{1}, …, t_{N} ∈ ℙ_{k}^{1}, for every collection of closed points x_{1}, …, x_{N} ∈ X_{v.f.}, and for every specification of an m-jet of a smooth curve in X at each point x_{i}, there exists a morphism

f : ℙ_{k}^{1} → X_{v.f.}

such that for every i = 1, …, n, f is unramified at t_{i}, f(t_{i}) equals x_{i} and the m-jet of t_{i} in ℙ_{k}^{1} maps isomorphically to the specified m-jet at x_{i}, and

f^{∗}T_{X} ≅ O_{ℙ_{k}^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ_{k}^{1}}(a_{n}), a_{1}, …, a_{n} ≥ a.

(iv) Every rational curve in X_{smooth} intersecting X_{v.f.} is contained in X_{v.f.}. Thus for every smooth, rationally chain connected variety, if X contains a very free rational curve then X_{v.f.} equals all of X.

(v) A proper, irreducible variety X is rationally chain connected if it is generically rationally chain connected, i.e., if there exists a morphism u as in the definition such that u^{(2)} is dominant (but not necessarily surjective).

(vi) For the morphism u : M ×_{k} ℙ_{k}^{1} → X, let m be a closed point of M such that u_{m} : ℙ^{1} → X has image in X_{smooth} and such that u^{(2)} is smooth at (m, t_{1}, t_{2}) for some t_{1}, t_{2} ∈ ℙ_{k}^{1}. Then the morphism u_{m} is very free. Thus an irreducible, quasi-projective variety X contains a very free curve if and only if there is a separably rationally connected open subset of X_{smooth}. Also, a smooth, quasi-projective variety X in characteristic 0 which is generically rationally connected contains a very free morphism.

(vii) For a surjective morphism f : X → Y of varieties over an algebraically closed field, if X is rationally connected, resp. rationally chain connected, then also Y is rationally connected, resp. rationally chain connected.

(viii) For a birational morphism f : X → Y of proper varieties over an algebraically closed field, if Y is rationally connected then X is rationally chain connected. If the characteristic is zero, then X is rationally connected.

Remark 3.6. Item (ii) is proved in Proposition 3.18. The generic case of Item (iii), which is all we will need, is proved in Proposition 3.19. The complete result was proved by Hassett and Tschinkel, [HT06]. Item (iv) follows from Corollary 3.20. The remaining items are not proved, nor are they used in the proof of the main theorem. For the most part they are proved by similar arguments; complete proofs are in [Kol96, §IV.3].

Rational connectedness is analogous to path connectedness in topology, and satisfies the analogues of many properties of path connectedness. One property of path connectedness is this: for a fibration of CW complexes, if the base space and the fibers are path connected, then also the total space is path connected. This led to two conjectures by Kollár, Miyaoka and Mori.

Conjecture 3.7. [Kol96, Conjecture IV.5.6] Let π : X → B be a surjective morphism of smooth, projective schemes over an algebraically closed field of characteristic 0. If both B and a general fiber of π are rationally connected, then X is also rationally connected.

Conjecture 3.7 is implied by the following conjecture about rationally connected fibrations over curves.

Conjecture 3.8. [Kol96, Conjecture IV.6.1.1] Let π : X → B be a surjective morphism of projective schemes over an algebraically closed field of characteristic 0. If B is a smooth curve and if a general fiber of f is rationally connected, then there exists a morphism s : B → X such that π ∘ s equals Id_{B}, i.e., s is a section of π.

Our next goal is to prove the following result.

Theorem 3.9. [GHS03] Conjecture 3.8 of Kollár-Miyaoka-Mori is true. Precisely, let k be an algebraically closed field of characteristic 0 and let π_{X} : X → B be a surjective morphism from a normal, projective k-scheme X to a smooth, projective, connected k-curve B. If the geometric generic fiber X_{\overline{η}_{B}} is a normal, integral scheme whose smooth locus contains a very free curve, then there exists a morphism s : B → X such that π_{X} ∘ s equals Id_{B}.

This was generalized by A. J. de Jong to the case that k is algebraically closed of arbitrary characteristic, [dJS03]. The key difference has to do with extensions of valuation rings in characteristic 0 and in positive characteristic. Given a flat morphism of smooth schemes in characteristic 0, π : U → B, and given codimension 1 points η_{D} of U and η_{Δ} of B with π(η_{D}) = η_{Δ}, the induced local homomorphism of stalks π_{U}^{∗} : Ô_{B,η_{Δ}} → Ô_{U,η_{D}}, is equivalent to

k(Δ) \begin{matrix}[ \\ [ & t & ] \\ ] \end{matrix} → k(D) \begin{matrix}[ \\ [ & r & ] \\ & ] \end{matrix} , t ↦ ur^{m}

for a unit u and a positive integer m, cf. the proof of Lemma 3.23 below. In particular, it is rigid in the sense that t ↦ ur^{m} + vr^{m+1} + … is equivalent to t ↦ ur^{m}. However, extensions of positive characteristic valuation rings are not rigid, e.g., t ↦ r^{p} + v_{1}r^{p+1} is equivalent to t ↦ r^{p} + v_{2}r^{p+1} only if v_{1} = v_{2}. But there is a weak rigidity of local homomorphisms, Krasner’s lemma in the theory of non-Archimedean valuations. This is a key step in the generalization to positive characteristic.

Of course when k has characteristic 0, then since X is normal the fiber X_{\overline{η}_{B}} is automatically normal. If X is also smooth (which can be achieved thanks to resolution of singularities in characteristic 0), then also X_{\overline{η}_{B}} is smooth. Then the hypothesis on X_{\overline{η}_{B}} is equivalent to rational connectedness.

3.1. Outline of the proof. The proof that follows is based on a proof by T. Graber, J. Harris and myself (not quite the version we chose to publish) together with several major simplifications due to A. J. de Jong. The basic idea is to choose a smooth curve C ⊂ X such that π_{X}|_{C} : C → B is finite, and then try to deform C as a curve in X until it specializes to a reducible curve in X, one component of which is the image of a section s of π_{X}. Here are some definitions that make this precise.

Definition 3.10. Let π_{C} : C → B be a finite morphism of smooth, projective k-curves. A linked curve with handle C is a reduced, connected, projective curve C_{link} with irreducible components

C_{link} = C ∪ L_{1} ∪ ⋯ ∪ L_{m}

together with a morphism

π_{C,link} : C_{link} → B

such that (i) π_{C,link} restricts to π_{C} on the component C, (ii) the restriction of π_{C,link} to each link component L_{i} is a constant morphism

with image b_{i}, where b_{1}, …, b_{m} are distinct closed points of B, (iii) and each link L_{i} is a smooth, rational curve intersecting C in a finite number

of nodes of C_{link}. If every link L_{i} intersects C in a single node of C_{link}, then (C_{link}, π_{C,link}) is called a comb and the links L_{i} are called teeth. For combs we will use the notation C_{comb} rather than C_{link}.

A one-parameter deformation of a linked curve (C_{link}, π_{C,link}) is a datum of a smooth, connected, pointed curve (Π, 0) and a projective morphism

(ρ, π_{C}) : C → Π ×_{k} B

such that ρ is flat and such that C_{0} := ρ^{−1}(0) together with the restriction of π_{C} equals the linked curve (C_{link}, π_{C,link}).

A one-parameter deformation specializes to a section curve if there exists a closed point ∞ ∈ Π and an irreducible component B_{i} of C_{∞} := ρ^{−1}(∞) such that

(i) C_{∞} is reduced at the generic point of B_{i} (ii) and the restriction of π_{C} to B_{i} is an isomorphism

π_{C}|_{B_{i}} : B_{i} ⟶^{≅} B.

Given a linked curve, a one-parameter deformation of the linked curve and a Bmorphism j : C_{link} → X, an extension of j is an open neighborhood of 0, 0 ∈ N ⊂ B and a B-morphism

j_{N} : C_{N} → X, C_{N} := ρ^{−1}(N)

restricting to j on C_{0} = C_{link}.

For the purposes of producing a section, the particular parameter space (Π, 0) of the one-parameter deformation is irrelevant. Thus, it is allowed to replace the one-parameter deformation by the new one-parameter deformation obtained from a finite base change (Π′, 0′) → (Π, 0). The following lemma is straightforward.

Lemma 3.11. Let (Π, 0, ∞) together with (ρ, π_{C}) : C → Π×_{k} B be a one-parameter deformation of (C_{link}, π_{C,link}) specializing to a section curve B_{i}. For every morphism of 2-pointed, smooth, connected curves

(Π′, 0′, ∞′) → (Π, 0, ∞),

the base change morphism

Π′ ×_{Π} C → Π′ ×_{k} B

is also a one-parameter deformation of (C_{link}, π_{C,link}) specializing to the section curve B_{i}.

The usefulness of these definitions is the following simple consequence of the valuative criterion of properness.

Lemma 3.12. Let (C_{link}, π_{link}) be a linked curve together with a B-morphism j : C_{link} → X. If there exists a one-parameter deformation of the linked curve specializing to a section curve and if there exists an extension of j, then there exists a section s : B → X of π_{X}.

Proof. Let R denote the stalk O_{C,η_{B_{i}}} of O_{C} at the generic point η_{B_{i}} of B_{i}. By the hypotheses on C and B_{i}, R is a discrete valuation ring with residue field κ = k(B_{i}) and fraction field K = k(C). The restriction of j_{N} to the generic point of C is a B-morphism

j_{K} : Spec K → X.

Because π_{X} : X → B is proper, by the valuative criterion of properness the Bmorphism j_{K} extends to a B-morphism

j_{R} : Spec R → X,

which in turn gives a B-morphism from the residue field Spec κ to X, i.e., a rational B-map

j_{B_{i}} : B_{i} ⊃ U → X, U ⊂ B_{i} a dense, Zariski open.

Finally, because B_{i} is a smooth curve, the valuative criterion applies once more and this rational transformation extends to a B-morphism

j_{B_{i}} : B_{i} → X.

Because π_{C}|_{B_{i}} : B_{i} → B is an isomorphism, there exists a unique B-morphism

s : B → X

such that j_{B_{i}} = s ∘ π_{C}|_{B_{i}}. The morphism s is a section of π_{X}. □

Thus the proof of the theorem breaks into three parts:

(i) find a “good” linked curve j : C_{link} → X, (ii) find a one-parameter deformation of the linked curve specializing to a sec-

tion curve, (iii) and find an extension of j to the one-parameter deformation.

The first step in finding j : C_{link} → X is to form a curve C_{init} which is an intersection of X with dim(X)−1 general hyperplanes in projective space. By Bertini’s theorem, if the hyperplanes are sufficiently general, then C_{init} will satisfy any reasonable transversality property. Moreover, there is a technique due to Kollár-Miyaoka-Mori – the smoothing combs technique – for improving C_{init} to another curve C ⊂ X still satisfying the transversality property and also satisfying a positivity property with respect to the vertical tangent bundle of π : X → B.

Unfortunately, even after such an improvement, there may be no one-parameter deformation of π_{X}|_{C} : C → B specializing to a section curve. However, after attaching sufficiently many link components over general closed points of B, there does exist a one-parameter deformation of C_{link} specializing to a section curve. This is one aspect of the well-known theorem that for a fixed base curve B and for a fixed degree d, if the number β of branch points is sufficiently large the Hurwitz scheme of degree d covers of B with β branch points is irreducible. (This was proved by Hurwitz when g(B) = 0, [Hur91], proved by Richard Hamilton for arbitrary genus in his thesis, and periodically reproved ever since, cf. [GHS02].) Because the general fibers of π : X → B are rationally connected, the inclusion C ⊂ X extends to a B-morphism j : C_{link} → X.

Finally the positivity property mentioned above implies j extends to the oneparameter deformation, at least after base change by a morphism Π′ → Π.

3.2. Hilbert schemes and smoothing combs. The smoothing combs technique of Kollár-Miyaoka-Mori depends on a result from the deformation theory of Hilbert schemes. Here is the setup for this result. Let Y → S be a flat, quasi-projective morphism and let

(ρ_{Hilb} : Hilb(Y/S) → S, Univ(Y/S) ⊂ Hilb(Y/S) ×_{S} T)

be universal among pairs (ρ : T → S, Z ⊂ T ×_{S} Y) of an S-scheme T and a closed subscheme Z ⊂ T ×_{S} Y such that Z → T is proper, flat and finitely presented. In other words, Hilb(Y/S) is the relative Hilbert scheme of Y over S.

In particular, for every field K the K-valued points of Hilb(Y/S) are naturally in bijection with pairs (s, Z) of a K-valued point s of S and a closed subscheme Z of Y_{s} := {s} ×_{S} Y. The closed immersion Z → Y_{s} is a regular embedding if at every point of Z the stalk of the ideal sheaf I_{Z/Y_{s}} is generated by a regular sequence of elements in the stalk of O_{Y_{s}}. In this case the conormal sheaf I_{Z/Y_{s}}/I_{Z/Y_{s}}^{2} is a locally free O_{Z}-module, and hence also the normal sheaf

N_{Z/Y_{s}} := Hom_{O_{Z}} (I_{Z/Y_{s}}/I_{Z/Y_{s}}^{2}, O_{S})

is a locally free O_{Z}-module.

Proposition 3.13. [Kol96, Theorem I.2.10, Lemma I.2.12.1, Proposition I.2.14.2] If Z ⊂ Y_{s} is a regular embedding and if h^{1}(Z, N_{Z/Y_{s}}) equals 0, then Hilb(Y/S) is smooth over S at (s, Z).

There is a variation of this proposition which is also useful. There is a flag Hilbert scheme of Y over S, i.e., a universal pair

(ρ_{fHilb} : fHilb(Y/S) → S, Univ_{1}(Y/S) ⊂ Univ_{2}(Y/S) ⊂ Hilb(Y/S) ×_{S} T)

among all pairs (ρ : T → S, Z_{1} ⊂ Z_{2} ⊂ T ×_{S} Y) of an S-scheme T and a nested pair of closed subschemes Z_{1} ⊂ Z_{2} ⊂ T ×_{S} Y such that for i = 1, 2, the projection Z_{i} → T is proper, flat and finitely presented. There are obvious forgetful morphisms

F_{i} : fHilb(Y/S) → Hilb(Y/S), F_{i}(s, Z_{1}, Z_{2}) = (s, Z_{i}).

Proposition 3.14. Let K be a field and let (s, Z_{1}, Z_{2}) be a K-point of fHilb(Y/S). If each closed immersion Z_{1} ⊂ Z_{2} and Z_{2} ⊂ Y_{s} is a regular embedding and if

h^{1}(Z_{2}, N_{Z_{2}/Y_{s}}) = 0, h^{1}(Z_{1}, N_{Z_{1}/Z_{2}}) = 0, and if h^{i}(Z_{2}, I_{Z_{1}/Z_{2}}·N_{Z_{2}/Y_{2}}) = 0 for i = 1, 2,

fHilb(Y/S) is smooth over S at (s, Z_{1}, Z_{2}), for each i = 1, 2, Hilb(Y/S) is smooth over S at (s, Z_{i}), and each forgetful morphism F_{i} : fHilb(Y/S) → Hilb(Y/S) is smooth at (s, Z_{1}, Z_{2}).

Proof. Since h^{1}(Z_{2}, N_{Z_{2}/Y_{s}}) equals 0, Hilb(Y/S) is smooth at (s, Z_{2}) by Proposition 3.13. It is easy to see that the forgetful morphism F_{2} is equivalent to the relative Hilbert scheme Hilb(Univ(Y/S)/Hilb(Y/S)) over Hilb(Y/S). Thus, applying Proposition 3.13 to this Hilbert scheme, the vanishing of h^{1}(Z_{1}, N_{Z_{1}/Z_{2}}) implies F_{2} is smooth at (s, Z_{1}, Z_{2}). Since a composition of smooth morphisms is smooth, also fHilb(Y/S) is smooth over S at (s, Z_{1}, Z_{2}). The long exact sequence of cohomology associated to the short exact sequence

0 ⟶ I_{Z_{1}/Z_{2}} · N_{Z_{2}/Y_{s}} ⟶ N_{Z_{2}/Y_{s}} ⟶ N_{Z_{2}/Y_{s}}|_{Z_{1}} ⟶ 0

implies that h^{1}(Z_{1}, N_{Z_{2}/Y_{s}}|_{Z_{1}}) equals h^{1}(Z_{2}, N_{Z_{2}/Y_{s}}), which is 0. Thus, the long exact sequence of cohomology associated to

0 ⟶ N_{Z_{1}/Z_{2}} ⟶ N_{Z_{1}/Y_{s}} ⟶ N_{Z_{2}/Y_{s}}|_{Z_{1}} ⟶ 0

implies that h^{1}(Z_{2}, N_{Z_{1}/Y_{2}}) equals 0. So again by Proposition 3.13, Hilb(Y/S) is smooth over S at (s, Z_{1}). Finally, F_{1} is a morphism of smooth S-schemes at (s, Z_{1}, Z_{2}). Thus, to prove F_{1} is smooth, it suffices to prove it is surjective on Zariski tangent vector spaces. This follows from the vanishing of h^{1}(Z_{2}, I_{Z_{1}/Z_{2}} · N_{Z_{2}/Y_{s}}). □

Another ingredient in the smoothing combs technique is a a simple result about elementary transforms of locally free sheaves on a curve: the higher cohomology of the sheaf becomes zero after applying elementary transforms at sufficiently many points.

Lemma 3.15. Let C be projective, at-worst-nodal, connected curve over a field k and let E be a locally free O_{C}-module.

(i) There exists a short exact sequence of coherent sheaves,

0 → F^{∨} → E^{∨} → T → 0

such that T is a torsion sheaf with support in C_{smooth} and such that h^{1}(C, F) equals 0.

(ii) Inside the parameter space of torsion quotients q : E^{∨} ↠ T with support in C_{smooth}, denoting

F^{∨} := Ker(E^{∨} → T ) and F := Hom_{O_{C}} (F^{∨}, O_{C}),

the subset parameterizing quotients for which h^{1}(C, F) = 0 is an open subset.

(iii) If h^{1}(C, F) equals 0, then for every short exact sequence of coherent sheaves

0 → G^{∨} → E^{∨} ⟶^{q′} S → 0

admitting a morphism r : S → T of torsion sheaves with support in C_{smooth} for which q = r ∘ q′, h^{1}(C, G) equals 0.

Proof. (i) By Serre’s vanishing theorem, there exists an effective, ample divisor D in the smooth locus of C such that h^{1}(C, E(D)) equals 0. Define F = E(D), define E → F to be the obvious morphism E → E(D), and define T to be the cokernel of F^{∨} → E^{∨}.

(ii) This follows immediately from the semicontinuity theorem, cf. [Har77, §III.12].

(iii) There exists an injective morphism of coherent sheaves F → G with torsion cokernel. Because h^{1}(C, F) equals 0 and because h^{1} of every torsion sheaf is zero, the long exact sequence of cohomology implies that also h^{1}(C, G) equals 0. □

It is worth noting one interpretation of the sheaf F associated to a torsion quotient T . Assume that T is isomorphic to a direct sum of skyscraper sheaves at n distinct points c_{1}, …, c_{n} of C_{smooth}. (Inside the parameter space of torsion quotients, those with this property is a dense, open subset.) For each point c_{i}, the linear functional E^{∨}|_{c_{i}} ↠ T |_{c_{i}} gives a one-dimensional subspace Hom_{k}(T |_{c_{i}}, k) ↪ E|_{c_{i}}. The sheaf F is precisely the sheaf of rational sections of E having at worst a simple pole at each point c_{i} in the direction of this one-dimensional subspace of E|_{c_{i}}. This is often called an elementary transform up of E at the point c_{i} in the specified direction. So Lemma 3.15 says that h^{1} becomes zero after sufficiently many elementary transforms up at general points in general directions.

This interpretation is useful because the normal sheaf of a reducible curve can be understood in terms of elementary transforms up. To be precise, let Y be a kscheme, let C be a proper, nodal curve, let C_{0} be a closed subcurve (i.e., a union of irreducible components of C), and let j : C → Y be a regular embedding such that Y is smooth at every node p_{1}, …, p_{n} of C which is contained in C_{0} and which is not a node of C_{0}. Then j_{0} : C_{0} → Y is also a regular embedding and both N_{C/Y} |_{C_{0}} and N_{C_{0}/Y} are locally free sheaves on C_{0}. For each i, there is a branch C_{i} of C at p_{i} other than C_{0}. Denote by T_{C_{i},p_{i}} the tangent direction of this branch in T_{Y,p_{i}}.

Lemma 3.16. [GHS03, Lemma 2.6] The restriction N_{C/Y} |_{C_{0}} equals the sheaf of rational sections of N_{C_{0}/Y} having at most a simple pole at each point p_{i} in the normal direction determined by T_{C_{i},p_{i}}.

Proof. The restrictions of N_{C/Y} |_{C_{0}} and N_{C_{0}/Y} to the complement of {p_{1}, …, p_{n}} are canonically isomorphic. The lemma states that this canonical isomorphism is the restriction of an injection N_{C_{0}/Y} ↪ N_{C/Y} |_{C_{0}} which identifies N_{C/Y} |_{C_{0}} with the sheaf of rational sections, etc. This local assertion can be verified in a formal neighborhood of each node p_{i}.

Locally near p_{i}, C → Y is formally isomorphic to the union of the two axes inside a 2-plane inside an n-plane, i.e., the subscheme of 𝔸_{k}^{n} with ideal I_{C/Y} = ⟨x_{1}x_{2}, x_{3}, …, x_{n}⟩. The branch C_{0} corresponds to just one of the axes, e.g., the subscheme of 𝔸_{k}^{n} with ideal I_{C_{0}/Y} = ⟨x_{2}, x_{3}, …, x_{n}⟩. The tangent direction of the other branch C_{i} is spanned by (0, 1, 0, …, 0). Thus it is clear that I_{C/Y} /I_{C_{0}/Y} ·I_{C/Y} is the submodule of I_{C_{0}/Y} /I_{C_{0}/Y}^{2} of elements whose fiber at 0 is contained in the annihilator of T_{C_{i},p_{i}}. Dualizing gives the lemma. □

The final bit of deformation theory needed has to do with deforming nodes. Let C be a proper, nodal curve and let jC → Y be a regular embedding. Let p be a node of C and assume that Y is smooth at p. There are two branches C_{1} and C_{2} of C at p (possibly contained in the same irreducible component of C). The sheaf

T := Ext_{O_{C}}^{1} (Ω_{C}, O_{C})

is a skyscraper sheaf supported at p and with fiber canonically identified to

T |_{p} = T_{C_{1},p} ⊗_{k} T_{C_{2},p}.

The following lemma is as much definition as lemma.

Lemma 3.17. There exists a quotient of coherent sheaves

N_{C/Y} ↠ T

such that for both i = 1, 2 the quotient N_{C/Y} |_{C_{i}}/N_{C_{i}/Y} equals T . A first-order deformation of C ⊂ Y, i.e., a global section of N_{C/Y} is said to smooth the node p to first-order if the image of the section in T_{C_{1},p} ⊗_{k} T_{C_{2},p} is nonzero. For a deformation

C ⊂ Π ×_{k} Y

of C ⊂ Y over a smooth pointed curve (Π, 0) (i.e., C_{0} = C), if the associated first-order deformation of C ⊂ Y smooths the node p to first-order, then p is not contained in the closure of the singular locus of the projection,

(Π − {0}) ×_{Π} C → (Π − {0})

i.e., a general fiber C_{t} of the deformation smooths the node.

This is a well-known result. A good reference for this result, and many other results about deformations of singularities, is [Art76], particularly §I.6. Here is a brief remark on the proof. Because C ⊂ Y is a regular embedding, the conormal sequence is exact on the left, i.e.,

0 ⟶ I_{C/Y} /I_{C/Y}^{2} ⟶ Ω_{Y} |_{C} ⟶ Ω_{C} ⟶ 0

is a short exact sequence. Applying global Ext, there is a connecting map

δ : H^{0}(C, N_{C/Y}) → Ext_{O_{C}}^{1} (Ω_{C}, O_{C}).

There is also a local-to-global sequence for global Ext inducing a map

Ext_{O_{C}}^{1} (Ω_{C}, O_{C}) → H^{0}(C, Ext_{O_{C}}^{1} (Ω_{C}, O_{C})) = H^{0}(C, T ) = T_{C_{1},p} ⊗_{k} T_{C_{2},p}.

The composition of these two maps is precisely the map on global sections associated to N_{C/Y} → T . induced map to H^{0}(C, Ext_{O_{C}}^{1} (Ω_{C}, O_{C})) = T is precisely the map above. The global Ext group is identified with the first-order deformations of C as an abstract scheme, and the Ext term is identified with the first-order deformations of the node. It is worth noting that even if the first-order deformation does not smooth the node, the full deformation C ⊂ Π ×_{k} Y may smooth the node if the total space C is singular at (0, p).

The first result using the smoothing combs technique is the following.

Proposition 3.18. Let Y be a smooth, irreducible, quasi-projective scheme over an algebraically closed field k. The very free locus Y_{v.f.} is an open subset of Y. More generally, for a smooth, quasi-projective morphism π : Y → B, the relative very free locus Y_{π,v.f.} is an open subset of Y.

Again in the absolute case, denote by t_{1}, resp. t_{2}, the closed point of ℙ_{k}^{1}, t_{1} = 0, resp. t_{2} = ∞. Let y_{1} and y_{2} be closed points of Y_{v.f.}, let a and k be nonnegative integers, and let there be given curvilinear k-jets in Y at each of y_{1} and y_{2}. If the given k-jets are general among all curvilinear k-jets at y_{1} and y_{2}, then there exists a morphism

f : (ℙ_{k}^{1}, t_{1}, t_{2}) → (Y_{v.f.}, y_{1}, y_{2})

mapping the k-jet of ℙ^{1} at t_{1} isomorphically to the given k-jet at y_{i} for i = 1, 2 and such that

f^{∗}T_{Y} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), a_{1}, …, a_{n} ≥ a.

Proof. By the definition of Y_{v.f.}, for each i = 1, 2 there exists a very free morphism

f_{i} : (ℙ^{1}, 0) → (Y_{v.f.}, y_{i}), f_{i}^{∗}T_{Y} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), a_{1}, …, a_{n} ≥ 1.

In particular, for each i = 1, 2, h^{1}(ℙ^{1}, f_{i}^{∗}T_{Y} (−\underline{0} − \underline{∞})) equals 0, where \underline{0}, resp. \underline{∞}, is the Cartier divisor of the point 0, resp. ∞, in ℙ^{1}. Since the normal sheaf of f_{i} is a quotient of f_{i}^{∗}T_{Y}, also h^{1}(ℙ^{1}, N_{f_{i}}(−\underline{0} − \underline{∞})) equals 0. Thus, applying Proposition 3.14 where Z_{1} = {0, ∞} and Z_{2} = ℙ^{1}, there exist deformations of the morphism f_{i} such that f_{i}(0) equals y_{i} and f_{i}(∞) is any point in a nonempty Zariski open subset of Y. The same argument holds in the relative case.

Since Y is irreducible, the open for i = 1 intersects the open for i = 2. Thus there exist very free morphisms f_{1} and f_{2} such that f_{1}(∞) = f_{2}(∞). Let C be the union of two copies C_{1} and C_{2} of ℙ^{1} attached at ∞ and let f : C → Y be the morphism whose restriction to each C_{i} equals f_{i}. Denote by

0 ⟶ N′_{C/Y} ⟶ N_{C/Y} ⟶ T ⟶ 0

the short exact sequence coming from Lemma 3.17. Using Lemma 3.16, there is an exact sequence

0 ⟶ N_{C/Y} |_{C_{1}}(−\underline{0} − \underline{∞}) ⟶ N′_{C/Y} (−y_{1} − y_{2}) ⟶ N_{C_{2}/Y} (−\underline{0}) ⟶ 0

and an exact sequence

0 ⟶ N_{C_{1}/Y} (−\underline{0} − \underline{∞}) ⟶ N_{C/Y} |_{C_{1}}(−\underline{0} − \underline{∞}) ⟶ κ_{∞} ⟶ 0

where κ_{∞} is the skyscraper sheaf on C_{1} supported at ∞. Applying the long exact sequence of cohomology, using that h^{1}(C_{i}, N_{C_{i}/Y} (−\underline{0} − \underline{∞})) equals 0 for i = 1, 2, and chasing diagrams, this finally gives that h^{1}(C, N′_{C/Y} (−y_{1} − y_{2})) also equals 0.

This has two consequences. First, this implies h^{1}(C, N_{C/Y} (−y_{1} −y_{2})) equals 0, and thus the space of deformations of C containing y_{1} and y_{2} is smooth by Proposition 3.14. And second, the map

H^{0}(C, N_{C/Y} (−y_{1} − y_{2})) → T_{C_{1},∞} ⊗ T_{C_{2},∞}

is surjective. Thus there exist first-order deformations of C containing y_{1} and y_{2} and smoothing the node at ∞. Since the space of deformations containing y_{1} and y_{2} is smooth, this first-order deformation is the one associated to a one-parameter deformation

C ⊂ Π ×_{k} Y

of [C] over a smooth, pointed curve (Π, 0) (e.g., choose Π to be a general complete intersection curve in the smooth deformation space containing the given Zariski tangent vector). By Lemma 3.17, for a general point t of Π, C_{t} is a smooth, connected curve containing y_{1} and y_{2}. Since the arithmetic genus of C is 0, the arithmetic genus of C_{t} is also 0, i.e., C_{t} ≅ ℙ_{k}^{1}. Let

f_{1} : ℙ_{k}^{1} → C_{t}

be an isomorphism with f_{1}(t_{i}) = y_{i} for i = 1, 2. Because h^{1}(C, N_{C/Y} (−y_{1} − y_{2})) equals 0, by the semicontinuity theorem also h^{1}(C_{t}, N_{C_{t}/Y} (−y_{1} −y_{2})) equals 0. This implies that

f_{1}^{∗}T_{Y} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), for integers a_{1}, …, a_{n} ≥ 1.

Next, for every integer a, let g_{a} : (ℙ^{1} → ℙ^{1} be the morphism z ↦ z^{a}. Then the composition f_{a} = f_{1} ∘ g_{1} is a morphism

f_{a} : (ℙ_{k}^{1}, t_{1}, t_{2}) → (Y_{v.f.}, y_{1}, y_{2})

with

f_{a}^{∗}T_{Y} = g_{a}^{∗}(f_{1}^{∗}T_{Y}) ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}) for integers a_{1}, …, a_{n} ≥ a,

namely the new integer a_{i}(f_{a}) equals a · a_{i}(f_{1}). Next, choosing a ≥ 2k + 1, this implies that

h^{1}(ℙ^{1}, f_{a}^{∗}T_{Y} (−(k + 1)(t_{1} + t_{2}))) = 0.

Applying Proposition 3.14 with ℙ^{1} ×_{k} Y in the place of Y, with the graph of f_{a} in the place of Z_{2} and with Z_{1} = (k + 1)(t_{1} + t_{2}) in the place of Z, deformations of f_{a} map the k-jet of ℙ^{1} at t_{1}, resp. at t_{2}, isomorphically to a general k-jet at y_{1}, resp. at y_{2}. □

The following proposition is the strongest generalization of Proposition 3.18 we will need. It is stated as a theorem about finding new sections of a rationally connected fibration under the hypothesis that one such section exists. In this sense it may seem premature (and dangerously close to circular logic), since Theorem 3.9 is not yet proved. In fact the proposition is used in the proof of Theorem 3.9 not for the original fibration, but only for a constant fibration

pr_{ℙ^{1}} : ℙ_{k}^{1} ×_{k} Y → ℙ_{k}^{1}

which obviously admits sections (constant sections). So there is nothing circular in the application of the proposition to the proof of Theorem 3.9.

Proposition 3.19 (Generic weak approximation). [KMM92], [HT06] Let B be a smooth, connected, projective curve over an algebraically closed field k. Let U be a smooth, quasi-projective k-scheme and let π : U → B be a good morphism (in the sense of Definition 3.22). Assume there exists a section s : B → U. Let (b_{1}, …, b_{M}, b′_{1}, …, b′_{M′}) be distinct closed points of B such that s(b_{i}) is in the very free locus U_{b_{i},v.f.} of the fiber U_{b_{i}} for each i = 1, …, M. Let k and a be nonnegative integers. For each i, let x_{i} be a closed point of U_{b_{i},v.f.} and let there be given an curvilinear k-jet in U at x_{i}. Assuming each of these k-jets is a general k-jet at x_{i}, there exists a section σ : B → U such that

(i) for each i = 1, …, M, σ(b_{i}) equals x_{i}, (ii) for each i = 1, …, M′, σ(b′_{i}) equals s(b′_{i}), (iii) for every invertible O_{B}-module L of degree ≤ a, h^{1}(B, N_{σ(B)/U} ⊗_{O_{B}} L^{∨})

equals 0, (iv) and σ maps the k-jet of b_{i} in B isomorphically to the given k-jet at x_{i} for

each i.

Proof. Choose a large integer N and enlarge the set of pairs ((b_{i}, x_{i}))|_{i=1,...,M} to a set ((b_{i}, x_{i}))_{i=1,...,N} having the same properties above and such that the collection (b_{i})_{i=M+1,...,N} is a general collection of N − M points in B (this is possible because for all but finitely many closed points of B, s(b) is contained in U_{b,v.f.}). By Prop 3.18, applied with k = 1, i.e., in the case that k-jets are simply tangent directions, for every i = 1, …, N there exists a morphism

f_{i} : (ℙ^{1}, 0, ∞) → (U_{b_{i},v.f.}, s(b_{i}), x_{i})

such that

f_{i}^{∗}T_{π,log} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), a_{1}, …, a_{n} ≥ 1

and the tangent direction of f_{i}(ℙ^{1}) at s(b_{i}) is a general tangent direction in T_{U_{b_{i}} ,s(b_{i})}. But of course the tangent space T_{U_{b_{i}} ,s(b_{i})} equals the normal space N_{s(B)/U}|_{s(b_{i})}. Thus the tangent direction of f_{i}(ℙ^{1}) at s(b_{i}) gives a general normal direction to s(B) in U at s(b_{i}).

Form the comb j : C_{comb} → U with handle s(B) and with each morphism f_{i} being a tooth L_{i} attached at s(b_{i}). By Lemma 3.16, N_{C_{comb}/U}|_{s(B)} equals the sheaf of rational sections of N_{s(B)/U} having at most a simple pole at each point s(b_{i}) in a general normal direction at s(b_{i}). Assuming the integer N is sufficiently large, Lemma 3.15 then implies that h^{1}(B, s^{∗}N_{C_{comb}/U}) equals 0. Moreover, fixing an auxiliary invertible sheaf M on B of degree g(B) + 1 and applying Lemma 3.15 to s^{∗}N_{s(B)/U}(−(b′_{1} + ⋯ + b′_{M′})) ⊗_{O_{B}} M^{∨}, for N sufficiently large also h^{1}(B, s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) ⊗_{O_{B}} M)^{∨} equals 0.

For every i, there is a short exact sequence

0 ⟶ f_{i}^{∗}N_{L_{i}/U_{b_{i}}} ⟶ f_{i}^{∗}N_{L_{i}/U} ⟶ f_{i}^{∗}N_{U_{b_{i}} /U} ⟶ 0.

Of course the normal sheaf N_{U_{b_{i}} /U} is just O_{U_{b_{i}}} since U_{b_{i}} is a smooth fiber of a morphism to a curve. Also the normal direction of s(B) at s(b_{i}) surjects onto the fiber of N_{U_{b_{i}} /U} at s(b_{i}). Thus the elementary transform up of N_{L_{i}/U} at s(b_{i}) in this normal direction surjects onto the elementary transform up of O_{ℙ}^{1} at ∞, i.e., it surjects onto O_{ℙ^{1}}(1). Thus, by Lemma 3.16, there is a short exact sequence

0 ⟶ f_{i}^{∗}N_{L_{i}/U_{b_{i}}} ⟶ f_{i}^{∗}N_{C_{comb}/U} ⟶ O_{ℙ^{1}}(1) ⟶ 0.

Twisting by O_{ℙ^{1}}(−2) and applying the long exact sequence of cohomology associated to the short exact sequence, h^{1}(ℙ^{1}, f_{i}^{∗}N_{C_{comb}/U}(−\underline{0} − \underline{∞})) equals 0. Combined with the result of the previous paragraph and joining the two types of normal sheaf via the short exact sequence

0 → ⊕_{i=1}^{N}N_{C_{comb}/U}|_{L_{i}}(−x_{i} −s(b_{i})) → N_{C_{comb}/U}(−(x_{1} +⋯+x_{N})−(b′_{1} +⋯+b′_{M′}))

→ N_{C_{comb}/U}|_{s(B)}(−(b′_{1} + ⋯ + b′_{M′})) → 0,

the long exact sequence of cohomology implies both that

h^{1}(C_{comb}, N_{C_{comb}/U}(−(x_{1} + ⋯ + x_{N}) − (b′_{1} + ⋯ + b′_{M′}))) equals 0,

and that the map

H^{0}(C_{comb}, N_{C_{comb}/U}(−(x_{1}+⋯+x_{N})−(b′_{1}+⋯+b′_{M′}))) → H^{0}(B, s^{∗}N_{C_{comb}/U}(−(b′_{1}+⋯+b′_{M′})))

is surjective.

Thus, by Proposition 3.14, the space of deformations of C_{comb} containing x_{1}, …, x_{N} and b′_{1}, …, b′_{M′} is smooth. And, by Lemma 3.17, to prove there exists a deformation smoothing every node of C_{comb}, it suffices to prove for every i there exists a section of s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) whose image in T_{s(B),s(b_{i})} ⊗_{k} T_{L_{i},s(b_{i})} is nonzero. Of course this skyscraper sheaf T_{s(b_{i})} is a quotient of the fiber of s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) at b_{i}. Thus it suffices to prove for every i that h^{1}(B, s^{∗}N_{C_{comb}/U}(−b_{i} − (b′_{1}+⋯+b′_{M′}))) equals 0. Recall the auxiliary invertible sheaf M of degree g(B)+1.

Because the invertible sheaf M(−b_{i}) has degree g(B), it is effective, say O_{B}(Δ_{i}). Thus there exists an injective O_{B}-module homomorphism

s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) ⊗_{O_{B}} M^{∨} ↪

s^{∗}N_{C_{comb}/U}(−(b′_{1} + … b′_{M′})) ⊗_{O_{B}} M^{∨}(Δ_{i}) = s^{∗}N_{C_{comb}/U}(−b_{i} − (b′_{1} + ⋯ + b′_{M′})) with torsion cokernel. Since h^{1}(B, s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) ⊗_{O_{B}} M^{∨}) equals 0, and since every torsion sheaf has h^{1} equal to 0, also h^{1}(B, s^{∗}N_{C_{comb}/U}(−b_{i} − (b′_{1} + ⋯ + b′_{M′}))) equals 0 for every i. Therefore there exist a one-parameter family of deformations (C_{t})_{t∈Π} of C_{comb} containing each of x_{1}, …, x_{M}, containing each of s(b′_{1}), …, s(b′_{M′}) and smoothing every node of C_{comb}, i.e., for t general, C_{t} is smooth.

Because π_{U} maps s(B) to B with degree 1, also π_{U} maps C_{t} to B with degree 1. Because C_{t} is smooth, this means the projection C_{t} → B is an isomorphism. Therefore there exists a section σ_{t} : B → U of π_{U} with image C_{t}. In particular, σ_{t}(b_{i}) = x_{i} for every i = 1, …, M and σ_{t}(b′_{i}) = s(b′_{i}) for every i = 1, …, M′. Because h^{1}(C_{comb}, N_{C_{comb}/U}(−(x_{1} + ⋯ + x_{N}))) equals 0, by semicontinuity also h^{1}(B, σ_{t}^{∗}N_{σ_{t}(B)/U}(−(x_{1} + ⋯ + x_{N}))) equals 0 for t general. In particular, if N ≥ a + g(B), then for every invertible sheaf L of degree ≤ a, L^{∨}(x_{1} + ⋯ + x_{N}) has degree ≥ g(B) and thus is effective, say O_{B}(Δ). Therefore there exists an injective sheaf homomorphism

σ_{t}^{∗}N_{σ_{t}(B)/U}(−(x_{1}+⋯+x_{N})) ↪ σ_{t}^{∗}N_{σ_{t}(B)/U}(−(x_{1}+⋯+x_{N})+Δ) = σ_{t}^{∗}N_{σ_{t}(B)/U}⊗_{O_{B}}L^{∨}

with torsion cokernel. So, by the same type of argument as above, h^{1}(B, σ_{t}^{∗}N_{σ_{t}(B)/U}⊗_{O_{B}} L^{∨}) equals 0 for every invertible sheaf L of degree ≤ a.

Finally, applying the last result when a = (k + 1)(M + M′) and L = O_{B}((k + 1)(b_{1} + ⋯ + b_{M} + b′_{1} + ⋯ + b′_{M′})), there exists a section σ : B → U of π_{U} as above and satisfying h^{1}(B, σ^{∗}N_{σ(B)/U}(−(k + 1)(b_{1} + ⋯ + b_{M} + b′_{1} + ⋯ + b′_{M′}))) equals 0. Therefore, by Proposition 3.14 once more, for a general deformation of σ(B) containing x_{1}, …, x_{M} and s(b′_{1}), …, s(b′_{M′}), the k-jet of the curve at each point x_{i} and s(b′_{i}) is a general curvilinear k-jet in U at that point. □

The main application is to the case when U = ℙ^{1} ×_{k} Y where Y is a smooth, irreducible, quasi-projective k-scheme whose very free locus Y_{v.f.} is nonempty.

Corollary 3.20. Every rational curve in Y intersecting Y_{v.f.} is contained in Y_{v.f.}. For every integer k, for every integer a, for every collection of closed points b_{1}, …, b_{M} of ℙ^{1}, for every collection of closed points y_{1}, …, y_{M} of Y_{v.f.}, and for every choice of a curvilinear k-jet in Y at each point y_{i}, if each k-jet is general among curvilinear k-jets at y_{i}, then there exists a morphism

f : (ℙ^{1}, b_{1}, …, b_{M}) → (Y, y_{1}, …, y_{M})

mapping the k-jet of ℙ^{1} at b_{i} isomorphically onto the given k-jet at y_{i} and such that

f^{∗}T_{Y} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), a_{1}, …, a_{n} ≥ a.

Proof. Let B = ℙ^{1}, let U = B ×_{k} Y and let π_{B} be the obvious projection. The sections of π_{B} are precisely the graphs of morphisms f : ℙ^{1} → Y. In particular, if f is a morphism whose image intersects Y_{v.f.}, then the section s = (Id_{B}, f) satisfies the hypotheses of Proposition 3.19. Thus, for every point b′ = b′_{1} of ℙ^{1}, there exists a section σ = (Id_{ℙ^{1}}, ϕ) with σ(b′) = s(b′) and h^{1}(B, σ^{∗}N_{σ(B)/U}(−2)) equals 0. In other words, ϕ : ℙ_{k}^{1} → Y is a morphism with ϕ(b′) = f(b′) and with h^{1}(ℙ^{1}, ϕ^{∗}T_{Y} (−2)) equals 0. Thus ϕ is a very free morphism whose image contains f(b′). Therefore every point in the image of f is contained in the very free locus, i.e., every rational curve in Y intersecting Y_{v.f.} is contained in Y_{v.f.}.

The rest of the corollary is just a straightforward translation of Proposition 3.19 to this context. □

There is one more result in this direction which is useful. The proof is similar to the arguments above.

Lemma 3.21. [Kol96, Lemma II.7.10.1] Let C_{comb} be a comb with handle C and teeth L_{1}, …, L_{n}. Let ρ : C → Π be a one-parameter deformation of C_{comb} over a pointed curve (Π, 0) whose general fiber C_{t} is smooth. Let E be a locally free sheaf on C. If E|_{L_{i}} is ample for every i and if h^{1}(C, (E|_{C}) ⊗_{O_{C}} M) equals 0 for every invertible O_{C}-module M of degree ≥ n, then h^{1}(C_{t}, E|_{C_{t}}) equals 0 for general t in Π.

3.3. Ramification issues. The argument sketched in Subsection 3.1 and the powerful smoothing combs technique from Subsection 3.2 form the core of the proof of Theorem 3.9. However there is a technical issue complicating matters. There may be codimension 1 points of X at which π_{X} : X → B is not smooth. In other words, finitely many scheme-theoretic fibers of π_{X} may have irreducible components occurring with multiplicity ≥ 1. This is a well-known issue when working with fibrations. Although there are sophisticated ways to deal with this (using log structures or Deligne-Mumford stacks), for the purposes of this proof it suffices to deal with this in a more naive manner.

Definition 3.22. Let B be a smooth k-curve, let X be a reduced, finite type kscheme and let π : X → B be a flat morphism. The good locus of π is the maximal open subscheme U of X such that U is smooth and such that for every point b of B the reduced scheme of the fiber π_{X}^{−1}(b) ∩ U is smooth. Denote the restriction of π_{X} to U by π_{U}. The morphism π is good if the good locus equals all of X.

The log divisor of π is the Cartier divisor D_{π,log} of U given by

D_{π,log} := ∑_{b∈B(k)} π_{U}^{∗}(b) − π_{U}^{∗}(b)_{red},

where π_{U}^{∗}(b)_{red} is the reduced Cartier divisor. Notice this is actually a finite sum over those closed points b of B for which π_{U}^{∗}(b) is nonreduced.

Lemma 3.23. The complement of U in X has codimension ≥ 2. If char(k) equals 0, then the pullback map on relative differentials

π_{U}^{∗} : π_{U}^{∗}Ω_{B/k} → Ω_{U/k}

factors uniquely through the inclusion

π_{U}^{∗}Ω_{B/k} ↪ π_{U}^{∗}Ω_{B/k}(D_{π,log})

and the cokernel

Ω_{π,log} := Coker(π_{U}^{∗}Ω_{B/k}(D_{π,log}) → Ω_{U/k})

is locally free.

Proof. To construct U, first remove the closure of the singular locus of the geometric generic fiber of π_{X} and next remove the singular locus from the reduced scheme of the finitely many singular fibers. Both of these sets have codimension 2 in X (the first by hypothesis).

The proof of the second part uses that char(k) = 0. It can be checked formally locally near every closed point x of U. Denote by b the image π_{X}(x) in B and denote by D the reduced structure on the irreducible component of π^{−1}(b) containing x. Since x is in U, D is a smooth Cartier divisor in U. Let r be a defining equation for D in U and let t be a defining equation for b in B. Near x, π^{∗}(b) = mD+ other terms. Thus, in Ô_{U,x},

π_{X}^{∗}t = a_{m}r^{m} + a_{m+1}r^{m+1} + …

where a_{m} is a unit. Because char(k) = 0, the power series

u = \sqrt[m]{a_{m} + a_{m+1}r + …}

is a well-defined unit in Ô_{U,x}. Thus, after replacing r by ur, there exists a regular system of parameters r, r_{2}, …, r_{n} for Ô_{U,x} such that π_{X}^{∗} is given by

π_{X}^{∗}t = r^{m}, π_{X}^{∗}(dt) = mr^{m−1}dr.

Thus π^{∗} locally factors through π_{X}^{∗}Ω_{B}((e − 1)D) = π_{X}^{∗}Ω_{B}(D_{π,log}). Moreover it is clear that locally the cokernel is the locally free sheaf generated by dr_{2}, …, dr_{n}. □

The locally free quotient Ω_{π,log} of Ω_{π} is called the sheaf of log relative differentials. Of course it equals the torsion-free quotient of Ω_{π}. But its true importance comes from the following lemma: given a base change V → B for which the normalized fiber product { \atop U ×_{B} V} is smooth over V, the sheaf of relative differentials of { \atop U ×_{B} V} /V equals the pullback of Ω_{π,log}. Thus the relative deformation theory of the { \atop U ×_{B} V} over V is already captured by the sheaf Ω_{π,log} on U. Before stating the lemma precisely, there is some setup.

Let

π : U → B, ϖ : V → B

be two good morphisms with respective log divisors D_{π,log} and E_{ϖ,log}. Let b be a closed point of B. Let D be a prime divisor of U in Supp(D_{π,log}) ∩ π^{−1}(b), and let E be a prime divisor of V in Supp(E_{ϖ,log}) ∩ ϖ^{−1}(b). Denote by m_{D} − 1, resp. m_{E} − 1, the coefficient of D in D_{π,log}, resp. the coefficient of E in E_{ϖ,log}. The normalized fiber product of U and V along D and E is the normalization { \atop U ×_{B} V} of U ×_{B} V along D ×_{{b}} E. Denote by

pr_{U} : U ×_{B} V → U, pr_{V} : U ×_{B} V → V

the two projections, and denote by

p̃r_{U} : { \atop U ×_{B} V} → U, p̃r_{V} : { \atop U ×_{B} V} → V

the compositions with the blowdown morphism. Denote by Exc the exceptional locus of the morphism, i.e.,

Exc := (p̃r_{U}^{−1}(D) ∩ p̃r_{V}^{−1}(E))_{reduced}.

From this point forward we explicitly assume that char(k) equals 0.

Hypothesis 3.24. The algebraically closed ground field k has characteristic 0.

The sheaves Ω_{π} and Ω_{π,log} agree on a dense open subset of U, namely U−Supp(D_{π,log}). Because p̃r_{V} and pr_{V} are isomorphic over a dense open subset of V (namely V −E) also Ω_{p̃r_{V}} agrees with p̃r_{V}^{∗} Ω_{π} on a dense open subset of { \atop U ×_{B} V}. Therefore also

Ω_{ϖ} agrees with p̃r_{V}^{∗} Ω_{π,log} on a dense open subset of { \atop U ×_{B} V}.

Lemma 3.25. The morphism

p̃r_{V} : { \atop U ×_{B} V} → V

is smooth at every point of Exc if and only if m_{D} divides m_{E}. In this case the reduced normalization equals the blowing up of U ×_{B} V along the closed subscheme pr_{U}^{−1}(D) × pr_{V}^{−1}((m_{E}/m_{D})E) and Exc is contained in the maximal open neighborhood of { \atop U ×_{B} V} on which Ωπ̃ agrees with (p̃r_{V})^{∗}Ω_{π,log}.

Proof. This is proved in much the same way as the second part of Lemma 3.23. For every closed point x of U and y of V with common image point b = π(x) = ϖ(y), there exist a regular system of parameters (r, r_{2}, …, r_{n}) for Ô_{U,x}, resp. (s, s_{2}, …, s_{p}) for Ô_{V,y}, and a regular parameter t for Ô_{B,b} such that

π^{∗}t = r^{m_{D}} and ϖ^{∗}t = s^{m_{E}},

and thus,

Ô_{U×_{B}V,(x,y)} = k \begin{aligned}[ \\ [ & r, r_{2}, …, r_{n}, s, s_{2}, …, s_{p} & ] \\ & ] \end{aligned} /⟨r^{m_{D}} − s^{m_{E}}⟩.

Denoting by m the greatest common factor of m_{D} and m_{E}, the stalk of the normalization equals

k \begin{aligned}[ \\ [ & u, r, r_{2}, …, r_{n}, s, s_{2}, …, s_{p} & ] \\ & ] \end{aligned} /⟨r − u^{m_{E}/m}, s − u^{m_{D}/m}⟩.

Thus it is formally smooth as a k \begin{aligned}[ \\ [ & s, s_{2}, …, s_{p} & ] \\ & ] \end{aligned}-algebra if and only if m_{D}/m equals 1, i.e., if and only if m_{D} divides m_{E}. In this case it is easy to see that the normalization is the blowing up at the ideal ⟨s, t^{m_{E}/m_{D}}⟩ and it is easy to see that the module of relative differentials is the free module generated by dr_{2}, …, dr_{n}, i.e., it is the pullback of Ω_{π,log}. □

Definition 3.26. A log preflexible curve is a connected, smooth, proper curve C ⊂ U such that

(i) C intersects the very free locus U_{π,v.f.} of π_{U}, (ii) π_{U}(C) equals B, (iii) and every intersection point of C with supp(D_{π,log}) is transverse, i.e., the

tangent direction of C at the intersection point is not contained in the tangent space of supp(D_{π,log}).

A linked log preflexible curve is a B-morphism from a linked curve j : C_{link} → U such that the handle C is log preflexible and for every link L_{i} the image in B of L_{i} is disjoint from the image in B of D_{π,log}.

A log preflexible curve C is a log flexible curve if

h^{1}(C, T_{π,log}|_{C}) equals 0, where T_{π,log} := Hom_{O_{U}} (Ω_{π,log}, O_{U}).

A linked log preflexible curve is a linked log flexible curve if

h^{1}(C_{link}, j^{∗}T_{π,log}) equals 0.

Lemma 3.27. There exists a log preflexible curve C. In fact, every intersection of X with dim(X) − 1 general hyperplanes is a log preflexible curve.

Proof. Because X − U has codimension 2 in X, a general complete intersection curve in X is disjoint from X −U, i.e., it is contained in U. By hypothesis, U_{π,v.f.} is a dense open subset of U and thus a general complete intersection curve intersects this open. Finally, by Bertini’s theorem a general complete intersection curve in U is smooth and intersects supp(D_{π,log}) transversally. □

An important consequence of the smoothing combs technique is the following result.

Proposition 3.28. There exists a log flexible curve in X. In fact, for every comb in X with log preflexible handle C and with sufficiently many very free teeth in fibers of π_{U} attached at general points of C and with general tangent directions, there exists a one-parameter deformation of the comb whose general member is a log flexible curve.

Proof. By hypothesis, C intersects the very free locus U_{π,v.f.} of the morphism π_{U}. By the same argument as in the proof of Proposition 3.18, U_{π,v.f.} is open. Therefore all but finitely many points of C are contained in U_{π,v.f.}. By Proposition 3.18 applied to 1-jets, i.e., to tangent directions, for each such point c there exists a very free rational curve in U_{π_{U} (c)} containing c and whose tangent direction at c is a general tangent direction in U_{π_{U} (c)}.

Let C_{comb} be a comb obtained by attaching to C a number of teeth L_{1}, …, L_{N} as in the previous paragraph at general points of C (in particular, points where C → B is unramified) and with general tangent directions in U_{π_{U} (c)}. These tangent directions are the same as normal directions to C in U. By the same argument as in the proof of Proposition 3.19, if N is sufficiently large there is a one-parameter deformation

C ⊂ Π ×_{k} U

of C_{comb} such that C_{t} is smooth for general t in Π. The properties (i), (ii) and (iii) of Definition 3.26 are all open properties and hold for C_{0} = C_{comb}, thus also hold for C_{t} so long as t is general.

For each tooth L_{i} in a fiber U_{b_{i}}, T_{π,log}|_{L_{i}} equals T_{U_{b_{i}}} |_{L_{i}}. Since L_{i} is very free, this is an ample locally free sheaf. Thus, by Lemma 3.21 with the pullback of T_{π,log} in the place of E, the restriction h^{1}(C_{t}, T_{π,log}|_{C_{t}}) equals 0 for t a general point of Π. Therefore, for t a general point of Π, C_{t} is a log flexible curve. □

Because the fibers of π are rationally connected, every log preflexible curve, resp. log flexible curve, extends to a linked log preflexible curve, resp. linked log flexible curve.

Lemma 3.29. For every linked curve C_{link} such that each point b_{i} = π_{C,link}(L_{i}) is disjoint from π_{U}(D_{π,log}), and for every B-morphism j_{0} : C → X mapping C isomorphically to a log preflexible curve, resp. log flexible curve, and mapping each fiber C_{b_{i}} into the very free locus U_{π,v.f.} of π_{U}, there exists a B-morphism j : C_{link} → X which is linked log preflexible, resp. linked log flexible, and restricting to j_{C} on C.

Proof. Let L_{i} be a link of C_{link}. Let L_{i} intersect C in m points t_{1}, …, t_{m} contained in the fiber over a general point b_{i} of B. Let x_{1}, …, x_{m} be the images j(t_{1}), …, j(t_{m}) in U_{b_{i},v.f.}. By Corollary 3.20, there exists a morphism

j_{i} : (L_{i}, t_{1}, …, t_{m}) → ((U_{b_{i},v.f.}, x_{1}, …, x_{m})

such that

j_{i}^{∗}T_{U_{b_{i}}} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}) for integers a_{1}, …, a_{n} ≥ m − 1.

Because of this,

h^{1}(L_{i}, j_{i}^{∗}T_{U_{b_{i}}} (−(t_{1} + ⋯ + t_{m}))) equals 0.

Define j : C_{link} → U to be the unique morphism such restricting to j_{0} on C and restricting to j_{i} on each link L_{i}. Because j_{i}(t_{k}) = j_{0}(t_{k}) for every link L_{i} and for every node t_{k} contained in L_{i}, this morphism is defined. It is clearly log preflexible.

Next assume that j_{0} is log flexible. The claim is that j is also log flexible. To see this, consider the short exact sequence

0 ⟶ ⊕_{i}j_{i}^{∗}T_{U_{b_{i}}} (−C_{b_{i}}) ⟶ j^{∗}T_{π,log} ⟶ j_{0}^{∗}T_{π,log} ⟶ 0.

By the hypothesis that j_{0} is log flexible, the third term has vanishing h^{1}. And by the construction of j_{i}, j_{i}^{∗}T_{U_{b_{i}}} (−C_{b_{i}}), i.e., j_{i}^{∗}T_{U_{b_{i}}} (−(t_{1} + ⋯ + t_{m})), has vanishing h^{1}. Thus, by the long exact sequence of cohomology, also h^{1}(C_{link}, j^{∗}T_{π,log}) equals 0. Therefore j : C_{comb} → U is a linked log flexible curve. □

3.4. Existence of log deformations. There is a definition of one-parameter deformation that takes the divisor D_{π,log} into account. Unfortunately, not every curve over B admits a log deformation specializing to a section curve, e.g., étale covers of B are rigid. However, after attaching a sufficient number of links, the linked curve does admit a log deformation specializing to a section curve.

Definition 3.30. Let (C_{link}, π_{C,link}) be a linked curve with handle C. Let D_{C} ⊂ C be an effective, reduced, Cartier divisor contained in the smooth locus of C_{link}. A one-parameter log deformation of (C_{link}, π_{C,link}, D_{C}) is a one-parameter deformation of (C_{link}, π_{C,link}),

(ρ, π_{C}) : C → Π ×_{k} B

together with an effective Cartier divisor D_{C} ⊂ C such that (i) the pullback of D_{C} to C_{0} = C_{link} equals D_{C}

(ii) and π_{C}(D_{C}) equals π_{C}(D_{C}), i.e., D_{C} is vertical over B.

Lemma 3.31. For every finite morphism of smooth, projective curves π_{C} : C → B and for every effective, reduced, Cartier divisor D_{C} of C, after attaching sufficiently many links to C over general points of B, there exists a one-parameter log deformation specializing to a section curve.

Proof. For all sufficiently positive integers e, for a general morphism g : C → ℙ^{1} of degree e, the induced morphism (π_{C}, g) : C → B ×_{k} ℙ^{1} is unramified and is injective except for finitely many double points, none of which intersect the image of D_{C}. Denote by Σ → B ×_{k} ℙ^{1} the blowing up along the finitely many double points of (π_{C}, g)(C). Then there is a B-morphism h : C → Σ which is an embedding.

For each point p of D_{C}, denote by m_{p} the multiplicity of p in the Cartier divisor π_{C}^{∗}(π_{C}(p)). Denote by ν_{p} : Σ′_{p} → Σ the m_{p}-fold iterated blowup of Σ first at p, then at the image of p in the strict transform of h(C), etc. Denote by E_{p} the final exceptional divisor of this sequence of blowups. The point of this construction is that the strict transform of h(C) intersects E_{p} at p, and E_{p} occurs with multiplicity m_{p} in the Cartier divisor Σ′_{p} ×_{B} {π_{C}(p)}. Denote by ν : Σ′ → Σ the fiber product over all points p in D_{C} of ν_{p} : Σ′_{p} → Σ. Denote by E the Cartier divisor in Σ′ being the sum over all p of the pullback of E_{p} from Σ′_{p}. Denote by π_{Σ′} : Σ′ → B the composition of Σ′ → Σ → B ×_{k} ℙ^{1} with pr_{B}. Denote by h′ : C → Σ′ the strict transform of h(C). The point of this construction is that E is a Cartier divisor in Σ′ which is vertical over B and such that h^{∗}E equals D_{C}.

Denote by d the degree of π_{C} and let t_{1}, …, t_{d} be closed points of ℙ^{1} such that the Cartier divisor B ×_{k} {t_{1}, …, t_{d}} of B ×_{k} ℙ^{1} is disjoint from all double points of (π_{C}, g)(C) and disjoint from (π_{C}, g)(D). Denote by T the strict transform of B ×_{k} {t_{1}, …, t_{d}} in Σ′. Form the invertible sheaf O_{Σ′}(h′(C) − T) and the pushforward E := π_{Σ′,∗}O_{Σ′}(h′(C) − T) on B. Because π_{Σ′} is flat and because O_{Σ′}(h′(C) − T) is locally free, E is torsion-free. For every point b B − π_{C}(D_{C}), Σ′_{b} is isomorphic to ℙ^{1} (via the projection Σ′ → B ×_{k} ℙ^{1} → ℙ^{1}). And Σ′_{b} ∩ h′(C) and Σ′_{b} ∩ T are divisors of the same degree d. Thus O_{Σ′}(h′(C)−T)|_{Σ′_{b}} is isomorphic to O_{Σ′_{b}} ≅ O_{ℙ^{1}}. Therefore E|_{b} is isomorphic to h^{0}(Σ′_{b}, O_{Σ′_{b}}), which is one-dimensional. Therefore E is an invertible sheaf.

By Riemann-Roch and Serre duality, for every sufficiently large degree, for a general effective divisor Δ on B of that degree, E ⊗_{O_{B}} O_{B}(Δ) is globally generated. Choose Δ to be disjoint from π_{C}(D_{C}) and from the image in B of the finitely many intersection points of h′(C) and T. Since E ⊗_{O_{B}} O_{B}(Δ) is globally generated, there exists a section which is nonzero at every point of Δ. Of course a nonzero section of this sheaf (up to scaling) is precisely the same thing as a divisor V on Σ′ such that

h′(C) + π_{Σ′}^{∗}Δ ∼ T + V.

For b in B − π_{C}(D_{C}), if the section is nonzero at b then V does not intersect Σ′_{b}. The same does not necessarily hold for points b of π_{C}(D_{C}) since b may lie in the support of R^{1}π_{Σ′,∗}O_{Σ′}(h′(C)−T). Therefore V is a sum of finitely many irreducible components of fibers of π_{Σ′} (possibly with multiplicity) lying over points not in Δ.

The linked curve (C_{link}, π_{C,link}) is h′(C) + π_{Σ′}^{∗}Δ together with the restriction of π_{Σ′}. Denote by Π the pencil of divisors in Σ′ spanned by the divisors h′(C) + π_{Σ′}^{∗}Δ and T + V, with these two divisors marked as 0 and ∞ respectively. Denote by C ⊂ Π×_{k} Σ′ the corresponding family of divisors. By Bertini’s theorem, the general member C_{t} is smooth away from the base locus. Now the only singular points of h′(C) + π_{Σ′}^{∗}Δ are the points h′(π_{C}^{−1}(Δ)). Since V does not intersect π_{Σ′}^{∗}Δ, these singular points are not in the base locus. Since C_{0} is nonsingular at every basepoint, the same is true for C_{t} for t general. Thus a general member C_{t} is smooth everywhere.

Define D_{C} to be the pullback to C of the Cartier divisor E in Σ′. Because E is vertical over B and because h^{∗}E equals D_{C}, the deformation C together with the effective Cartier divisor D_{C} is a one-parameter log deformation of (C_{link}, π_{C,link}, D_{C}). And it specializes at t = ∞ to a union of section curves and vertical curves. □

3.5. Completion of the proof. We are finally prepared for the proof of Theorem 3.9.

Proof of Theorem 3.9. By Proposition 3.28, there exists a log flexible curve j_{0} : C → U. Denote by D_{C} the reduced scheme of the C ∩ D_{π,log}. By Lemma 3.31, after attaching finitely many links to C over the points of a general divisor Δ of B, the linked curve C_{link} together with D_{C} admits a one-parameter log deformation

(ρ, π_{C}) : C → Π ×_{k} B, D_{C} ⊂ C

of (C_{link}, D_{C}) specializing to a section curve (in fact C_{∞} is a union of section curves and vertical curves).

By Proposition 3.18, the relative very free locus U_{π,v.f.} is open in U. Thus C ∩U_{π,v.f.} is open in C. So its complement is finitely many points in C. Thus a general divisor Δ is disjoint from the finite set π_{U}(D_{π,log}) and from the finite set π_{C}(C−C∩U_{π,v.f.}). Then, by Lemma 3.29, there exists an extension of j_{0} to a linked log flexible curve

j : C_{link} → U.

Form the fiber product

U_{C} := C ×_{π_{C},B,π_{U}} U.

Since π_{U} is flat, also the projection

pr_{C} : U_{C} → C

is flat. Since π_{C} is surjective, the geometric generic fiber of pr_{C} equals the geometric generic fiber of π_{U}, which is integral. Since pr_{C} is flat with integral geometric generic fiber, U_{C} is integral. Define

ν : Ũ_{C} → U_{C}

to be the blowing up of U_{C} along the closed subscheme D_{C} ×_{B} D_{π,log}. Since U_{C} is integral, also Ũ_{C} is integral. And the composition

Ũ_{C} → U_{C} → C → Π

is surjective. Since Π is a smooth curve, the morphism

ρ̃: Ũ_{C} → Π

is flat.

Consider the graph,

Γ_{j} : C_{link} = C_{0} → C_{0} ×_{B} U = U_{C,0}.

Because the links of C_{link} do not intersect D_{π,log}, the image of Γ_{j} is smooth at every point of intersection with D_{C} ×_{B} D_{π,log}. Since ν is birational, Γ_{j} gives a rational transformation from C_{link} to Ũ_{C,0}. Since ν is proper, and since C_{link} is smooth at every point of intersection with D_{C} ×_{B} D_{π,log}, the valuative criterion of properness implies this rational transformation is actually a regular morphism

Γ̃_{j} : C_{0} → Ũ_{C,0}.

Clearly this is a section of the projection morphism

pr_{C_{0}} : Ũ_{C,0} → C_{0}.

For every point t in C_{link} − D_{C}, the morphism π_{U} : U → B is smooth at j(t). Therefore also U_{C,0} → C_{0} is smooth at Γ_{j}(t). And since ν is an isomorphism over Γ_{j}(t), also pr_{C_{0}} : Ũ_{C,0} → C_{0} is smooth at Γ̃_{j}(t). Also the vertical tangent bundle equals the pullback of the vertical tangent bundle of π_{U} : U → B, which also equals T_{π,log} (since j(t) is not in D_{π,log}).

Let t be a point of D_{C} and let D_{t} be the unique irreducible component of D_{π,log} containing j(t). Give D_{t} the reduced structure. Because j_{0}(C) is transverse to D_{t} at j_{0}(t), the ramification index m_{C} − 1 of π_{C} : C → B at t equals the ramification index m_{D} − 1 of π_{U} along D_{t}. Therefore, by Lemma 3.25, the projection

pr_{C_{0}} : Ũ_{C,0} → C_{0}

is smooth over the preimage of {t} × D_{t} for every t and the vertical tangent bundle equals the pullback of T_{π,log}. Since Γ_{j}(t) is in {t} × D_{t}, this implies that pr_{C_{0}} is smooth at every point of the image of Γ̃_{j} and the vertical tangent bundle of pr_{C_{0}} equals the pullback of T_{π,log}.

Since Γ̃_{j} is a section with image in the smooth locus of pr_{C_{0}}, the normal sheaf N equals the restriction of the vertical tangent bundle. Therefore Γ̃_{j}^{∗}N equals j^{∗}T_{π,log}. Since j : C_{link} → U is log flexible, h^{1}(C_{link}, j^{∗}T_{π,log}) equals 0. Therefore, by Proposition 3.13, the relative Hilbert scheme Hilb(Ũ_{C}/Π) is smooth over Π at the point 0′ := [Image(Γ̃_{j})]. Thus for a general complete intersection curve Π′ containing 0′, the morphism Π′ → Π is smooth at 0′.

Replace Π′ by the unique irreducible component containing 0′, and then replace this by its normalization. The result is that Π′ is a smooth, projective, connected curve together with a morphism Π′ → Hilb(Ũ_{C}/Π) so that the induced morphism Π′ → Π is smooth at 0′. In particular it is flat, so surjective. Let ∞′ denote a closed point of Π′ mapping to ∞. Then (Π′, 0′, ∞′) → (Π, 0, ∞) is a flat morphism of 2-pointed smooth curves. Thus, by Lemma 3.11, the base change Π′ ×_{Π} C is a one-parameter deformation of C_{link} over (Π′, 0′, ∞′) specializing to a section curve.

Denote by

Z ⊂ Π′ ×_{Π} Ũ_{C}

the pullback of the universal closed subscheme Univ(Ũ_{C}/Π) by the morphism Π′ → Hilb(Ũ_{C}/Π). The composition with pr_{C} is a projective morphism

Z ⊂ Π′ ×_{Π} Ũ_{C} → Π′ ×_{Π} C

of flat Π′-schemes. Moreover, the fiber over 0′ ∈ Π′ is an isomorphism since the projection Γ̃_{j}(C_{flex}) → C_{flex} is an isomorphism. Therefore the morphism is an isomorphism over N ×_{Π} C for some open neighborhood N of 0′ in Π′. (This is wellknown; a complete proof is given in [dJS03, Lemma 4.7].) Invert this isomorphism and compose it with the morphism

Π′ ×_{Π} Ũ_{C} → Ũ_{C} → U_{C} → U.

The result is precisely an extension

j_{N} : N ×_{Π} C → X

of j for the one-parameter deformation Π′ ×_{Π} C. Therefore, by Lemma 3.12, there exists a section s : B → X of π_{X}. □

3.6. Corollaries. There are a number of consequences of Theorem 3.9 and its generalization to positive characteristic in [dJS03]. Many of these consequences were recognized before Conjecture 3.8 was proved.

Corollary 3.32. [Kol96, Conjecture IV.5.6] Conjecture 3.7 is true. Moreover, for every smooth, projective, irreducible variety X over an algebraically closed field of characteristic 0, there exists a dense open X^{0} ⊂ X and a projective, smooth morphism q_{0} : X^{0} → Q^{0} such that every fiber of q_{0} is rationally connected, and every projective closure of Q^{0} is nonuniruled.

Corollary 3.33. [GHS03, Corollary 1.7] The uniruledness conjecture implies Mumford’s conjecture. To be precise, assume that for every smooth, projective, irreducible variety X over an algebraically closed field k of characteristic 0, if X is nonuniruled then h^{0}(X, ω_{X}^{⊗n}) is nonzero for some n > 0. Then for every smooth, projective, irreducible variety X over k, if X is not rationally connected then h^{0}(X, Ω_{X}^{⊗n}) is nonzero for some n > 0.

The next corollary is a fixed point theorem. In characteristic 0 it can be proved using the Atiyah-Bott fixed point theorem. But in positive characteristic it is a new result. There are examples due to Shioda proving one cannot replace “separably rationally connected” by “rationally connected”, cf. [Shi74].

Corollary 3.34. [Kol03] Let Y be a smooth, projective, separably rationally connected variety over a field k and let f : Y → Y be a k-automorphism. If char(k) is positive, say p, assume in addition that f has finite order n not divisible by p^{2}. Then the fixed locus of f is nonempty.

Of course it suffices to prove the case when k is algebraically closed, since the fixed locus of the base change equals the base change of the fixed locus. First assume f has finite order n. If n is prime to char(k), let B′ denote ℙ^{1} and let ℤ/nℤ act on ℙ^{1} by multiplication by a primitive n^{th} root of unity. Note this action fixes ∞ and has trivial generic stabilizer. If char(k) = p is positive and if n = pm where m is prime to p, let B′ be the normal, projective completion of the affine curve

𝕍(y^{m} − (x^{p} − x)) ⊂ 𝔸_{k}^{2}.

Let ζ be a primitive m^{th} root of unity, and let a generator of ℤ/mℤ act by (x, y) ↦ (x, ζy). Similarly, let a generator of ℤ/pℤ act by (x, y) ↦ (x + 1, y). Clearly these actions commute, thus define an action of ℤ/nℤ on B′. Note this action fixes the unique point ∞ not in the affine chart above, and the action has trivial generic stabilizer.

Let ℤ/nℤ act diagonally on Y ×_{k} B′, and let X be the quotient. Also let B the quotient of the ℤ/nℤ-action on B′. The projection π : X → B satisfies the hypotheses of Theorem 3.9 (or its generalization in [dJS03]). Therefore there exists a section. This is the same as ℤ/nℤ-equivariant k-morphism f : B′ → Y. In particular, since ∞ is a fixed point in B′, f(∞) is a fixed point in Y.

Next assume k has characteristic 0. By general limit arguments there exists an integral, finitely generated ℤ-algebra R, a ring homomorphism R ↪ k, a smooth, projective morphism Y_{R} → Spec R whose relative very free locus is all of Y_{R}, and an R-automorphism f_{R} : Y_{R} → XY R such that the base change Y_{R} ⊗_{R} k equals Y and the base change of f_{R} equals f. Since the intersection of the graph of f_{R} and the diagonal of Y_{R} ×_{R} Y_{R} is a proper scheme over Spec R, the image in Spec R is a closed subscheme of Spec R. To prove this closed subscheme equals all of Spec R, and thus contains the image of Spec k, it suffices to prove it contains a Zariski dense set of closed points.

Choose an f-invariant very ample sheaf, choose a basis for the space of global sections, and let A be the N × N matrix with entries in R giving the action of f on global sections with respect to this basis. The set of maximal ideal in Spec R with residue field of characteristic p > N are Zariski dense in Spec R. Every invertible matrix over a characteristic p field with order divisible by p^{2} has a Jordan block with eigenvalue 1 and size divisible by p. Thus, since p > N, the finite order of f_{R} modulo the prime is not divisible by p^{2}. Therefore, by the previous case, the reduction of f_{R} modulo the prime has nonempty fixed locus. Therefore the original automorphism f has nonempty fixed locus.

Corollary 3.34 is particularly interesting in combination with a result which is attributed to Campana.

Proposition 3.35 (Campana). Let Y be a quasi-projective, rationally chain connected variety over an algebraically closed field k. The algebraic fundamental group of Y is finite. Moreover, if k = ℂ, then the topological fundamental group of Y is finite.

The basic point is this: there exists a choice of u : M ×_{k} C_{n} → Y such that the induced morphism u^{(2)} : M ×_{k} C_{n} ×_{k} C_{n} → Y ×_{k} Y is generically finite. For a general point y of Y, the fiber over {y} × Y has an irreducible component N which is generically finite over Y. Thus there exists a morphism

v : N ×_{k} C_{n} → Y

and two points 0, ∞ ∈ C_{n} such that

v_{0} : N = N × {0} → Y

the constant map with image {Y } and

v_{∞} : N = N × {∞} → Y

is dominant and generically finite. Let w : Ỹ → Y be a connected, unramified, cover in either the algebraic category or in the complex analytic category. Let ỹ be a point over y. The morphism v_{0} lifts to a constant morphism ṽ_{0}. Since C_{n} is simply connected, this implies that there is a unique morphism

ṽ: N ×_{k} C_{n} → Ỹ

whose restriction to N ×{0} equals ṽ_{0} and which is compatible with v. In particular, v factors through w. Therefore the degree of w is bounded by the degree of v so that Y has fundamental group bounded by the degree of v.

Corollary 3.36. [Cam91] [Kol03] Let X be a smooth, projective, separably rationally connected variety over an algebraically closed field k. The algebraic fundamental group of X is trivial. If k = ℂ, then the topological fundamental group of X is also trivial.

This was proved over ℂ by Campana. The general case was proved by Kollár. By Proposition 3.35, the universal cover X̃ → X is finite. Moreover, by the proof of the proposition, X̃ is also smooth, projective and separably rationally connected. If the fundamental group of X is nonzero, then it contains a cyclic subgroup ℤ/nℤ such that p^{2} does not divide n. Of course the action of this group on X̃ is fixed-point-free. But Corollary 3.34 implies there exists a fixed point. Thus X is simply connected. Kollár has generalized this considerably to prove a result for open subschemes of rationally connected varieties.

Theorem 3.9 also plays an important role in the proof of a “converse” to Theorem 3.9.

Theorem 3.37. [GHMS05] Let π : X → B be a surjective morphism of normal, projective, irreducible varieties over an algebraically closed field k of characteristic zero. Assume that for some sufficiently large, algebraically closed field extension K/k, for every k-morphism C → B from a smooth, projective, K-curve to X, the pullback π_{C} : C ×_{B} X → C has a section. Then there exists a closed subvariety Y ⊂ X such that the geometric generic fiber of π|_{Y} : Y → B is nonempty, irreducible and rationally connected.

One corollary of this theorem, in fact the motivation for proving it, was to answer a question first asked by Serre and left unresolved by Theorem 3.9: could it be that a smooth, projective variety X over the function field of a curve has a rational point if it is O-acyclic, i.e., if h^{i}(X, O_{X}) equals 0 for all i > 0? One reason to ask this is that the corresponding question has a positive answer if “function field” is replaced by “finite field” (thanks to N. Katz’s positive characteristic analogue of the Atiyah-Bott fixed point theorem).

Corollary 3.38. [GHMS05] There exists a surjective morphism π : X → B of smooth, projective varieties over ℂ such that B is a curve and the geometric generic fiber of π is an Enriques surface, but π has no section. Thus, to guarantee a fibration over a curve has a section, it is not sufficient to assume the geometric generic fiber is O-acyclic.

In fact G. Lafon found an explicit morphism π as in Corollary 3.38 where B is ℙ_{ℂ}^{1}, or in fact ℙ_{k}^{1} for any field k with char(k) ≠ 2, and there does not even exist a power series section near 0 ∈ ℙ_{k}^{1}, cf. [Laf04].

4. The Period-Index theorem

Theorem 3.9 is a generalization of Tsen’s theorem, Corollary 2.15, because a sufficiently general complete intersection 𝕍(F_{1}, …, F_{r}) ⊂ ℙ^{n} with d_{1} + ⋯ + d_{r} ≤ n is smooth, projective and separably rationally connected (the proof of this is nontrivial, as is the specialization argument reducing Tsen’s theorem to the case of complete intersections which are sufficiently general). Is there a similar generalization of the Tsen-Lang theorem?

Joint work with Harris, [HS05], proves the spaces of rational curves on general low degree hypersurfaces are rationally connected. This was later generalized in joint work with A. J. de Jong: complete intersections X = 𝕍(F_{1}, …, F_{r}) ⊂ ℙ^{n} with d_{1}^{2} + ⋯ + d_{r}^{2} ≤ n + 1 are rationally simply connected in the sense that the space of “good” rational curves in X containing two fixed, general points is itself a rationally connected variety. This is analogous to simple connectedness in topology: a path connected topological space is simply connected if the space of paths connecting two fixed points is itself path connected.

Moreover, de Jong gave a heuristic argument suggesting that for a rationally simply connected fibration over a surface, the only obstruction to existence of a rational section is the elementary obstruction. Given a geometrically integral scheme X defined over a field K, the elementary obstruction to existence of a K-point is the existence of a Gal(\overline{K}/K)-invariant splitting of the homomorphism of Abelian Galois modules,

\overline{K}^{∗} ↪ Frac(X ⊗_{K} \overline{K})^{∗},

where \overline{K} is the separable closure of K and Frac is the function field. If there exists a K-point of X, evaluation at this point gives a Galois-invariant splitting. The elementary obstruction was introduced by Colliot-Thélène and Sansuc, [CTS87]. Its vanishing implies the vanishing of other known obstructions. In particular, it implies the vanishing of a Brauer obstruction

δ : Pic(X ⊗_{K} \overline{K})^{Gal(\overline{K}/K)} → Br(K)

measuring whether or not a Galois-invariant invertible sheaf \overline{L} on X ⊗_{K} \overline{K} is the pullback of an invertible sheaf L on X.

At the moment, in order to give a rigorous proof, de Jong’s heuristic argument requires several additional hypotheses on the rationally simply connected fibration. One case where the hypotheses hold is when all fibers of the fibration are Grassmannian varieties. Although this is very special, it is also quite interesting since it gives a second proof of de Jong’s period-index theorem.

Theorem 4.1. [dJS05] Let K be the function field of a surface over an algebraically closed field k. Let (X, L) be a pair of a K-scheme and an invertible O_{X}-module L. If (X ⊗_{K} \overline{K}, L⊗_{K} \overline{K}) is isomorphic to (Grass(r, \overline{K}^{n}), O(1)), then X has a K-point.

Corollary 4.2 (de Jong’s Period-Index theorem). [dJ04] Let A be a central simple K-algebra with A ⊗_{K} \overline{K} ≅ Mat_{n×n}(\overline{K}). Let r < n be an integer such that r[A] equals 0 in Br(K). Then there exists a left ideal I ⊂ A such that dim_{K}(I) = rn. In particular, if A = D is a division algebra then [D] has order n in Br(K), i.e., the period of D equals the index of D.

Corollary 4.2 follows from Theorem 4.1 by setting X to be the generalized Brauer-Severi variety parameterizing left ideals in A of rank rn. Since A ⊗_{K} \overline{K} equals Mat_{n×n}(\overline{K}), X ⊗_{K} \overline{K} equals Grass(r, \overline{K}^{n}). The Brauer obstruction to the existence of an invertible sheaf L with L⊗_{K} \overline{K} ≅ O(1) is precisely the element r[A] in Br(K).

The first reduction is “discriminant avoidance”, i.e., reduction to the case that the variety X is the generic fiber of a smooth, projective morphism over a smooth, projective surface. Let T be a quasi-compact, integral scheme and let G be a smooth, affine group scheme over T whose geometric fibers are reductive.

Lemma 4.3. For every integer c there exists a datum (U, \overline{U}, T_{U}) of a projective, flat T -scheme \overline{U} with integral geometric fibers, an open subset U of \overline{U} and a G-torsor T_{U} over U such that

(i) U is smooth over T, (ii) the complement \overline{U} − U has codimension ≥ c in \overline{U}, (iii) and for every G-torsor T_{k} over an infinite field K over T, there exists a

T -morphism i : Spec K → U and an isomorphism of G-torsors over K, i^{∗}T_{U} ≅ T_{K}.

The idea is to form the GIT quotient \overline{U} of a linear action of G on (ℙ_{T}^{N}, O(1)). If the linear representation is “sufficiently large”, then G acts properly and freely on an open subset V of ℙ_{T}^{N} of codimension ≥ c. Take U to be the quotient of V. Then U is smooth over T and \overline{U} − U has codimension ≥ c. Finally, for every field K over O_{T} and for every G-torsor T_{K} over K, the twist ℙ_{T}^{N} ×_{T} T_{K}/G is isomorphic to ℙ_{K}^{N}. Thus there exists a K-point. If K is infinite, then the set of K-points is Zariski dense so that there exists a point in the image of V. This point is only well-defined up to the action of G, but the associated morphism i : Spec K → U is well-defined. Chasing definitions, i^{∗}T_{U} is isomorphic to T_{K}.

Proposition 4.4. Let k be an algebraically closed field of characteristic 0, let S be a smooth, projective surface over k, let π : X → S be a smooth, projective morphism, and let L be an invertible O_{X} -module. Let K/k be the fraction field of S and let X be the generic fiber of π.

(i) If (X ⊗_{K} \overline{K}, L ⊗_{K} \overline{K}) is isomorphic to (Grass(r, \overline{K}^{n}), O(1)), then X has a K-point.

(ii) Item (i) implies Theorem 4.1

The point of this proposition is Item (ii), i.e., to prove Theorem 4.1 it suffices to assume that X is the generic fiber of a proper and everywhere smooth morphism. The point is that for every algebraically closed field k, there exists the spectrum of a DVR, T, whose residue field is k and whose fraction field has characteristic 0. Now let G be the automorphism group scheme of (Grass(r, O_{T}^{n}), O(1)). This satisfies the hypotheses of Lemma 4.3. Taking c = 3, there exists a datum (U, \overline{U}, T_{U}) as in Lemma 4.3 such that \overline{U} − U has codimension ≥ 3. For the original field K/k, there exists a morphism i : Spec K → U inducing the pair (X, L). Because tr.deg.(K/k) = 2, the closure of Image(i) in \overline{U} has dimension ≤ 2. Because \overline{U} − U has codimension ≥ 3, there exists a locally closed subscheme S of \overline{U} such that

(i) S → T is flat,

(ii) the closed fiber S_{0} of S is irreducible with generic point i(Spec K), (iii) and the generic fiber S_{η} of S is a closed subscheme of \overline{U}_{η} completely con-

tained in U_{η}. Using specialization arguments, to prove Theorem 4.1 for the restriction of T_{U} to the generic point of the closed fiber S_{0}, it suffices to prove Theorem 4.1 for the restriction of T_{U} to the generic point of the geometric generic fiber S\overline{η}. By construction, this satisfies the additional hypotheses in Proposition 4.4.

Thus, assume the additional hypotheses of Proposition 4.4 are satisfies. After replacing S by the blowing up at the base locus of a Lefschetz pencil of divisors, and replacing X by its base change, assume there exists a flat, proper morphism ρ : S → B with smooth, connected generic fiber. Denote by B^{0} the maximal open subscheme of B over which ρ is smooth, and denote S^{0} = B^{0} ×_{B} S and X^{0} = B^{0} ×_{B} X. There is a pair

(ρ_{Sect} : Section(X^{0}/S^{0}/B^{0}) → B^{0}, σ : Section(X^{0}/S^{0}/B^{0}) ×_{B^{0}} S^{0} → X^{0})

which is universal among all pairs (T, σ_{T}) of a B^{0}-scheme T and an S_{0}-morphism σ_{T} : T ×_{B^{0}} S^{0} → X^{0}. The universal pair can be constructed in terms of the relative Hilbert scheme. In Grothendieck’s terminology, it is Π_{S^{0}/B^{0}}X_{0}, cf. [Gro62, p. 195-13].

There is a relative Picard scheme Pic(S^{0}/B^{0}) of S^{0} over B^{0}. Associated to the invertible sheaf L on X^{0}, there is an invertible sheaf σ^{∗}L on Section(X^{0}/S^{0}/B^{0})×_{B^{0}} S^{0}. This induces an Abel morphism

α : Section(X^{0}/S^{0}/B^{0}) → Pic(S^{0}/B^{0}).

Of course Pic(S^{0}/B^{0}) breaks up according the relative degree of the line bundle,

Pic(S^{0}/B^{0}) = ⊔_{d∈ℤ}Pic^{d}(S^{0}/B^{0}).

Pulling this back via the Abel morphism gives a decomposition

Section(X^{0}/S^{0}/B^{0}) = ⊔_{d∈ℤ}Section^{d}(X^{0}/S^{0}/B^{0})

together with Abel morphisms

α_{d} : Section^{d}(X^{0}/S^{0}/B^{0}) → Pic^{d}(S^{0}/B^{0}).

For all d ≥ 0, there are sections of the projection Pic^{d}(S^{0}/B^{0}) → B^{0}. The image of this section is a curve B′_{0} isomorphic to the smooth curve B^{0}. If the generic fiber of α_{d} is a dense open subset of a rationally connected variety, then Theorem 3.9 together with the generic version of weak approximation, Proposition 3.19, implies there exists a rational section of the restriction of α_{d} over B′_{0} (this uses a slight specialization argument, because the restriction of α_{d} may not be a rationally connected fibration). Thus it suffices to prove that for d ≫ 0,

(i) the fiber of α_{d} over the geometric generic point of Pic^{d}(S^{0}/B^{0}) is nonempty, (ii) the fiber is also irreducible, (iii) the fiber is also isomorphic to an open subset of a rationally connected

variety.

Moreover, and this will be important, it suffices to prove there exists a canonically defined open subscheme W ⊂ Section^{d}(X^{0}/S^{0}/B^{0}) such that (i)–(iii) hold for α_{d}|_{W}.

Note that (i)–(iii) are really statements about the morphism X\overline{η_{B}} → S\overline{η_{B}} of fibers over the geometric generic point of B. Thus, it is again a question about a fibration over a projective curve, namely the curve C = S\overline{η_{B}} over the algebraically closed field κ = \overline{k(B)}. So Proposition 4.4 (i) follows from.

Proposition 4.5. Let κ be an algebraically closed field of characteristic 0. Let C be a smooth, projective, connected curve over κ. Let π : X_{C} → C be a smooth, projective morphism and let L be an invertible sheaf on X_{C}. Assume the fiber of (X_{C}, L) over the geometric generic point of C is isomorphic to (Grass(r, \overline{κ(C)}), O(1)). Then for d ≫ 0 there exists a canonically defined open subset W_{d} ⊂ Section^{d}(X_{C}/C/Spec κ) such that the geometric generic fiber of

α_{d} : W_{d} ↪ Section^{d}(X_{C}/C/Spec κ) → Pic^{d}(C/Spec κ)

satisfies (i), (ii) and (iii) above.

In order to prove Corollary 4.2, it suffices to prove this in the special case that X_{C} → C is the parameter scheme for rank rn left ideals in an Azumaya algebra A over C with A ⊗_{O_{C}} \overline{κ(C)} ≅ Mat_{n×n}(\overline{κ(C)}). An Azumaya algebra over a scheme T is a coherent O_{T} -algebra which is étale locally isomorphic to Mat_{n×n}(O_{T}) for some integer n.

Because of Tsen’s theorem, Corollary 2.15, and Proposition 2.16(i), there exists a locally free O_{C}-module E of rank n such that A ≅ End(E). The locally free sheaf E is only well-defined up to the operation E ↦ E ⊗_{O_{C}} N, for any invertible sheaf N. The choice of a locally free sheaf E and an isomorphism of algebras gives an isomorphism of Sect(X_{C}/C/Spec κ) with the parameter scheme of locally free quotients E ↠ Q of rank r. To see this, associate to each quotient the left ideal of endomorphisms that factor as

E ↠ Q ⟶^{ϕ} E

as ϕ varies over all O_{C}-module homomorphisms. Replacing E by E ⊗_{O_{C}} N gives a new isomorphism sending the original quotient to the twist E ⊗_{O_{C}} N ↠ Q ⊗_{O_{C}} N. The real effect of this change has to do with the Abel map. Up to a constant translation by a point of Pic(C/Spec κ), which does not change (i)–(iii), the Abel map is identified with the map sending a quotient to det(Q) in Pic(C/Spec κ). After replacing E by E ⊗_{O_{C}} N, the constant changes by adding [N^{⊗r}]. Thus, it is clear that the Abel map really should only be considered as well-defined up to an additive constant.

If d is sufficiently large, there exist quotients such that Q is stable. Note that Q is stable if and only if Q ⊗_{O_{C}} N is stable. Therefore the open subset W_{d} of Section(X_{C}/C/Spec κ) parameterizing stable quotients is well-defined and canonical. Fix an integer d_{0}. For every integer e, the moduli space of stable, rank r locally free sheaves on C of degree d_{0} is isomorphic to the moduli space for degree d_{0} + re via the map sending Q to Q(D), where D is any fixed Cartier divisor of degree e. For each fixed locally free sheaf Q of rank r < n, if e is sufficiently large, there exists a surjection E ↠ Q(D) for all Cartier divisors of degree e. Because the moduli space of stable bundles is quasi-compact, it follows that there exists a single integer e_{0} such that for every e ≥ e_{0} and every stable locally free sheaf Q of rank r and degree d_{0}, for every Cartier divisor D of degree e there exists a surjection E ↠ Q(D). By the same sort of argument, if e is sufficiently large then h^{1}(C, Hom_{O_{C}} (E, Q(D))) equals 0.

Repeating this argument with d_{0} replaced by each of d_{0}, d_{0} +1, d_{0} +2, …, d_{0} +r−1, there exists an integer d_{1} such that for every d ≥ d_{1} (i.e., d = d_{0} + re, etc.), for every stable, locally free sheaf Q of rank r and degree d, there exists a surjection

E ↠ Q

and also h^{1}(C, Hom_{O_{C}} (E, Q)) equals 0. In other words, the forgetful morphism from the space of quotients E ↠ Q to the space of stable sheaves Q is smooth and surjective, and the geometric fibers are each isomorphic to an open subset of an affine space Hom_{O_{C}} (E, Q). Moreover, the fiber of the Abel map α_{d} is the inverse image of the space of stable sheaves with fixed determinant. As is well-known, the moduli space of stable sheaves over C of fixed rank r and fixed determinant is a unirational variety of dimension (r^{2} − 1)(g(C) − 1). Thus the fiber of the Abel map fibers over a rationally connected variety and the fibers are rationally connected. By Corollary 3.32, it follows that a general fiber of

α_{d}|_{W} : W_{d} ↪ Section^{d}(X_{C}/C/Spec κ) → Pic^{d}(C/Spec κ)

is isomorphic to an open subset of a rationally connected variety, i.e., (i), (ii) and (iii) hold. Therefore Corollary 4.2 is true.

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Department of Mathematics, Stony Brook University, Stony Brook, NY 11794

E-mail address: jstarr@math.sunysb.edu