Moving Algebraic Cycles of Bounded Degree

Eric M. Friedlander^{∗} and H. Blaine Lawson^{∗∗}

The Chow Moving Lemma is a theorem which asserts that a given algebraic s-cycle on a smooth algebraic variety X can be moved within its rational equivalence class to intersect properly a given r-cycle on X provided that r + s ≥ dim(X) (cf. [Chow], [S2]). In the past few years, there has been considerable interest in studying spaces of algebraic cycles rather than simply cycles modulo an equivalence relation. With this in mind, it is natural to ask whether one can move a given “bounded family” of s-cycles on the smooth variety X to intersect properly a given “bounded family” of r-cycles. The main point of this paper is to formulate and prove just such a result. In Theorem 3.1, we demonstrate that for any integer e and any smooth projective variety X, one can simultaniously and algebraically “move” all effective s-cycles of degree ≤ e on X so that each such cycle meets every effective r-cycle of degree ≤ e on X in proper dimension.

The primary motivation for this Moving Lemma for Cycles of Bounded Degree was the possibility of a duality theorem between cohomology and homology theories defined in terms of homotopy groups of cycle spaces. Using Theorem 3.1, we have proved such a duality theorem for complex quasi-projective varieties in [F-L2]. We prove our Moving Lemma for varieties over an arbitary infinite field, permitting a proof in [F-V] of a duality theorem for “motivic cohomology and homology”.

The reader will find that our Moving Lemma has numerous good properties. First of all, the move is given as an algebraic move (parametrized by a punctured projective line) on Chow varietes. Although this move is “good” only for s-cycles of bounded degree, it is defined on all effective s-cycles. Moreover, the move starts at “time 0” by expressing an effective s-cycle Z as a difference of effective s-cycles both of which have intersection properties no worse than Z. Finally, our Moving Lemma is applicable to smooth quasiprojective varieties, for it is stated for a possibly singular projective variety X resulting in a conclusion of proper intersection off the singular locus of X.

The classical motivation for the moving lemma was to define an intersection product on algebraic cycles modulo rational equivalence, thereby establishing the Chow ring A_{∗}(X). Apparently, the classical literature overlooked the question of whether or not intersection of cycles defined via a moving lemma is independent of the move (e.g., [Chev], [Chow], [R], [S2]). One direct consequence of our Moving Lemma is a proof for smooth quasiprojective varieties that the intersection product is indeed well defined independent of the choice of move (Theorem 3.4). Of course, the intersection product now has an intrinsic formulation for all smooth algebraic varieties due to Fulton and MacPherson [Fu]. We

∗Partially supported by the N.S.F. and NSA Grant # MDA904-93-H-3029 ∗∗Partially supported by the N.S.F. should emphasize that our arguments are projective in nature, and thus apply directly only to quasi-projective varieties.

We gratefully acknowledge helpful comments from William Fulton and Vladimir Voevodsky. We are especially indebted to Ofer Gabber who foresaw the validity of such a moving lemma for families and directed us to local estimates arising in the proof of the key Theorem 1.1.

0. Definitions, Conventions, and a Brief Outline

We adopt the following conventions thoughout. We fix a ground field k, choose an algebraic closure \overline{k} ⊃ k, and consider projective spaces P^{n} over k of various dimensions n. A projective variety over k is a reduced and absolutely irreducible algebraic kscheme which admits a (Zariski) closed embedding in some projective space P^{n}. A quasiprojective variety over k is a Zariski open subset of some projective algebraic variety over k whose complement has defining ideal also defined over k. The set of geometric points X(\overline{k}) of such a quasi- projective variety X is the set of morphisms Spec(\overline{k}) → X of schemes over k. For a point x on a quasi-projective variety X, we denote by O_{X,x} the stalk at x ∈ X of the structure sheaf of X and by 𝔪_{x} the maximal ideal of this local ring. An algebraic cycle Z of dimension r on a quasi-projective variety X is an integral combination of closed, r-dimensional subvarieties of X, which are (reduced, irreducible but not necessarily absolutely irreducible and) defined over k. We say Z is effective if the integer coefficients are all positive. Given a cycle Z on X, we define its support to be the algebraic subset |Z| ⊂ X consisting of the union of the irreducible components of Z. If Y and Z are cycles on X of dimension r and s respectively with r + s ≥ m = dim(X), we say that Y and Z intersect properly if each component of |Y | ∩ |Z| has dimension ≤ r + s − m. If X is smooth and if Y and Z intersect properly, then the intersection product Y • Z is a well- defined cycle of dimension r + s − m on X (See [Fu]).

Let X ⊂ P^{n} be a closed embedding over k of an m- dimensional projective variety X. For integers d ≥ 0 and r with 0 ≤ r < dim(X), we denote by C_{r,d}(X) the Chow variety of effective r-cycles of degree d on X (cf. [S1]). The disjoint union

C_{r}(X) = ∐_{d≥0} C_{r,d}(X)

has the structure of an abelian monoid and is called the Chow monoid. The geometric points C_{r}(X)(\overline{k}) of this Chow monoid constitute the (discrete) monoid of effective r-cycles on X\overline{k}, the base-change of X to Spec(\overline{k}). (See [F,1.2] for a discussion of the the k′-rational points of C_{r}(X) for any k′/k.)

Given an r-cycle Y and an s-cycle Z on a projective variety X ⊂ P^{n} of dimension m with r + s ≥ m, the classical Chow Moving Lemma (cf. [R]) asserts the existence of a rational equivalence between Z and a necessarily ineffective s-cycle Z′ which intersects Y properly at all smooth points of X. Roughly speaking the proof proceeds as follows. One considers a linear projection π_{L} : P^{M} ⟶ P^{m} whose vetex L ≅ P^{M−m−1} does not meet X and constructs the projecting cone C_{L}(Z) = π_{L}^{∗}(π_{L∗}(Z)) of Z. Since π_{L}|_{X} is a finite morphism, C_{L}(X)•X is a well-defined s-cycle on X which can be moved to intersect Y properly by moving the cone C_{L}(Z) in P^{M}. The idea then is to choose L so that the residual cycle

R_{L}(Z) = C_{L}(Z) • X − Z

intersects Y properly at least at all points outside the ramification locus of p_{L} =^{def} π_{L}|_{X}. Furthermore, any component of excess intersection of |R_{L}(Z)| ∩ |Y | lies in ram(p_{L}) ∩ |Z| ∩ |Y |. Thus, if we choose L_{1}, …, L_{m+1} so that ∩_{i}ram(p_{L_{i}}) is contained in the singular locus of X, then after m + 1 iterations the residual cycle R_{L_{m+1}} ∘ R_{L_{m}} ∘ ⋯ ∘ R_{L_{1}}(Z) will meet Y properly at all regular points of X.

In our paper we shall carry through this argument so that it applies simultaneously to all effective cycles Y, Z of degree bounded by any specified positive integer e. To do this we consider the reimbeddings X ↪ P^{M_{d}} via the Veronese embeddings P^{M} ↪ P^{M_{d}} of degree d. For each d we consider the subset U_{X}(d) of linear subspaces of codimension m + 1 in P^{M_{d}} which do not meet X. The main point is to show that the codimension of “bad L’s” in U_{X}(d), i.e., those for which R_{L}(Z) does not have an improved intersection with Y, goes to infinity as d goes to infinity. This is achieved in §1.

In §2 we construct the moving of the projecting cones. This is conventionally done by projective transformations of the ambient P^{M}. Here we introduce a different method which is related to the movings introduced in [L] and algebraicized in [F] to prove the “algebraic suspension theorem” in Lawson homology. This argument essentially verifies that effective s-cycles of bounded degree in P^{n} can be moved to intersect properly a hyperplane. Here we generalize this argument so that it applies to all effective r-cycles of bounded degree, where s + r ≥ n, rather than to a single hyperplane.

In §3 we present our main results. In addition to Theorems 3.1 and 3.4 mentioned above, we prove in Theorem 3.5 that the intersection product on homotopy groups of cycle spaces on a smooth, projective complex variety (cf. [F-G]) can be represented by the intersection of families of cycles which have been moved to intersect properly. In Theorem 3.7, we present a basic ingredient of duality theorems for cycle spaces. It is the assertion that for a flat map X → B from a smooth variety, families of s-cycles on X with s ≥ dimB can be moved to be equidimensional over B.

1. Residual cycles

Consider an m-dimensional projective variety X provided with a closed embedding X ⊂ P^{n} over k. Choose a positive integer d and define

U_{X} ≡ U_{X}(d) ⊂ P(Γ(O_{P^{n}}(d)^{m+1}))

to be the Zariski open set of those F = (f_{0}, …, f_{m}) with the property that the zero locus L_{F} = {t ∈ P^{n} : F(t) = 0} misses X and thus has codimension m + 1 in P^{n}. (More precisely, F is a k-rational point of the indicated Zariski open subset.) Each such F determines a finite morphism (defined over k)

p_{F} : X ⟶ P^{m}  (1.0.1)

which can be viewed as follows. Let v : P^{n} ⟶ P^{M} denote the Veronese embedding of degree d (so M = \binom{n+d}{d} − 1). Then each F ∈ U_{X} determines a surjective linear projection

π_{F} : P^{M} − −− > P^{m}  (1.0.2)

with the property that p_{F} = π_{F} ∘ v|_{X}. The condition F(x) ≠ 0 for x ∈ X is equivalent to

the condition that the linear subspace L(F) =^{def} the vertex of the projection π_{F}, does not meet v(X).

Suppose now that Y, Z ⊂ X are closed algebraic subsets of pure dimension r and s respectively, where r + s ≥ m. Let Y × Z − Δ denote (Y × Z) ∩ (X^{2} − diag(X)) and denote by Y ⋆_{F} Z the following closed subset of Y × Z − Δ:

Y ⋆_{F} Z ≡ {(y, z) : y ≠ z, p_{F}(y) = p_{F}(z)}

for any F ∈ U_{X}. We begin by investigating the condition on F ∈ U_{X} that Y ⋆_{F} Z should have pure dimension r + s − m.

An interesting special case of the following theorem is the case in which Y equals Z, a cycle of dimension r ≥ m/2 on the projective variety X of dimension m. Then, our theorem asserts for “most” projections of sufficiently high degree that the projection is injective on components of Z off a subset of dimension no greater than 2r − m.

Theorem 1.1 Let X ⊂ P^{M} be a closed embedding over k of an m-dimensional projective variety X. Consider closed algebraic subsets Y, Z ⊂ X of pure dimension r, s respectively, and assume that r + s ≥ m. Then for any k-rational point F ∈ U_{X}(d), each irreducible component of Y ⋆_{F} Z has dimension ≥ r + s − m.

Fix any integer N > 0, and suppose for each closed point w = (y, z) ∈ Y × Z − Δ and for each φ ∈ Γ(O_{P^{n}}(d)) with φ(y) ≠ 0 ≠ φ(z) that the map

Γ(O_{P^{n}}(d)) ⟶ O_{Y,y}/𝔪_{y}^{N+1} × O_{Z,z}/𝔪_{z}^{N+1},

sending f to the restrictions of f/φ, is surjective . Then the subset

B(d)_{{Y,Z}} ⊂ U_{X}(d)

of those F for which some component of Y ⋆_{F} Z has dimension > r + s − m is a (Zariski) closed subset of codimension at least N − r − s.

Proof. Fix w = (y, z) ∈ Y × Z − Δ and denote by B_{w} ⊂ U_{X} ≡ U_{X}(d) the subset of those F = (f_{0}, …, f_{m}) for which some component of Y ⋆_{F} Z has dimension > r + s − m at w. Observe that the germ at w of the variety Y ⋆_{F} Z is defined by the equations:

f_{i}(y′)f_{j}(z′) = f_{j}(y′)f_{i}(z′)  for 0 ≤ i, j ≤ m  (1.1.1)

in a neighborhood of w = (y, z) in Y ×Z. Now there exists a linear combination φ = ∑ a_{i}f_{i} of f_{0}, …, f_{m} with the property that φ(y) ≠ 0 ≠ φ(z). For notational convenience we assume that φ = f_{0}, and we restrict attention to those F ∈ U_{X} for which f_{0}(y) ≠ 0 ≠ f_{0}(z).

The general case will follow since U_{X} is covered by a finite collection of sets of this type. (In fact the set of linear combinations f_{i} + f_{j} for 0 ≤ i ≤ j ≤ m will do.)

With this assumption on f_{0} one sees directly that equations (1.1.1) are equivalent to the equations:

g_{F}^{i}(y′, z′) = \frac{f_{i}(y′)}{f_{0}(y′)} − \frac{f_{i}(z′)}{f_{0}(z′)} = 0  for 1 ≤ i ≤ m  (1.1.2)

on the germ of Y ×Z at w. In particular, the germ at w of Y ⋆_{F}Z has dimension ≥ r+s−m.

We now proceed inductively to estimate the dimension of the bad set B_{w}. We phrase the problem in terms of the local rings. In what follows, 𝔪 will always denote the maximal ideal of the local ring in question. Set R(w, F)_{0} = O_{Y ×Z,w}, and let R(w, F)_{i} denote the local ring O_{Y ×Z,w}/⟨g_{F}^{1}, …, g_{F}^{i}⟩ for 1 ≤ i ≤ m. If R(w, F)_{i} has pure dimension r + s − i for some i < m, then the condition on g_{F}^{i+1} that R(w, F)_{i+1} have dimension ≤ r + s − i − 1 is the condition that g_{F}^{i+1} be non-zero in R(w, F)_{i,α} ≡ R(w, F)_{i}/P_{i,α} for each minimal prime P_{i,α} of R(w, F)_{i}. This is guaranteed if the class of g_{F}^{i+1} is non-zero in the quotient R(w, F)_{i,α}/𝔪^{N+1} for all α.

Let us now fix φ ∈ Γ(O_{P^{n}}(d)) with φ(y) ≠ 0 ≠ φ(z) and set B_{w,φ} = {F ∈ B_{w} : f_{0} = φ}. Consider the subset B_{w,φ}^{i+1} ⊂ B_{w,φ} of those F for which R(w, F)_{i} has pure dimension r + s − i but R(w, F)_{i+1} is not of dimension r + s − (i + 1). By the paragraph above we know that if F ∈ B_{w,φ}^{i+1}, then for some α, f_{i+1} lies in the kernel K_{i+1,α} ≡ ker(G_{α}^{i+1}) of the linear map

G_{α}^{i+1} : Γ(O_{P^{n}}(d)) ⟶ R(w, F)_{i,α}/𝔪^{N+1}  (1.1.3)

defined by taking the image in R(w, F)_{i,α}/𝔪^{N+1} of the germ at w = (y, z) of the function

G^{i+1}(F)(y′, z′) = \frac{f_{i+1}(y′)}{φ(y′)} − \frac{f_{i+1}(z′)}{φ(z′)}.  (1.1.4)

We make the following claim which is proved below.

Claim:

codim(K_{i+1,α}) ≥ N.  (1.1.5)

We now show how this claim proves the theorem. Consider the closed subvariety

W ≡ {(y, z, F) : (y, z) ∈ (Y ⋆_{F} Z)^{cl}} ⊂ Y × Z × U_{X},

where (Y ⋆_{F} Z)^{cl} ⊂ Y × Z denotes the closure of Y ⋆_{F} Z. By upper semi-continuity of the dimension of the fibres of W → U_{X}, we conclude that B(d)_{{Y,Z}} ⊂ U_{X} is a closed subvariety. Moreover, if B̃ ⊂ W denotes pr_{3}^{−1}(B(d)_{{Y,Z}}), then the fibre of B̃ above w ∈ (Y × Z) − Δ equals B_{w}. Therefore if we prove that

The codimension of B_{w} in {w} × U_{X} is ≥ N,  (1.1.6)

it will follow that the codimension of pr_{3}(B̃) = B(d)_{{Y,Z}} in U_{X} is at least N − r − s as claimed.

To prove (1.1.6) we proceed as follows. Set Γ = Γ(O_{P^{n}}(d)) and let p : U_{X} ⊂ Γ^{×(m+1)} ⟶ Γ denote projection onto the first coordinate. Then for φ ∈ Γ with φ(y) ≠ 0 ≠ φ(z) we see that

B_{w,φ} = p^{−1}(φ) ∩ B_{w} ≅ Γ^{×m} ∩ B_{w}.

It will suffice to prove that the codimension of B_{w,φ} in Γ^{×m} is ≥ N. Note that

B_{w,φ} = ⋃_{i=1}^{m} B_{w,φ}^{i} ,

and that B_{w,φ}^{1} ⊂ ⋃_{α} K_{1,α}. Now for i > 1, the K_{i,α} depend on F^{k} for k < i. So fix i > 1, let p : Γ^{×m} → Γ^{×i} be projection onto the first i coordinates, and let G^{i} denote the complement of p(B_{w,φ}^{1} ∪ ⋯ ∪ B_{w,φ}^{i}) ⊂ Γ^{×i}. (Note that G^{i} consists of all F_{i} = (F^{1}, …, F^{i}) such that R(w, F)_{i} has pure dimension r + s − i). Then for each F_{i} ∈ G^{i} we have that

p^{−1}(F_{i}) ∩ B_{w,φ}^{i+1} ⊂ {⋃_{α} K_{i+1,α}} × Γ^{×(m−i−1)} ⊂ Γ^{×(m−i)}

where the union over α is finite. Hence (1.1.6) follows from our claim (1.1.5).

Thus it remains to prove our claim (1.1.5) for i = 0, ..., m − 1. We proceed as follows. Denote by R(y, F)_{i,α} the quotient of the local ring O_{Y,y} by the pre-image of P_{i,α} under the composition O_{Y,y} ⟶ O_{Y ×Z,w} ⟶ R(w, F)_{i,α}. We define R(z, F)_{i,α} analogously. These rings correspond to the projections to Y and Z of the subvariety with generic point P_{i,α}. For i < m one of these projections must have positive dimension (since the subvariety itself does). Hence, for i < m at least one of R(y, F)_{i,α}, R(z, F)_{i,α} has Krull dimension ≥ 1. We assume wlog that this is true of R(y, F)_{i,α}. Then (by further projection onto a generic line) we know that

dim{R(y, F)_{i,α}/𝔪^{N+1}} ≥ N.  (1.1.7)

Let Γ = Γ(O_{P^{n}}(d)) as above and note that the map G^{i+1} : Γ ⟶ O_{Y ×Z,w} defined in (1.1.4) factors through the subspace O_{Y,y} × O_{Z,z} ⊂ O_{Y ×Z,w}. Passing to quotients gives a linear map

Ĝ^{i+1} : Γ ⟶ O_{Y,y}/𝔪_{y}^{N+1} × O_{Z,z}/𝔪_{z}^{N+1} (⊂ O_{Y ×Z,w}/𝔪_{w}^{N+1})

which by our assumption is surjective. Write Ĝ^{i+1} = (Ĝ_{1}^{i+1}, Ĝ_{2}^{i+1}) with respect to the splitting above and let Γ_{0} = ker(Ĝ_{2}^{i+1}). Then Ĝ_{1}^{i+1} : Γ_{0} ⟶ O_{Y,y}/𝔪_{y}^{N+1} ×{0} is surjective. This produces a commutative diagram

Γ_{0} ⟶ O_{Y,y}/𝔪_{y}^{N+1} × O_{Z,z}/𝔪_{z}^{N+1} ⟶ R(w, F)_{i,α}/𝔪^{N+1}
Ĝ_{1}^{i+1} ↘  ↑  ↑ φ

O_{Y,y}/𝔪_{y}^{N+1}  ⟶^{π} R(y, F)_{i,α}/𝔪^{N+1}

where Ĝ_{1}^{i+1} and π are surjective and where φ is injective (since it corresponds to pull-back of functions from the image of projection of the subvariety). Our claim now follows from (1.1.7). This completes the proof.

In order to apply Theorem 1.1, we show in the following lemma that the surjectivity hypothesis of the proposition is satisfied for all sufficiently large degrees d.

Lemma 1.2. Retain the notation of Theorem 1.1. For each N > 0, there exists some d_{N} such that for all d ≥ d_{N} the map

Γ(O_{P^{n}}(d)) ⟶ O_{Y,y}/𝔪_{y}^{N+1} × O_{Z,z}/𝔪_{z}^{N+1}  (1.2.1)

sending f to the restrictions of f/φ is surjective for all closed points (y, z) ∈ Y × Z − Δ, all φ ∈ Γ(O_{P^{n}}(d)) with φ(y) ≠ 0 ≠ φ(z).

Proof. Since O_{P^{n},y} → O_{Y,y}, O_{P^{n},y} → O_{Z,z} are surjective local homomorphisms, it suffices to consider the special case Y = P^{n} = Z. Moreover, base-changing by a field extension Spec(k′) → Spec(k) merely tensors the map (1.2.1) with k′ over k. Consequently, it suffices to assume that both y, z are rational points of P^{n}. By the two-point homogeneity of P^{n} under P GL_{n+1}, it thus suffices to verify the existence of d_{N} for a single pair of (rational) points (y, z) ∈ P^{n} × P^{n}.

The map sending f to the restrictions of f/φ in O_{Y,y}/𝔪_{y}^{N+1} and O_{Z,z}/𝔪_{z}^{N+1} has image whose dimension is independent of the choice of φ with φ(y) ≠ 0 ≠ φ(z) since the quotient of any two such choices of φ is invertible in both O_{Y,y} and O_{Z,z}. Hence, it will suffice to consider the case where φ = X_{0}^{d} for some X_{0} ∈ Γ(O_{P^{n}}(1)).

Let R = k[X_{1}/X_{0}, …, X_{n}/X_{0}] denote the coordinate algebra of the affine variety A^{n} = P^{n} − {X_{0} = 0}, where we assume that X_{0}(y) ≠ 0 ≠ X_{0}(z). Since 𝔪_{y}^{N+1}, 𝔪_{z}^{N+1} are coprime in R, the Chinese Remainder Theorem implies that

R ⟶ R/𝔪_{y}^{N+1} × R/𝔪_{z}^{N+1}

is surjective. Let d_{N} be the minimal degree d for which polynomials of degree ≤ d_{N} in R map via this surjective map onto R/𝔪_{y}^{N+1} × R/𝔪_{z}^{N+1}. Then for any d ≥ d_{N}, (1.2.1) is surjective.

Let X be a projective variety provided with a closed embedding X ⊂ P^{n} over k and consider non-negative integers r, d. For any non-negative integer e, set

C_{r,≤e}(X) = ∐_{d≤e} C_{r,d}(X).

In the next proposition we consider the Veronese embeddings of X of degree d. We consider all cycles of fixed degree ≤ e (in the original embedding) and examine those projections which are “bad” for pairs of such cycles. The main assertion is that the codimension of these bad projections goes to infinity with d.

Proposition 1.3. Let X ⊂ P^{n} be a closed subvariety of dimension m, let r, s be nonnegative integers with r + s ≥ m, and let e be a positive integer. There exist (Zariski) closed subsets B(d)_{e} of the quasi-projective variety U_{X}(d),

B(d)_{e} ⊂ U_{X}(d) ⊂ P(Γ(O_{P^{n}}(d)))^{m+1},

for d > 0 with

lim_{d→∞} codimB(d)_{e} = ∞

which satisfy the following property: for any effective r-cycle Y on X of degree ≤ e and any effective s-cycle Z on X also of degree ≤ e, |Y | ⋆_{F} |Z| has pure dimension r + s − m whenever F is a k-rational point of U_{X}(d) − B(d)_{e}.

Proof. For notational brevity, let C_{r,e} denote C_{r,≤e}(X). We define

W_{e} ⊂ X^{2} × C_{r,e} × C_{s,e} × U_{X}(d)

to be the closed subariety of those quintuples (y, z, Y, Z, F) for which y × z lies in the closure of some component of Y ⋆_{F} Z. The principle of upper semi-continuity applied to W_{e} over C_{r,e} × C_{s,e} × U_{X}(d) implies that

V_{e} ⊂ C_{r,e} × C_{s,e} × U_{X}(d)

is a closed subvariety, where V_{e} consists of those triples (Y, Z, F) such that Y ⋆_{F} Z has some irreducible component of dimension > m − r − s. Thus,

B(d)_{e} = pr_{3}(V_{e}) ⊂ U_{X}(d)

is a closed subvariety of U_{X}(d).

Observe that the fibre of V_{e} above (Y, Z) ∈ C_{r} × C_{s} is B_{{Y,Z}} (in the notation of Proposition 1.1). Let E_{r} = dim{C_{r,e}}, E_{s} = dim{C_{s,e}}. By Theorem 1.1 and Lemma 1.2, for any N we may find d_{N} so that B(d)_{{Y,Z}} ⊂ U_{X}(d) has codimension at least N − E_{r} − E_{s} − r − s for d ≥ d_{N}. Hence, B_{e}(d) has codimension ≥ N − r − s for d ≥ d_{N}.

Finally, if Y, Z are effective r, s cycles of degrees ≤ e and if Y ⋆_{F}Z has some component of dimension > r + s − m, then (Y, Z, F) determines a point of V_{e} so that F ∈ B(d)_{e}.

Chow’s classical technique of moving Z to intersect Y properly entails consideration of a Veronese embedding P^{n} → P^{M} and a linear plane L ⊂ P^{M} of codimension m + 1 missing X. We let π_{L} : P^{M} − −− > P^{m} denote the linear projection with center L and p_{L} : X → P^{m} denote the finite morphism given as the restriction of π_{L} to X. Departing from the classical construction (cf [R]), we define the projecting cone of a cycle Z on X to be the cycle

C_{L}(Z) =^{def} π_{L}^{∗}(p_{L∗}(Z)) ⊂ P^{M},

where p_{L∗} denotes proper push-forward and π_{L}^{∗} denotes the algebraic join operation (−)#L sending a cycle W on P^{m} to W #L on P^{m}#L = P^{M} (cf. [L,2.10]). This differs from the classical definition which is given by the discontinuous construction of sending an irreducible cycle Z to π_{L}^{∗}(p_{L}(Z)), where p_{L}(Z) is the irreducible, reduced image of Z.

Since p_{L} is a finite map, the intersection product C_{L}(Z) • X is defined for all cycles Z on X. We recall that p_{L} : X → P^{m} is flat at any smooth point x ∈ X (cf. [Mat,20.D]). Thus, if no component of Z is contained in the singular locus of X, then C_{L}(Z) • X can alternatively be described as the closure in X of p_{L}^{ns∗}(p_{L∗}(Z)), where p_{L}^{ns} : X_{ns} → P^{m} is the restriction to p_{L} to the complement of the singular locus of X.

One considers the residual cycle

R_{L}(Z) =^{def} C_{L}(Z) • X − Z.

This is a linear construction on cycles, sending effective cycles to effective cycles (cf. Lemma 1.6).

Given Y and Z, one wants to choose L so that R_{L}(Z) has an improved intersection with Y. A key to this is the following lemma proved in [R] for the classical definition of residual cycle involving the classical projecting cone π_{L}^{∗}(p_{L}(Z)) • X. Denote by ram(p_{L}) the ramification locus of the finite map p_{L} : X ⟶ P^{m}. (In particular ram(p_{L}) contains all the singular points of X.)

Proposition 1.4 (cf. [R; Lemma 6]) Let Z, X and L be as above and consider a geometric point \underline{y} : Spec(\overline{k}) → |R_{L}(Z)| above a closed point y ∈ |R_{L}(Z)|. Then either:

(i) There exists a geometric point \underline{z} : Spec(\overline{k}) → |Z| over a closed point z ∈ |Z|

with \underline{z} ≠ \underline{y} and p_{L}(\underline{z}) = p_{L}(\underline{y}), or:
(ii) y ∈ ram(p_{L}).

Proof. Since the construction of the residual cycle is additive, we may assume that Z is irreducible. By definition of R_{L}(Z), if y ∉ Z, then there exists a point z ∈ Z with z ≠ y and p_{L}(z) = p_{L}(y). Thus, we may assume y ∈ Z.

If [Z : p_{L}(Z)] > 1 (i.e., if the function field of Z is a non-trivial extension of that of p_{L}(Z)), then either (i) is satisfied for some \underline{z} : Spec(\overline{k}) → Z or in a formal neighborhood of \underline{y} at least two sheets of Z (above a formal neighborhood of p_{L}(y) ∈ P^{m}) intersect at y. This latter situation implies that in a formal neighborhood of y ∈ X at least two sheets of X intersect at y so that y ∈ ram(p_{L}).

If [Z : p_{L}(Z)] = 1 and if Z is a component of R_{L}(Z), then as shown in [R; Lemma 6] p_{L} must ramify along all of Z. Namely, a “general” point of Z is necessarily smooth on Z and would also be smooth on X if p_{L} does not ramify along Z. At such a point, p_{L} is etale and so the multiplicity of Z in p_{L}^{ns∗}(p_{L∗}(Z)) is 1; this contradicts the assumption that Z is a component of R_{L}(Z).

Finally, assume that [Z : p_{L}(Z)] = 1 and that Z is not a component of R_{L}(Z). Since y ∈ |R_{L}(Z)|, y lies in some component R of R_{L}(Z) not equal to Z. Then, in a formal neighborhood of y ∈ |R_{L}(Z)| at least two sheets (one determined by Z and another determined by R) intersect at y so that y ∈ ram(p_{L}).

As in (1.0.1) and (1.0.2), for each (m + 1)-tuple F = (f_{0}, …, f_{m}) of homogeneous forms of degree d on P^{n}, we consider the associated linear projection π_{F} : P^{M} −−− > P^{m} with center L(F). If X ⊂ P^{n} is a subvariety of dimension m and if F ∈ U_{X}(d), we denote by p_{F} : X → P^{m} the finite map given by restricting π_{F} to X ⊂ P^{M} (embedded via the Veronese embedding). For any cycle Z on X we set

C_{F}(Z) = π_{F}^{∗}(p_{F∗}((Z))) = C_{L(F)}(Z)  and  R_{F}(Z) = C_{F}(Z) • X − Z.

Corollary 1.5. Consider a projective variety X of dimension m provided with a closed embedding X ⊂ P^{n}. Let Y be an r-cycle and Z an s-cycle on X with r + s − m ≥ 0. Assume F ∈ U_{X}(d) satisfies the condition that |Y | ⋆_{F} |Z| has no component of excess dimension (i.e., > r + s − m). Then any component of |Y | ∩ |R_{F}(Z)| of excess dimension must be contained in |Y | ∩ |Z| ∩ ram(p_{F}).

Proof. By definition of |Y | ⋆_{F} |Z|, any y ∈ (|Y | ∩ |R_{F}(Z)|) − |Z| necessarily lies in pr_{|Y |}(|Y |⋆_{F}|Z|). By Proposition 1.5, any y ∈ (|Y |∩|R_{F}(Z)|)−ram(p_{F}) admits a geometric point (\underline{y}, \underline{z}) : Spec(\overline{k}) → |Y | ⋆_{F} |Z| which projects to \underline{y} over y, so that y also lies in pr_{|Y |}(|Y | ⋆_{F} |Z|). Thus, the complement of |Y | ∩ |Z| ∩ ram(p_{F}) in |Y | ∩ |R_{F}(Z)| lies in pr_{|Y |}(|Y | ⋆_{F} |Z|) whose components have dimensions less than or equal to that of the maximal dimension of the components of |Y | ⋆_{F} |Z|.

We shall use the following property of the construction of residual cycles. We recall that a continuous algebraic map f : X → Y with Y projective is a set-theoretic map on geometric points which is induced by a correspondence Γ_{F} ⊂ X × Y (cf. [F]).

Lemma 1.6. Let F be a k-rational point of U_{X}(d) ⊂ P(Γ(O_{P^{n}}(d)))^{m+1}. Then R_{F}(Z) is an effective s-cycle on X whenever Z is itself an effective s-cycle on X. Moreover, sending Z to R_{F}(Z) determines a continuous algebraic map

R_{F}(−) : C_{s,≤e}(X) → C_{s,≤e′}(X),

where e′ is a positive integer depending upon d, e and the degree of X.

Proof. The fact that R_{F}(−) sends effective cycles on X to effective cycles on X follows immediately from the observation that each of the operations in the definition of R_{F}(−) is a well defined operation on effective cycles.

We employ various functoriality properties proved in [F]. The Veronese embedding ν : P^{n} → P^{M} of degree d determines a morphism

ν_{∗} : C_{s,≤e}(X) → C_{s,≤d^{s}e}(ν(X)).

The linear projection p_{F} : ν(X) → P^{m} determines a morphism

p_{F∗} : C_{s,≤d^{s}e}(ν(X)) → C_{s,≤d^{s}e}(P^{m}).

The join construction determines a continuous algebraic map

π_{F}^{∗} : C_{s,≤d^{s}e}(P^{m}) → C_{s+M−m,≤d^{s}e}(P^{M}).

Furthermore, intersection product in P^{M} determines a continuous algebraic map

(−) • X : C_{s+M−m,≤d^{s}e}(P^{M}) → C_{s,≤d^{s}ef}(P^{M})

where f equals the degree of ν(X). Finally, the additive structure of the Chow monoid is also given by continuous algebraic maps. Thus, sending Z to R_{F}(Z) ≡ π_{F}^{∗}(p_{P^{m}∗}(Z))•X−Z is a continuous algebraic map.

In the following theorem, we verify that an iteration of the residual cycle construction enables one to arrange (by choosing the degrees of projections sufficiently large) that the iterated residual cycle for Z meets Y properly for all effectives cycles Y, Z of degree ≤ e.

Theorem 1.7 Let X ⊂ P^{n} be a closed subvariety of dimension m, and let U ⊂ X be the Zariski open subset consisting of the smooth points of X. For any (m+1)-tuple of positive integers d = (d_{0}, …, d_{m}), there is a (Zariski) open dense subset

ℜ_{X}(d) ⊂ ∏_{i=0}^{m} U_{X}(d_{i}) ⊂ ∏_{i=0}^{m} P(Γ(O_{P^{n}}(d_{i}))^{m+1})

consisting of (m + 1)-tuples F^{∗} ≡ (F^{0}, …, F^{m}) with the property that

U ∩ ram(p_{F^{0}}) ∩ … ∩ ram(p_{F^{m}}) = ∅.

For any non-negative integers r, s with r + s ≥ m and any positive integer e, there exists a (Zariski) closed subset

𝔅(d)_{e} ⊂ ℜ_{X}(d)

such that

(a) The codimension of B(d)_{e} can be made arbitrarily large for all appropriately chosen d which are sufficiently large (cf. (1.8) below).

(b) For all effective r-cycles Y and s-cycles Z of degree ≤ e on X and for all k-rational points F^{∗} ∈ ℜ_{X}(d) − 𝔅(d)_{e}, each component of

|Y | ∩ |R_{F^{∗}}(Z)| , R_{F^{∗}}(Z) ≡ R_{F^{m}} ∘ ⋯ ∘ R_{F^{0}}(Z)

of “excess” dimension (i.e., > r + s − m) is contained in the singular locus X − U.

Proof. We consider the (closed) incidence correspondence J ⊂ U_{X}(d_{0})×X …×U_{X}(d_{m})× X of those (F^{0}, x_{0}, …, F^{m}, x_{m}) with x_{j} ∈ ram(p_{F^{j}}) for all j. Then the complement of R_{X}(d) in U_{X}(d) ≡ ∏_{j=0}^{m} U_{X}(d_{j}) is given by

U_{X}(d) − R_{X}(d) = pr{J) ∩ (U_{X}(d) × Δ(X)},

where pr is the proper projection pr : U_{X}(d_{0}) × X … × U_{X}(d_{m}) × X → U_{X}(d) and Δ : X → X^{m+1} is the diagonal embedding. Hence, R_{X}(d) is open.

For any smooth point of x ∈ X, those F ∈ U_{X}(d) such that p_{F} ramifies at x is a proper closed subset (defined by the condtion that L_{F} meet the tangent planes of X at x). Thus, the subset of those F for which p_{F} ramifies everywhere is a proper closed subset of U_{X}(d) (empty, if k has characteristic 0). We readily see that R_{X}(d) is nonempty by observing that F^{∗} ∈ R_{X}(d) provided that p_{F^{0}} does not ramify everywhere and such that p_{F^{j}} does not ramify everywhere along any non-empty irreducible component of ram(p_{F^{j−1}}) ∩ ⋯ ∩ ram(p_{F^{0}}) ∩ U, for each j with 0 < j ≤ m.

The asserted 𝔅(d)_{e} is the intersection with ℜ_{X}(d) of the Zariski closed subset

𝔅′(d)_{e} ⊂ U_{X}(d)

constructed as the union over j, 0 ≤ j ≤ m, of closed subsets

U_{X}(d_{0}) × … U_{X}(d_{j−1}) × B(d_{0}, …, d_{j})_{e} × … U_{X}(d_{m})

for suitably defined closed subsets B(d_{0}, …, d_{j})_{e} ⊂ U_{X}(d_{j}). Namely, B(d_{0})_{e} ⊂ U_{X}(d_{0}) is the closed subset given by Proposition 1.3 for d_{0}, e. By Lemma 1.6, the residual cycles R_{F}(Z) constructed for F_{0} ∈ U_{X}(d_{0}) and effective cycles Z on X of degree bounded by e are of degrees bounded by some e_{1} (depending upon both e and d_{0}). We define B(d_{0}, d_{1})_{e} ⊂ U_{X}(d_{1}) to be the closed subset given by Proposition 1.3 for d_{1}, e_{1}. Continuing inductively, we conclude by Lemma 1.6 that the residual cycles R_{F_{j−1}} ∘ R_{F_{0}}(Z) constructed for F_{0} ∈ U_{X}(d_{0}), …, F_{j−1} ∈ U_{X}(d_{j−1}) and effective cycles Z on X of degree bounded by e are of degrees bounded by some e_{j} (depending upon e and d_{0}, …, d_{j−1}). We define B(d_{0}, …, d_{j})_{e} ⊂ U_{X}(d_{j}) to be the closed subset given by Proposition 1.3 for d_{j}, e_{j}.

By Corollary 1.5, any component of excess dimension of |Y | ∩ |R_{F^{∗}}(Z)| must be contained in |Y | ∩ |Z| ∩ ram(p_{F^{m}}) ∩ … ∩ ram(p_{F^{0}}) provided that F^{∗} ∉ B′(d)_{e}. Thus, for F^{∗} ∈ ℜ_{X}(d), F^{∗} ∉ B′(d)_{e}, any component of excess dimension of |Y | ∩ |R_{F^{∗}}(Z)| must be contained in ram(p_{F^{m}}) ∩ … ∩ ram(p_{F^{0}}) ⊂ X − U.

By Proposition 1.3, for any integer c we may choose each d_{j} (depending as above on e and the d_{i}’s for i < j) so that B(d_{0}, …, d_{j})_{e} ⊂ U_{X}(d_{j}) has codimension at least c. This insures that

𝔅(d)_{e} ≡ 𝔅′(d)_{e} ∩ ℜ(d) ⊂ ℜ(d)

also has codimension at least c.

Note 1.8. Note from the paragraph above that each d_{j} may be chosen arbitrarily, so long as it is sufficiently large. This lower bound depends only on c, e and the choice of the previous integers d_{1}, …, d_{j−1}.

In §3 we will use Theorem 1.7 in conjunction with the following proposition which expresses a cycle Z in terms of projecting cones and the cycle R_{F^{∗}}(Z). This proposition follows immediately from Corollary 1.5 and Lemma 1.6.

Proposition 1.9. Let X ⊂ P^{n} be a closed subvariety of dimension m, let r, s be nonnegative integers with r + s ≥ m, and let e be a positive integer. Assume that d is chosen so that the codimension of B(d)_{e} ⊂ R_{X}(d) is at least 1 and that the field k is sufficiently large that there exists a k-rational point F^{∗} ∈ ℜ_{X}(d) − 𝔅(d)_{e}. Choose some such F^{∗}. Consider the equality

Z = (−1)^{m+1}R_{F^{∗}}(Z) + ∑_{i=1}^{m}(−1)^{i}π_{F^{i}}^{∗}{p_{F^{i}∗}{R_{F^{i−1}} ∘ ⋯ ∘ R_{F^{0}}(Z)}} • X  (1.9.1)

and let ψ_{F^{∗}}^{+}(Z) , ψ_{F^{∗}}^{−}(Z) denote the positive and negative parts of the right hand side of (1.9.1). Then:

(a) For each effective s-cycle Z on X, (ψ_{F^{∗}}^{+}(Z) , ψ_{F^{∗}}^{−}(Z)) is a pair of effective s-cycles on X with the property that

Z = ψ_{F^{∗}}^{+}(Z) − ψ_{F^{∗}}^{−}(Z).

(b) For all effective cycles Y, Z on X of dimension r, s which are of degrees ≤ e, any component of excess dimension of either |Y | ∩ |ψ_{F^{∗}}^{+}(Z)| or |Y | ∩ ψ_{F^{∗}}^{−}(Z)| lies in |Y | ∩ |Z|.

(c) This construction determines a continuous algebraic map

ψ_{F^{∗}} = (ψ_{F^{∗}}^{+}, ψ_{F^{∗}}^{−}) : C_{s}(X) → C_{s}(X)^{2}.

Moreover, there is an integer e′ such that

ψ_{F^{∗}}(C_{s,≤e}(X)) ⊂ C_{s,≤e′}(X)^{2}.

2. Moves in Projective Space

In this section, we consider effective r-cycles Y and effective s-cycles Z on P^{m} of degree bounded by a fixed positive integer e, where r + s ≥ m. We present a construction on all s-cycles on P^{m} which moves each such Z so that the resulting cycle intersects properly each such Y. The method we employ generalizes a technique introduced in [L] to move s-cycles Z of bounded degree to intersect properly a given hyperplane L ≡ P^{m−1} ⊂ P^{m}. One can interpret this method as taking the “algebraic suspension” z_{1}#Z ⊂ P^{m+1} = z_{0}#P^{m} and observing that z_{1}#Z intersects z_{0}#L properly provided that z_{1} ≠ z_{0}. One then observes that for most divisors D ⊂ P^{m+1} of sufficiently high degree the projection of (z_{1}#Z) • D off z_{0} meets L properly.

Throughout this section, we consider non-negative integers r, s ≤ m with r + s ≥ m. We set t ≡ m − r. We shall fix a linear embedding

P^{m} ⊂ P^{m+t} , (m + t = 2m − r)

and some fixed linear subspace L_{0} ⊂ P^{m+t} of dimension t − 1 missing P^{m}. Thus,

P^{m+t} = L_{0}#P^{m},

where −#− denotes the “algebraic join”.

We next introduce our “parameter space of moves” for s-cycles in P^{m}.

Definition 2.1. For any t-tuple of positive integers N = (N_{1}, …, N_{t}), let

L(N) ⊂ Grass_{t−1}(P^{m+t}) × ∏_{i=1}^{t} P(Γ(O_{P^{m+t}}(N_{i})))

denote the open set consisting of those (L, D), where D = (D_{1}, …, D_{t}) which satisfy

(a) L ∩ P^{m} = ∅ = L ∩ L_{0},
(b) |D_{1}| ∩ … ∩ |D_{t}| ∩ L = ∅ = |D_{1}| ∩ … ∩ |D_{t}| ∩ L_{0}.

For any L ∈ Grass_{t−1}(P^{m+t}), we denote by L_{L}(N) the Zariski dense open subset of ∏_{i=1}^{t} P(Γ(O_{P^{m+t}}(N_{i}))) given as the fibre of the projection L(N) → Grass_{t−1}(P^{m+t}) above L.

Observe that for any k-rational point (L, D) ∈ L(N), the intersection product D_{1} • ⋯ • D_{t} is a well defined n-cycle on P^{m+t} with support |D_{1}| ∩ ⋯ ∩ |D_{t}| which meets L#Z properly and which misses L_{0}, the center of the projection π_{0}.

In the following proposition, we establish a constructible set of arbitrarily high codimension of “good moves” for effective s-cycles Z of degree ≤ e which satisfy a single intersection condition depending upon a chosen (t − 1)- dimensional linear subspace L. We recall that a subset of a quasi-projective variety is said to be constructible if it can be written as a finite disjoint union of (Zariski) locally-closed subsets. The key property of constructible subsets is that any map f : X → Y of quasi-projective varieties sends a constructible subset C ⊂ X to a constructible subset f(C) ⊂ Y.

Proposition 2.2. Let L be a (t − 1)-dimensional linear subspace of P^{m+t} satisfying L ∩ P^{m} = ∅ = L ∩ L_{0}. For a given positive integer e and a t-tuple of positive integers N = (N_{1}, …, N_{t}), there exists a constructible subset

B_{L}(N)_{e} ⊂ L_{L}(N)

such that

(a) For a constant K depending only upon X and e,

codimB_{L}(N)_{e} ≥ min(N_{i}) − K,

(b) For every k-rational point D = (D_{1}, …, D_{t}) ∉ B_{L}(N)_{e}, the cycles

|L_{0}#Y | ∩ |L#Z| ∩ |D_{1}| ∩ ⋯ ∩ |D_{i−1}| and |D_{i}|

meet properly for all i ≥ 0 whenever Y is an effective r-cycle and Z is an effective s-cycle both of degree ≤ e on P^{m} with the property that L_{0}#Y intersects L#Z properly.

Proof. Let

J ⊂ C_{r,≤e}(P^{m}) × C_{s,≤e}(P^{m}) × P^{m+t}

denote the incidence correspondence of triples (Y, Z, x) satisfying x ∈ |L_{0}#Y | ∩ |L#Z|. We define

G_{L,e} ⊂ C_{r,≤e}(P^{m}) × C_{s,≤e}(P^{m})

to be the open subset above which the fibre of the projection

J → C_{r,≤e}(P^{m}) × C_{s,≤e}(P^{m})

has proper dimension (i.e., r + s − m + t = s). Note that G_{L,e} consists of pairs of cycles (Y, Z) such that L_{0}#Y and L#Z intersect properly. We consider the closed subset

B̃_{L}(N)_{e} ⊂ G_{L,e} × L_{L}(N)

consisting of those tuples (Y, Z, D) satisfying the condition that L_{0}#Y and L#Z intersect properly but

|L_{0}#Y | ∩ |L#Z| ∩ |D_{1}| ∩ ⋯ ∩ |D_{i−1}| , |D_{i}|

intersect improperly for some i ≤ t. Finally, we set

B_{L}(N)_{e} = pr_{3∗}[(B̃_{L})(N)_{e}].

We proceed to verify that the fibre B̃_{Y,Z} ⊂ L_{L}(N) of B̃_{L}(N)_{e} above any (Y, Z ∈ G_{L,e} has codimension at least N if each N_{i} ≥ N. This will immediately imply the codimension of B_{L}(N)_{e} ⊂ L_{L}(N) is greater than some prescribed positive integer c provided that

N ≥ c + dim(C_{r,≤e}(P^{m})) + dim(C_{s,≤e}(P^{m})).

The condition that W_{j} ≡ |L_{0}#Y | ∩ |L#Z| ∩ |D_{1}| ∩ … ∩ |D_{j−1}| meet |D_{j}| improperly is equivalent to the condition that the restriction map

O_{P^{m+t}}(N_{j}) ⟶ O_{W_{j}}(N_{j}) ⟶ O_{W_{j,ω}}(N_{j})

sends D_{j} ∈ Γ(O_{P^{m+t}}(N_{j})) to 0 for some irreducible component W_{j,ω} ⊂ W_{j}. Since W_{j,ω} has positive dimension, this composition has rank at least N. To see this note that if we restrict to some affine open A^{m} ⊂ P^{m} containing a dense open subset of W_{j,ω}, then the dimension of the subspace of regular functions on A^{m} ∩ W_{j,ω} of degree ≤ N is at least N. We conclude that the closed subvariety B̃_{Y,Z,j} ⊂ L_{L}(N) consisting of those D with D_{j} meeting W_{j} improperly and D_{j′} meeting W_{j′} properly for j′ < j has codimension at least N. Thus,

B̃_{Y,Z} = ∐_{j} B̃_{Y,Z,j} ⊂ L_{L}(N)

also has codimension at least N.

Proposition 2.2 suggests the following construction of moving cycles in P^{m}, sending the s-cycle Z to φ_{L,D}(Z).

Proposition 2.3 Let π_{0} : P^{m+t} − −− > P^{m} denote the linear projection with center L_{0}. For any e > 0 and any t-tuple N = (N_{1}, …, N_{t}), the assignment

φ_{L,D}(Z) =^{def} π_{0∗}((L#Z) • D_{1} • … • D_{t})

determines a continuous algebraic map

Φ : C_{s}(P^{m}) × L(N) ⟶ C_{s}(P^{m})

with Φ(C_{s,≤e}(P^{m}) × L(N)) ⊂ C_{s,≤e|N|}(P^{m}), where |N| =^{def} ∏_{i} N_{i}. This construction has the following properties:

(a) For any k-rational point (L, D) ∈ L(N) and any effective s-cycle Z on P^{m}, φ_{L,D}(Z) is also an effective s-cycle on P^{m}.

(b) For any k-rational point (L, D) ∈ L(N) with the property that D_{1} • … • D_{t} =

|N| · P^{m},

φ_{L,D}(Z) = |N| · Z

for all effective s-cycles Z.

(c) If (L, D) is a k-rational point of L(N) such that D ∉ B_{L}(N)_{e} (cf. (2.2)), if Y, Z are effective cyles on P^{m} of dimension r, s and degree ≤ e, and if L_{0}#Y meets L#Z properly, then

Y , φ_{L,D}(Z)

intersect properly.

Proof. If Z on P^{m} is an s-effective cycle of degree d, then L#Z is an effective (s+t)-cycle P^{m+t} also of degree d. Consequently, (L#Z) • D_{1} • … • D_{t} and thus also φ_{L,D}(Z) have degree d·|N|. Moreover, sending ⟨Z, (L, D)⟩ to φ_{L,D}(Z) determines a continuous algebraic map, for Φ is a composition of operations each of which is given by a continuous algebraic map (cf. [F]).

As observed following Definition 2.1, the intersection product D_{1} • ⋯ • D_{t} is a well defined n-cycle on P^{m+t} with support |D_{1}| ∩ ⋯ ∩ |D_{t}| which meets L#Z properly and which misses L_{0} for any k-rational point (L, D) ∈ L(N). Thus, φ_{L,D}(Z) is a well defined s-cycle on P^{m}.

To prove that φ_{L,D}(Z) = |N| · Z whenever D_{1} • … • D_{t} = |N| · P^{m}, observe that P^{m} ⊂ P^{m+t} meets L#Z properly (since L∩P^{m} = ∅) and transversely at all regular points of Z.

If (L, D) ∈ L(N), then L_{0}#Y, L#Z have pure dimension m, s+t respectively. The hypothesis that L_{0}#Y meets L#Z properly thus implies that |L_{0}#Y |∩|L#Z| has dimension s. We thus may apply Proposition 2.2 to conclude that if D ∉ B(N)_{e}, if Y, Z are effective cycles of degrees ≤ e, and if L_{0}#Y meets L#Z properly, then (L_{0}#Y)•(L#Z)•D_{1}•…•D_{t} has pure dimension r + s − m. Because L_{0} ∩ |D_{1}| ∩ … ∩ |D_{t}| = ∅, π_{0∗} restricts to a (well defined) finite map on |(L#Z) • D_{1} • … • D_{t}| We apply the projection formula to π_{0∗} to conclude that

Y • π_{0∗}[(L#Z) • D_{1} • … • D_{t}] = π_{0∗}[(L_{0}#Y) • (L#Z) • D_{1} • … • D_{t}].

Hence, Y_{,} φ_{L,D}(Z) meet properly.

The following lemma will enable us to apply Proposition 2.3 to all effective cycles of bounded degree.

Lemma 2.4. Let L be a k-rational point of Grass_{t−1}(P^{m+t}) satisfying L ∩ P^{m} = ∅ = L ∩ L_{0}. Then

(L_{0}#Y), (L#Z) intersect properly

in P^{m+t} for every r-cycle Y, s-cycle Z on P^{m} such that either |L_{0}#Y | ∩ L = ∅ or Y intersects Z properly.

Proof. We first check that if Y, Z intersect properly, so do (L_{0}#Y), (L#Z) for any choice of L. Observe that

|L_{0}#Y | ∩ P^{m} = |Y |, |L#Z| ∩ P^{m} = |Z|.

Thus, if some component of |L_{0}#Y | ∩ |L#Z| has dimension > s, then some component of |L_{0}#Y | ∩ |L#Z| ∩ P^{m} = |Y | ∩ |Z| would have dimension > s − t.

On the other hand, if |L_{0}#Y | ∩ L = ∅, then L#z intersects L_{0}#Y in only finitely many points for any z ∈ P^{m}. Thus, |L_{0}#Y | ∩ |L#Z| has dimension at most s.

Combining Proposition 2.3 and Lemma 2.4, we now verify for any given e that we can “move” all effective s-cycles Z of degree ≤ e so that the resulting cycles intersect properly every effective r-cycle Y of degree ≤ e.

As in Proposition 1.9, we require the existence of k-rational points avoiding certain “bad” closed subsets of positive codimension. The existence of such k-rational points is automatic for an infinite field, but for a finite field all rational points of a given variety might lie on a subvariety of positive codimension. On the other hand, if k is a finite field and if there are only finitely many “bad” subvarieties of positive codimension, there will exist some finite extension k′ of k such that there exist k′-rational points avoiding these “bad sets”. Moreover, any extension k′′ of k′ will evidently have this same property.

Since our definitions of “good” and “bad” sets are independent of k, we conclude that all “sufficiently large” finite fields k of a given residue characteristic admit the existence of k-rational points missing the “bad” sets. We shall use the expression “k is a sufficiently large finite field” to mean that k admits the existence of k-rational points avoiding the bad sets (of positive codimension) occuring in our constructions.

Proposition 2.5. Fix positive integers e and c. Assume that either k is an infinite field or a sufficiently large finite field. Consider a sequence N_{1}, N_{2}, N_{3}, … of t-tuples of positive integers, and set e_{1} = e and e_{j+1} = |N_{j}|e_{j} for j > 1. Assume that the N_{j}’s are chosen so that min_{1≤i≤t} N_{j+1,i} > K(e_{j}) + c where K(e_{j}) is the constant in Proposition 2.2a. Then there exist an integer E ≤ dimC_{s,≤e}(P^{m}) and (t − 1)-dimensional linear subspaces L_{1}, …, L_{E} of P^{m+t}, each missing L_{0} and P^{m}, with the following properties:

(a) For each j,

codim{L_{L_{j}}(N_{j}) − B_{L_{j}}(N_{j})_{e_{j}}} ≥ c,

(b) For any k-rational point

(D_{1}, …, D_{E}) ∈ ∏_{j=1}^{E} L_{L_{j}}(N_{j}) − B_{L_{j}}(N_{j})_{e_{j}}

and any effective cycles Y, Z on P^{m} of dimensions r, s and degrees ≤ e,

Y , φ_{L_{E},D_{E}} ∘ ⋯ ∘ φ_{L_{1},D_{1}}(Z)

intersect properly.

Proof. Note to begin that assertion (a) follows immediately from Proposition 2.2a, so it remains to prove assertion (b).

For each (of the finitely many) irreducible components A_{γ} of A = C_{r,≤e}(P^{m}) of maximal dimension d_{e}, choose some k-rational point a(γ) ∈ A_{γ} (possible since k is assumed either infinite or sufficiently large). Using once again the fact that k is infinite or sufficiently large, we may choose some (t − 1)-dimensional linear plane L_{1} in P^{m+t} missing P^{m}, L_{0}, and each L_{0}#Y_{a(γ)}. Let A_{1} ⊂ A denote the closed subvariety of those Y ’s such that L_{1} ∩ |L_{0}#Y | ≠ ∅, and observe that each component of A_{1} has dimension strictly less than d_{e}. By Proposition 2.3 and Lemma 2.4, Y and φ_{L_{1},D_{1}}(Z) intersect properly for any effective r-cycle Y of degree ≤ e and any effective s-cycle Z of degree ≤ e which does not lie in A_{1} whenever D_{1} is k-rational point of L_{L_{1}}(N_{1}) − B_{L_{1}}(N_{1})_{e_{1}}.

Proceeding inductively, assume for some j ≥ 1 that we have defined the closed subvariety A_{j} ⊂ A of codimension at least j which satisfies the condition that

Y , φ_{L_{j},D_{j}} ∘ ⋯ ∘ φ_{L_{1},D_{1}}(Z)

intersect properly for any effective r-cycle Y of degree ≤ e and any effective s-cycle Z of degree ≤ e which does not lie in A_{j}. For each (of the finitely many) irreducible components A_{j,γ} of maximal dimension choose as above some point a(γ) ∈ A_{j,γ} and choose L_{j+1} missing P^{m}, L_{0}, and each L_{0}#Y_{a(γ)}. We set A_{j+1} ⊂ A_{j} equal to the closed subvariety of that Y ’s such that L_{j+1} ∩ |L_{0}#Y | = ∅. Applying Proposition 2.3 and Lemma 2.4 once again, we conclude that

Y , φ_{L_{j},D_{j}} ∘ ⋯ ∘ φ_{L_{1},D_{1}}(Z)

intersect properly for any effective r-cycle Y of degree ≤ e and any effective s-cycle Z of degree ≤ e which does not lie in A_{j}, whenever D_{j} is k-rational point of L_{L_{j}}(N_{j}) − B_{L_{j}}(N_{j})_{e_{j}}.

Finally, we see that this process stops after E steps, where E ≤ d_{e} + 1.

The following theorem is our “Moving Lemma for Cycles of Bounded Degree” in the special case in which the variety X is projective space P^{m} itself.

Theorem 2.6. Fix positive integers e and c. Assume that either k is an infinite field or a sufficiently large finite field. Choose N_{1}, …, N_{E} and L_{1}, …, L_{E} as in Proposition 2.5 and let

\underline{0} =^{def} (D_{1}, …, D_{E}) ∈ ∏_{j=1}^{E} L_{L_{j}}(N_{j})

be a k-rational point such that D_{j,1} • ⋯ • D_{j,t} = |N_{j}| · P^{m} for each j, 1 ≤ j ≤ E. (For example, suppose that D_{j,i} = N_{j,i}H_{j,i} where H_{j,1}, …, H_{j,t} are k-rational hyperplanes in general position.) Let

ℓ ⊂ ∏_{j=1}^{E} P(Γ(O_{P^{m+t}}(N_{j}))) ≡ ∏_{j=1}^{E} ∏_{i=1}^{t} P(Γ(O_{P^{m+t}}(N_{j,i})))

be any k-rational line through \underline{0} chosen so that

ℓ^{∘} ≡ ℓ ∩ ∏_{j=1}^{E} L_{L_{j}}(N_{j}) − B_{L_{j}}(N_{j})_{e_{j}} ⊂ ℓ

is dense. Then there exists a continuous algebraic map

Θ : C_{s}(P^{m}) × ℓ^{∘} → C_{s}(P^{m})

with the following properties. Set θ_{p} = Θ|_{C_{s}(P^{m})×{p}}.

(a) θ\underline{0}(Z) = ∏_{j=1}^{E} |N_{j}| · Z for any effective s-cycle Z.

(b) For any k-rational point p ∈ ℓ^{∘}, θ_{p} determines a linear map on effective s-cycles on P^{m}.

(c) For any effective s-cycle Z, the restriction of Θ to {Z}×ℓ^{∘} determines a rational equivalence Z_{θ} ⊂ P^{m} × ℓ whose fibre above a point p ∈ ℓ^{∘} is the cycle with Chow point θ_{p}(Z).

(d) For all effective cycles Y, Z on P^{m} of dimensions r, s and degrees ≤ e and all k-rational points p ∈ ℓ^{∘} − \underline{0},

Y , θ_{p}(Z)

intersect properly.

Proof. Iterating the continuous algebraic map Φ of Proposition 2.3, we obtain the continuous algebraic map

Θ̃ : C_{s}(P^{m}) × L(N_{1}) × ⋯ × L(N_{1}) → C_{s}(P^{m}).

We define Θ to be the restriction of Θ̃ to C_{s}(P^{m}) × ℓ^{∘}.

Property (a) is given by Proposition 2.3.b, whereas property (b) follows from Proposition 2.3.a. To verify property (c), we let λ denote the generic point of ℓ and k(λ) denote the function field of ℓ. Then Θ(Z, λ) determines a cycle on P^{m} rational over some radiciel extension of k(λ) (cf. [F]). On the other hand, since the operations used to define Θ send cycles to cycles, we see that the cycle Z_{λ} with Chow point Θ(Z, λ) is in fact a cycle on P_{k(λ)}^{m}. The closure of this cycle, Z_{θ} on P^{m} × ℓ, is the asserted rational equivalence. (Observe that Z_{θ} is necessarily flat over ℓ so that the fibre p ∈ ℓ is given by specialization of Z_{λ}.) Finally, property (d) follows from Proposition 2.5.

3. Main Theorem and Consequences

Using Theorem 1.7, Proposition 1.9, and Theorem 2.6, we now prove our “Moving Lemma for Cycles of Bounded Degree”.

Theorem 3.1. Let X ⊂ P^{n} be a projective variety of dimension m. Assume that either k is an infinite field or a sufficiently large finite field. Let r, s, e be non-negative integers with r + s ≥ m. Then there exists a Zariski open neighborhood O ⊂ P^{1} of a k-rational point 0 ∈ P^{1} and a continuous algebraic map

Ψ = (Ψ^{+}, Ψ^{−}) : C_{s}(X) × O → C_{s}(X)^{2}

satisfying the following properties. Set ψ_{p}^{±} = Ψ^{±}|_{C_{S}(X)×{p}}. Then:

(a) For some positive integer M and some k-rational point F^{∗} ∈ ℜ_{X}(d) − 𝔅(d)_{e}, chosen as in 1.9, one has

ψ_{0}^{+} = (M + 1) · ψ_{F^{∗}}^{+} + M · ψ_{F^{∗}}^{−} , ψ_{0}^{−} = (M + 1) · ψ_{F^{∗}}^{−} + M · ψ_{F^{∗}}^{+}

where ψ_{F^{∗}}^{+}, ψ_{F^{∗}}^{−} : C_{s}(X) → C_{s}(X) are the continuous algebraic maps of Proposition 1.9.a. In particular, for every effective s-cycle Z,

Z = ψ_{0}^{+}(Z) − ψ_{0}^{−}(Z).

(b) For any k-rational point p ∈ O, ψ_{p}^{±} determines a pair of linear maps on effective s- cycles

(c) For any effective s-cycle Z on X, the restriction of Ψ to {Z} × O determines a rational equivalence Z_{ψ} ⊂ P^{n} × P^{1} whose fibre above a point p ∈ O is ψ_{p}^{+}(Z) − ψ_{p}^{−}(Z).

(d) For all effective cycles Y, Z on X of dimensions r, s and degrees ≤ e and all krational points p ∈ O−{0}, any component of excess dimension (i.e., > r+s−m) of either |Y | ∩ |ψ_{p}^{+}(Z)| or |Y | ∩ |ψ_{p}^{−}(Z)| lies in the singular locus of X.

Proof. Choose F^{∗} as in Proposition 1.9 and recall the equality

Z = (−1)^{m+1}R_{F^{∗}}(Z) + ∑_{i=1}^{m}(−1)^{i}π_{F^{i}}^{∗}{p_{F^{i}∗}{R_{F^{i−1}} ∘ ⋯ ∘ R_{F^{0}}(Z)}} • X  (1.9.1)

and the fact that ψ_{F^{∗}}^{±} are given as the positive and negative parts of the right hand side of (1.9.1). Let e′ ≥ e be a positive integer bounding the degrees of p_{F_{∗}^{i}}(Y) and of p_{F_{∗}^{i}}(R_{F^{i−1}} ∘ ⋯ ∘ R_{F^{0}}(Z)) as Y and Z vary over effective r-cycles and s-cycles on X (respectively) of degrees ≤ e, and i varies from 1 to m.

We now apply Theorem 2.6 with e replaced by e′ and with some fixed c > 1. In fact we apply the theorem twice. We choose two sequences N = (N_{1}, …, N_{E}) and N′ = (N′_{1}, …, N′_{E′}) with N_{j,i} and N′_{j,i} sufficiently large as required in Proposition 2.5, so that

∏_{j=1}^{E} |N_{j}| = M + 1 , ∏_{j=1}^{E′} |N′_{j}| = M

for some positive integer M. To see that this is possible recall that the integers N_{j,i} can be chosen arbitrarily subject to the inequalities of 2.5. Consequently we may choose the N_{j,i}’s and the N′_{j,i} so that ∏_{j} |N_{j}| and ∏_{j} |N′_{j}| are relatively prime. Then there exist positive integers a and a′ so that a ∏_{j} |N_{j}| − a′ ∏_{j} |N′_{j}| = ±1. We may assume, by reordering the choice if necessary, that the difference is +1, and we then replace N_{E,t} with aN_{E,t} and N′_{E′,t} with a′N′_{E′,t}.

By Proposition 2.5 we obtain sequences (L_{1}, L_{2}, …, L_{E}) and (L′_{1}, L′_{2}, …, L′_{E′}) of (t − 1)-dimensional linear subspaces, for which the conclusions of 2.5 hold.

We now apply Theorem 2.6 (twice) for the {N_{j}}, {L_{j}} (and the {N′_{j}}, {L′_{j}}) chosen above. We fix k-rational points \underline{0} and \underline{0′} and k-rational lines ℓ and ℓ′ as in Theorem 2.6, and we let

Θ\underline{N} : C_{s}(P^{m}) × ℓ^{∘} → C_{s}(P^{m})  and  Θ′\underline{N′} : C_{s}(P^{m}) × (ℓ′)^{∘} → C_{s}(P^{m})

be the continuous algebraic maps constructed in 2.6 for this data.

We now choose a dense open subset O ⊂ P^{1} and point 0 ∈ O together with open immersions f : O ↪ ℓ^{∘} and f′ : O ↪ (ℓ′)^{∘} sending 0 to \underline{0} and \underline{0′} respectively. We let Θ_{\underline{N},p} denote the restriction of Θ\underline{N} to C_{s}(P^{m}) × \{f(p)\} for any k-rational point p ∈ O. We define Ψ\underline{N} by

Ψ\underline{N}(Z, p) = (−1)^{m+1}(M +1)·R_{F^{∗}}(Z)+∑_{i=1}^{m}(−1)^{i}π_{F^{i}}^{∗}\{Θ_{\underline{N},p}\{p_{F^{i}∗}(R_{F^{i−1}} ∘⋯∘R_{F^{0}}(Z))\}\}•X,

and define Ψ\underline{N}′ similarly. Finally, we define Ψ^{±} to be the positive and negative parts of Ψ\underline{N} − Ψ\underline{N}′, so that

Ψ^{+}(Z, p) − Ψ^{−}(Z, p) = Ψ\underline{N}(Z, p) − Ψ\underline{N}′(Z, p).

The fact that Ψ^{+}, Ψ^{−} are continuous algebraic maps follows immediately from Proposition 1.9 and Theorem 2.6. Property (a) is part of our definition of Ψ^{±}. Property (b) follows immediately from Proposition 1.9 and Theorem 2.6.b. The proof of property (c) is merely a repetition of the proof of Theorem 2.6.c.

To verify property (d), observe that Theorem 1.7.b implies that any component of excess dimension of |Y |∩|R_{F^{∗}}(Z)| lies in the singular locus of X. On the other hand, since p_{F^{i}} is a finite map,

Y , π_{F^{i}}^{∗}{Θ_{N,p}{p_{F^{i}∗}(R_{F^{i−1}} ∘ ⋯ ∘ R_{F^{0}}(Z))}} • X

intersect properly for any i, 1 ≤ i ≤ m, if and only if

p_{F^{i}∗}(Y) , Θ_{N,p}{p_{F^{i}∗}(R_{F^{i−1}} ∘ ⋯ ∘ R_{F^{0}}(Z))}

intersect properly. These do intersect properly whenever Y, Z are effective cycles of degrees ≤ e and p ∈ O − {0} by Theorem 2.6.d. The analogous remarks apply with N replaced by N′ . This completes the proof.

Remark 3.2. The proof of Theorem 3.1 in fact proves the more general result that one can “move” effective s-cycles Z of degrees ≤ e to meet properly (off the singular locus of X) all effective cycles Y of dimension ≥ m − s and degree ≤ e. Namely, we verify that the arguments of section 1 apply when C_{r,≤e}(X) is replaced by

C_{≥n−s,≤e}(X) ≡ ∐_{r≥m−s,d≤e} C_{r,d}(X).

Similarly, the arguments of section 2 apply when we replace C_{r,≤e}(P^{n}) by C_{≥s,≤e}(P^{n}) and t = n − r by s.

When k = C our Main Theorem can be phrased differently and in a way that may appeal to complex geometers. In this case the p-cycles form a topological abelian group where the topology is compactly generated by the images of C_{p,≤e}(X) × C_{p,≤e}(X) under the natural projection

π : C_{p}(X) × C_{p}(X) ⟶ Z_{p}(X)

We set Z_{p,≤e}(X) =^{def} π{C_{p,≤e}(X) × C_{p,≤e}(X)}

Corollary 3.3. Let X ⊂ P_{C}^{n} be a complex projective variety of dimension m. Let r, s, e be non-negative integers with r + s ≥ m. Then there exists a Zariski open neighborhood O of {0} in C, and a continuous algebraic map

Ψ̃ : C_{s}(X) × O ⟶ C_{s}(X)^{2}

such that π ∘ Ψ induces by linearity a continuous map

Ψ : Z_{s}(X) × O ⟶ Z_{s}(X)

satisfying the following properties. Set ψ_{p} = Ψ|_{Z_{s}(X)×{p}} for p ∈ O.

(a) ψ_{0} = Id.

(b) For any Z ∈ Z_{s}(X) and any p ≠ 0 in O, the restriction

Ψ|_{{Z}×O} : {Z} × O ⟶ Z_{s}(X),

determines a rational equivalence between Z and ψ_{p}(Z).

(c) For any p ∈ O, ψ_{p} is a continuous group homomorphism.

(d) For any Z ∈ Z_{s,≤e}(X), Y ∈ Z_{r,≤e}(X) and any p ≠ 0 in O, each component of excess dimension (i.e., > r + s − m) of the intersection

|Y | ∩ |ψ_{p}(Z)|

is contained in the singular locus of X.

Remark. We may paraphrase Corollary 3.3 in terms of families of cycles. Let us say that a collection {Y_{α}; α ∈ A} of r-cycles on X is a family of cycles of bounded degree if there exists some positive integer e and some function g : A → Z_{r,≤e}(X) such that Y_{α} = g(α) for all α ∈ A. Then Corollary 3.3 asserts that for a given family {Y_{α}; α ∈ A} of r-cycles of bounded degree and a given family {Z_{β}; β ∈ B} of s-cycles of bounded degree, Ψ provides a continuous move of all s-cycles on X with the following property: for any α ∈ A, β ∈ B, p ∈ O − {0} each component of |Y_{α}| ∩ |ψ_{p}(Z_{β})| of excess intersection lies in the singular locus of X.

Moreover, Corollary 3.3 further asserts how such a move is algebraic, thereby moving one “algebraic family” to another.

As demonstrated in [R], [S], any s-cycle Z can be moved to intersect properly any rcycle Y on a smooth, m-dimensional variety X with r +s ≥ m. This does not immediately imply that the intersection product is well defined, for one must verify that if Z′, Z′′ are both obtained from Z by moving Z and both intersect Y properly, then Y • Z′, Y • Z′′ are rationally equivalent. As we see in the proof of the following theorem, the fact that such an intersection product is well defined is a straight-forward consequence of Theorem 3.1

Theorem 3.4 Let X be a smooth, quasi-projective variety of dimension m and let Y, Z be cycles on X of dimension r, s with r + s ≥ m. If both Z′ and Z′′ are rationally equivalent to Z and intersect Y properly, then Y • Z′, Y • Z′′ are also rationally equivalent.

Proof. We first consider the case in which k is either an infinite field or a “sufficiently large” finite field as in the statement of Theorem 3.1. We may assume that the rational equivalences relating Z to Z′ and Z′′ are given by irreducible, smooth, rational curves. Thus, setting T = Speck[x, y]/xy, there is a (flat) family Z_{T} = {Z_{t} : t ∈ T } on T × X of cycles on X parametrized by T giving a rational equivalence between Z = Z_{0,0}, Z′ = Z_{1,0}, and Z′′ = Z_{0,1}. Choose some projective closure \overline{X} of X.

Write Z_{T} = Z_{T}^{+} − Z_{T}^{−}, where Z_{T}^{+}, Z_{T}^{−} are flat families of effective cycles over T. Let B_{i}, i = 1, 2 denote the two branches of T and let Z_{B_{i}}^{+} (respectively, Z_{B_{i}}^{−}) denote the

restriction of Z_{T}^{+} (resp., Z_{T}^{−}) to B_{i}. Then the closures \overline{Z}_{B_{i}}^{±} in B_{i} × \overline{X} of Z_{B_{i}}^{±} are flat over B_{i}. Let \overline{Z}_{i}^{±} denote the fibres of \overline{Z}_{B_{i}}^{±} above (0, 0), and let \overline{Z}_{0}^{±} denote the closures in \overline{X} of the fibres of Z_{T}^{±} above (0, 0). Then

\overline{Z}_{i}^{+} = \overline{Z}_{0}^{+} + A_{i}^{+} , \overline{Z}_{i}^{−} = \overline{Z}_{0}^{−} + A_{i}^{−}

where A_{1}^{+}, A_{1}^{−}, A_{2}^{+}, A_{2}^{−} are effective cycles on \overline{X} supported on \overline{X} − X.

We define

Ẑ_{T}^{+} = (\overline{Z}_{B_{1}}^{+} + (A_{2}^{+} × T)) + (\overline{Z}_{B_{2}}^{+} + (A_{1}^{+} × T))

and observe that Ẑ_{T}^{+} determines a continuous algebraic map φ^{+} : T → C_{s}(\overline{X}) because the restrictions to B_{1}, B_{2} are flat and their futher restrictions to (0, 0) are equal. We similarly define Ẑ_{T}^{−} determining φ^{−} : T → C_{s}(\overline{X}), and set

φ = (φ^{+}, φ^{−}) : T → C_{s}(\overline{X})^{2}.

We now apply Theorem 3.1 with e greater than the degrees of some closure of the positive and negative parts of Y as well as the degrees of Ẑ_{t}^{+}, Ẑ_{t}^{−} for all t ∈ T to obtain

Ψ̃ ∘ (φ × 1) : T × ℓ ⟶ C_{s}(\overline{X})^{2},

for some and some k rational line ℓ. Then for all k-rational points p ∈ ℓ − {0}, the restriction

ψ̃_{R} : R ⟶ C_{s}(\overline{X})^{2}  (3.4.1)

of Ψ̃ ∘ (φ × 1) to

R ≡ ({(1, 0)} × ℓ) ∪ (T × {p}) ∪ ({(0, 1)} × ℓ)

provides a rational equivalence between (Ẑ_{T})_{1,0} and (Ẑ_{T})_{0,1} through cycles each of which meet Y properly when restricted to X. Thus, sending t ∈ R to Y • (ψ̃_{R}(t))|_{X}) determines a rational equivalence between Y • Z′, Y • Z′′.

Suppose now that k is an arbitary finite field and k → k′ is a finite field extension with k′ sufficiently large. Let π : X_{k′} → X denote the finite, flat map given by base extension, where X_{k′} ≡ X ×_{Spec(k)} Spec(k′); this is a map of algebraic k schemes, but not a map of quasi-projective algebraic varieties over k. By the preceding argument, the rational equivalence class of π^{∗}(Y • Z′ − Y • Z′′) is 0 in the Chow group of r + s − mcycles on X_{k′}. Since π_{∗} ∘ π^{∗} equals multiplication by d′ = [k′ : k] on cycles, π_{∗} ∘ π^{∗} also equals multiplication by d′ on rational equivalence classes of cycles. By considering π^{∗}(Y • Z′ − Y • Z′′) for two such such sufficiently large field extensions k → k′, k → k′′ of relatively prime degrees d′, d′′ over k, we conclude that Y • Z′, Y • Z′′ are rationally equivalent on X.

In [F-G], an intersection product was defined on the Lawson homology groups of a smooth, complex, quasi-projective variety X of dimension m:

L_{r}H_{2r+i}(X) ⊗ L_{s}H_{2r+j}(X) ⟶ L_{r+s−m}H_{2(r+s−m)+i+j}(X), r + s ≥ m.

This product was defined using an argument involving the Fulton-MacPherson “Deformation to the normal cone” technique together with a homotopy lifing argument. The following consequence of Theorem 3.1 gives a more concrete construction of this intersection product for X projective as well as smooth. Even in this case, the constructon of [F-G] remains useful in establishing the numerous good properties of this product.

Theorem 3.5 Let X be a smooth, complex projective variety of dimension m and let r, s be non-negative integers with r + s ≥ m. Let Z_{r}(X) denote the topological abelian group of r-cycles on X obtained as the “naive” group completion of the abelian monoid C_{s}(X) provided with the analytic topology. Then the pairing on homotopy groups induced by the intersection product of [F-G]

π_{i}(Z_{r}(X)) ⊗ π_{j}(Z_{s}(X)) ⟶^{•} π_{i+j}(Z_{r+s−m}(X))

can be represented as follows: given homotopy classes α ∈ π_{i}(Z_{r}(X)), β ∈ π_{j}(Z_{s}(X)), there exist representative (base point preserving) maps a : S^{i} → Z_{r}(X), b : S^{j} → Z_{s}(X) such that a(t), b(u) intersect properly for all t ∈ S^{i}, u ∈ S^{j} and such that the map sending t ∧ u to the intersection of a(t) and b(u) is a continuous map a · b : S^{i+j} → Z_{r+s−m}(X) which represents α • β.

Proof. Since the topology on Z_{k}(X) is compactly generated by the images of C_{k,≤e}(X)^{2} for e > 0, we may choose e sufficiently large that arbitrary representatives a, b′ of α, β lift to continuous maps ã : S̃^{i} → C_{r,≤e}(X)^{2}, b̃′ : S̃^{j} → C_{s,≤e}(X)^{2}.

The continuous algebraic map Ψ of Theorem 3.1 chosen for this e determines a continuous map (for the analytic topology)

Ψ : Z_{s}(X) × O → Z_{s}(X)

with the property that ψ_{0} (i.e., the restriction of Ψ to Z_{s}(X) × {0}) is the identity. Define

b = Ψ ∘ (b′ × i_{p}) : S^{j} → Z_{s}(X) × O → Z_{s}(X)

for some p ∈ O − {0}, where i_{p} : S^{j} → O is the constant map with value p. By Theorem 3.1.d and our choice of e, the r-cycles a(t), b(u) intersect properly for all t ∈ S^{i}, u ∈ S^{j}.

The intersection of cycles meeting properly is continuous, so that a, b determine a continuous map a · b : S^{i+j} → Z_{r+s−m}(X). The fact that this map represents α • β follows from [F-G;3.5.a].

Remark 3.6. Our techniques fail to provide a representation of the intersection product on Lawson homology group of complex, smooth varieties X which are quasi-projective but not projective. This is because the “moves” we construct for s-cycles on a smooth completion \overline{X} of X do not restrict to moves on s- cycles on \overline{X} − X.

We finish with a theorem that captures the fundamental assertion of the duality theory in [F-L2]. It says that on a flat family of varieties, any cycle (of dimension greater than that of the base) can be moved so that it becomes itself a family of cycles in this family of varieties. The assertion holds in fact for any algebraic collection of cycles on the family.

Theorem 3.7 Let X be a smooth projective variety, f : X → B a flat morphism of varieties over k, and s an integer ≥ dim(X) − dim(B). Assume that k is an infinite field. Then for any positive integer e, there exist a Zariski open neighbrohood O ⊂ P^{1} of a k-rational point 0 ∈ P^{1} and continuous algebraic maps

Ψ = (Ψ^{+}, Ψ^{−}) : C_{s}(X) × O → C_{s}(X)^{2}

satisfying the following properties (where ψ_{p}^{±} denotes the restriction of Ψ^{±} to C_{s}(X)×{p}):

(a) Z = ψ_{0}^{+}(Z) − ψ_{0}^{−}(Z) for every effective s-cycle Z on X.

(b) For any effective s-cycle Z, the restriction of Ψ to {Z}×O determines a rational equivalence Z_{O} ⊂ X × P^{1} whose fibre equals ψ_{p}^{+}(Z) − ψ_{p}^{−}(Z).

(c) For every effective s-cycle Z of degree ≤ e, the cycles ψ_{p}^{+}(Z), ψ_{p}^{−}(Z) intersect properly each fibre f^{−1}(b), b ∈ B of f whenever p is a rational point of O −{0}.

Proof. Apply Theorem 3.1 with e replaced by the larger of the integer e of the statement of this theorem and the maximum of the degrees of the fibres f^{−1}(b), b ∈ B of f.

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