VERY TWISTING FAMILIES OF POINTED LINES ON GRASSMANNIANS

A. J. DE JONG AND JASON MICHAEL STARR

Abstract. This excerpt is a section from an article in progress. This section proves that for the Grassmannians, i.e., the homogeneous spaces of Picard number one for the classical simple algebraic groups, there exists a very twisting family of pointed lines.

1. Very twisting lines on Grassmannians

This is an extract from an article in progress relating rational connectedness of spaces of rational curves to existence of sections of families over surfaces. Two other articles, [dJS05b] and [dJS05a], also deal with aspects of this work.

This extract is concerned with a very limited problem: extending the basic arguments from [HS05] to Grassmannians and isotropic Grassmannians. This might seem superfluous since Kim and Pandharipande prove rationality of the spaces of rational curves on every projective homogeneous spaces, [KP01]. However, for sections of families over surfaces, one needs also the existence of very twisting families of lines, which is what this note proves.

Let κ be an algebraically closed field. Let (X, O_{X}(1)) be a quasi-projective κvariety together with an ample invertible sheaf. Denote by X_{sm} the smooth locus of X. The scheme \overline{M}_{0,1}(X_{sm}, 1) represents the functor of pointed lines in X. On \overline{M}_{0,1}(X_{sm}, 1), there is a rank 2 locally free sheaf E, a rank 1 locally direct summand L of E, and a morphism g : ℙ(E) → X_{sm} pulling back O_{X_{sm}}(−1) to the universal rank 1 locally direct summand O_{ℙ(E)}(−1) of π^{∗}E. The universal line is the ℙ^{1}- bundle π : ℙ(E) → \overline{M}_{0,1}(X_{sm}, 1), the universal section of π, σ : \overline{M}_{0,1}(X_{sm}, 1) → ℙ(E), pulls back O_{ℙ(E)}(−1) to the locally direct summand L, and the universal map is g.

On \overline{M}_{0,1}(X_{sm}, 1) there is an important invertible sheaf ψ^{∨}, defined as σ^{∗}O_{ℙ(E)}(Image(σ)) = σ^{∗}T_{π}, the pullback of the normal bundle of σ, which is also the pullback of the vertical tangent bundle of π. Equivalently, ψ^{∨} satisfies a canonical isomorphism,

ψ^{∨} ≅ det(E) ⊗ (L^{∨})^{⊗2}.

The evaluation morphism, denoted ev : \overline{M}_{0,1}(X_{sm}, 1) → X_{sm}, is g ∘ σ. The open subset where ev is smooth is denoted U. On this open set, an important locally free sheaf is the vertical tangent bundle T_{ev} of ev, i.e., the dual of the sheaf of relative differentials of ev. The morphism (π, g) : ℙ(E) → \overline{M}_{0,1}(X_{sm}, 1) × X_{sm} is a regular embedding, and thus has a locally free normal sheaf N. The open subset U is the maximal open set over which N is π-relatively globally generated, and T_{ev} has an equivalent definition,

Date: February 27, 2006.

T_{ev} ≅ π_{∗}(N(−Image(σ))).

Definition 1.1. [HS05] A very twisting family of pointed lines on X is a morphism ζ : ℙ^{1} → \overline{M}_{0,1}(X_{sm}, 1) such that,

(i) U contains Image(ζ), (ii) ζ^{∗}T_{ev} is ample, and (iii) the degree of ζ^{∗}ψ^{∨} is nonnegative.

Using the canonical isomorphisms, (i)–(iii) are equivalent to,

(i’) the restriction of N to the fiber of π over each point of Image(ζ) is globally

generated, (ii’) ζ^{∗}π_{∗}(N(−σ(ℙ^{1}))) is ample, and (iii’) ζ^{∗}det(E) ⊗ (L^{∨})^{⊗2} has nonnegative degree.

The open subset U intersects each irreducible component of \overline{M}_{0,1}(X_{sm}, 1) whose lines cover a dense subset of X_{sm}, cf. [KMM92, 1.1].

Lemma 1.2. If T_{X_{sm}} is globally generated, then U equals \overline{M}_{0,1}(X_{sm}, 1). In particular, if X is a homogeneous space G/P, then U equals \overline{M}_{0,1}(X, 1).

Proof. Since T_{X_{sm}} is globally generated, g^{∗}T_{X_{sm}} is globally generated, and thus the quotient N is globally generated. For a homogeneous space G/P, T_{X} is globally generated by T_{e}G ⊗_{κ} O_{X}. □

Unfortunately, there typically exist rational curves in \overline{M}_{0,1}(G/P, 1) on which ψ^{∨} or T_{ev} has negative degree. Thus, if there exists a very twisting family, it is a special rational curve. There are some obvious special rational curves in \overline{M}_{0,1}(X, 1); in some cases these give very twisting families.

Definition 1.3. Let λ : G_{m} → G be a 1-parameter subgroup. Let s : G × \overline{M}_{0,1}(G/P, 1) → \overline{M}_{0,1}(G/P, 1) be the canonical action. Let p ∈ \overline{M}_{0,1}(G/P, 1) be a point. There is an induced morphism ζ^{o} : G_{m} → \overline{M}_{0,1}(G/P, 1) by ζ(t) = s(λ(t), p). This extends uniquely to a morphism ζ : ℙ^{1} → \overline{M}_{0,1}(G/P, 1), by the valuative criterion of properness. An orbit curve is a morphism ζ thus obtained.

Question 1.4. Does there exist a very twisting family ζ : ℙ^{1} → \overline{M}_{0,1}(G/P, 1)?

If the answer is affirmative, a second question is whether there exists a very twisting orbit curve.

We answer Question 1.4 when X is the Grassmannian Flag(k, V) of rank k subspaces of an n-dimensional vector space V and when X is the Grassmannian Flag_{k}(V, β) of rank k isotropic subspaces of an n-dimensional vector space V with a symmetric or skew-symmetric bilinear pairing β. Thus we answer the question when G is one of the classical simple groups SL_{n}, SO_{n}, Sp_{2n} and P is a maximal parabolic group.

The exceptional cases. There are some exceptional cases: there does not exist a very twisting family of pointed lines if X equals a finite set, ℙ^{1}, or ℙ^{1} × ℙ^{1}. In these case ev is finite, and thus T_{ev} is the zero sheaf.

(i) For the classical Grassmannian, the single exceptional case is (n, k) = (2, 1).

(ii) In the skew-symmetric case, the single exceptional case is (n, k) = (2, 1).

(iii) In the symmetric case, the exceptional cases are (n, k) = (2, 1), (3, 1), (4, 1), and (4, 2).

Theorem 1.5. For every pair of positive integers (n, k) satisfying n ≥ 2k and not on the exceptional list above, there is a very twisting family of pointed lines to X. In many cases, there is a very twisting orbit curve.

2. Pointed lines on classical Grassmannians

Let V be a rank n κ-vector space, and let k be an integer, 0 < k < n. Denote by X the Grassmannian Flag(k, V), and denote by S(k, V) the universal rank k locally direct summand of V ⊗_{κ} O_{X}. Denote by O_{X}(1) the ample invertible sheaf giving the Plücker embedding, i.e., the ample generator of the Picard group of X. Denote by \overline{M} the scheme \overline{M}_{0,1}(X, 1). Denote by M the flag variety Flag(k −1, k, k +1; V), and denote by S_{k−1} ⊂ S_{k} ⊂ S_{k+1} ⊂ V ⊗ O_{M} the universal (k − 1, k, k + 1)-flag of locally direct summands of V.

Lemma 2.1. On ℙ(E), the locally free sheaf g^{∗}S(k, V)^{∨} is globally generated. On \overline{M}, π_{∗}[g^{∗}S(k, V)^{∨} ⊗O_{ℙ(E)}(−1)] is an invertible sheaf, and π_{∗}[g^{∗}S(k, V)^{∨}] is locally free of rank k + 1. The tautological map V^{∨} ⊗ O\overline{M} → π_{∗}[g^{∗}S(k, V)^{∨}] is surjective.

Proof. Since V^{∨} ⊗ O_{X} generates S(k, V)^{∨}, V^{∨} ⊗ O_{ℙ(E)} generates g^{∗}S(k, V)^{∨}. So the restriction of g^{∗}S(k, V)^{∨} to every fiber of π is isomorphic to O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{k}) for integers 0 ≤ a_{1} ≤ ⋯ ≤ a_{k}. By definition, O_{X}(1) = ⋀^{k} S(k, V)^{∨} has degree 1 on every fiber of π. Thus the restriction of g^{∗}S(k, V)^{∨} to every fiber of π is isomorphic to F := O_{ℙ^{1}}(1) ⊕ O_{ℙ^{1}}^{⊕(k−1)}. So, firstly, π_{∗}[g^{∗}S(k, V)^{∨}] is locally free of rank h^{0}(ℙ^{1}, F) = k + 1, and π^{∗}[g^{∗}S(k, V)^{∨} ⊗ O_{ℙ(E)}(−1)] is locally free of rank h^{0}(ℙ^{1}, F (−1)) = 1. Since the only subspace of H^{0}(ℙ^{1}, F) generating F is all of H^{0}(ℙ^{1}, F), and since V^{∨} ⊗ O_{ℙ(E)} generates g^{∗}S(k, V)^{∨}, also V^{∨} ⊗ O\overline{M} → π_{∗}[g^{∗}S(k, V)^{∨}] is surjective. □

There is a (1, 2)-flag of locally direct summands of π_{∗}[g^{∗}S(k, V)^{∨}],

π_{∗}[g^{∗}S(k, V)^{∨}⊗O_{ℙ(E)}(−1)]⊗(E/L)^{∨} ⊂ π_{∗}[g^{∗}S(k, V)^{∨}⊗O_{ℙ(E)}(−1)]⊗E^{∨} ⊂ π_{∗}[g^{∗}S(k, V)^{∨}].

Dually, there is a (k − 1, k)-flag of locally direct summands of (π_{∗}[g^{∗}S(k, V)^{∨}])^{∨},

(π_{∗}[g^{∗}S(k, V)^{∨} ⊗ O_{ℙ(E)}(−1)] ⊗ E^{∨})^{⊥} ⊂
(π_{∗}[g^{∗}S(k, V)^{∨} ⊗ O_{ℙ(E)}(−1)] ⊗ (E/L)^{∨})^{⊥} ⊂
(π_{∗}[g^{∗}S(k, V)^{∨}])^{∨}.

Because V^{∨} ⊗ O_{M} → π_{∗}[g^{∗}S(k, V)^{∨}] is surjective, (π_{∗}[g^{∗}S(k, V)^{∨}])^{∨} is canonically a rank k + 1 locally direct summand of V ⊗ O\overline{M}. This defines a (k − 1, k, k + 1)-flag of locally direct summands of V ⊗ O\overline{M}, denoted E_{k−1} ⊂ E_{k} ⊂ E_{k+1} ⊂ V ⊗ O\overline{M},

\begin{aligned}E_{k−1} & := & (π_{∗}[g^{∗}S(k, V)^{∨} ⊗ O_{ℙ(E)}(−1)] ⊗ E^{∨})^{⊥}, \\ E_{k} & := & (π_{∗}[g^{∗}S(k, V)^{∨} ⊗ O_{ℙ(E)}(−1)] ⊗ (E/L)^{∨})^{⊥}, \\ E_{k+1} & := & (π_{∗}[g^{∗}S(k, V)^{∨}])^{∨}. \end{aligned}

By the universal property of the flag variety, there exists a unique morphism ι′ : \overline{M} → M pulling back S_{k−1} ⊂ S_{k} ⊂ S_{k+1} ⊂ V ⊗ O_{M} to E_{k−1} ⊂ E_{k} ⊂ E_{k+1} ⊂ V ⊗ O\overline{M}.

Proposition 2.2. The morphism ι′ is an isomorphism. Moreover, ev : \overline{M} → Flag(k, V) is the composition of ι′ with the tautological projection Flag(k − 1, k, k + 1; V) → Flag(k, V).

Proof. On M, denote by E′ the rank 2 locally free sheaf S_{k+1}/S_{k−1}. Denote by L′ the rank 1 locally direct summand S_{k}/S_{k−1}. Denote by π′ : ℙ(E′) → Flag(k − 1, k, k + 1; V) the associated ℙ^{1}-bundle. There is a unique section σ′ : Flag(k − 1, k, k + 1; V) → ℙ(E′) pulling back the universal rank 1 locally direct summand O_{ℙ(E′)}(−1) of (π′)^{∗}E′ to L′. On ℙ(E′), there is a rank k locally direct summand S′_{k} of V ⊗ O_{ℙ(E′)} defined to be the preimage in (π′)^{∗}S_{k+1} of the universal rank 1 locally direct summand O_{ℙ(E′)}(−1) ⊂ (π′)^{∗}(S_{k+1}/S_{k−1}). By the universal property of X, there exists a unique morphism g′ : ℙ(E′) → X pulling back the locally direct summand S(k, V) to S′_{k}. By definition of σ′, (σ′)^{∗}S′_{k} equals S_{k} as a locally direct summand of V ⊗ O_{F}. Therefore g′ ∘ σ′ is the tautological projection Flag(k − 1, k, k + 1; V) → Flag(k, V).

By the definition of O_{X}(1), (g′)^{∗}O_{X}(−1) is isomorphic to ⋀^{k} S′_{k}. By definition of S′_{k}, this is isomorphic to (π′)^{∗} ⋀^{k−1} S_{k−1}⊗O_{ℙ(E′)}(−1). In particular, the restriction of (g′)^{∗}O_{X}(1) to every fiber of π′ is isomorphic to O_{ℙ^{1}}(1). Thus (π′, σ′, g′) is a family of pointed lines in X. By the universal property of \overline{M}_{0,1}(X, 1), there exists a unique morphism ι : M → \overline{M} pulling back (π, σ, g) to (π′, σ′, g′). It follows easily that ι′ ∘ ι is the identity map Id_{M}.

To prove that ι ∘ ι′ is Id\overline{M}, it suffices to find an isomorphism h : ℙ(E) → (ι′)^{∗}ℙ(E′) such that (ι′)^{∗}σ′ equals h ∘ σ and g equals (ι′)^{∗}g′ ∘ h. By construction of ι′, there is a canonical isomorphism of (ι′)^{∗}E′ with E ⊗ (π_{∗}[g^{∗}S(k, V)^{∨} ⊗ O_{ℙ(E)}(−1)])^{∨}. By the universal property of ℙ(E′), there exists a unique isomorphism h : ℙ(E) → (ι′)^{∗}ℙ(E′) pulling back the universal locally direct summand (ι′)^{∗}O_{ℙ(E′)}(−1) of (ι′)^{∗}(π′)^{∗}E′ to the locally direct summand,

O_{ℙ(E)}(−1)⊗(π^{∗}π_{∗}[g^{∗}S(k, V)⊗O_{ℙ(E)}(−1)])^{∨} ⊂ π^{∗}E⊗(π^{∗}π_{∗}[g^{∗}S(k, V)⊗O_{ℙ(E)}(−1)])^{∨} ≅ π^{∗}(ι′)^{∗}E′,

obtained from the universal locally direct summand O_{ℙ(E)}(−1) ⊂ π^{∗}E. Since (ι′)^{∗}L′ equals L ⊗ (π_{∗}[g^{∗}S(k, V)^{∨} ⊗ O_{ℙ(E)}(−1)])^{∨} as locally direct summands of (ι′)^{∗}E′, (ι′)^{∗}σ′ equals h ∘ σ. To prove (ι′)^{∗}g′ ∘ h equals g, it suffices to prove that (ι′)^{∗}S′_{k} equals g^{∗}S(k, V) as locally direct summands of V ⊗ O_{ℙ(E)}. By definition of ι′, the locally direct summand (ι′)^{∗}(π′)^{∗}S_{k−1} of V ⊗ O_{ℙ(E)} equals π^{∗}E_{k−1}, which is contained in g^{∗}S(k, V). Forming the corresponding quotients, it suffices to prove that g^{∗}S(k, V)/π^{∗}E_{k−1} equals (ι′)^{∗}O_{ℙ(E′)}(−1) as locally direct summands of (ι′)^{∗}(π′)^{∗}E′, i.e., of π^{∗}(E_{k}/E_{k−1}). As g^{∗}S(k, V)/π^{∗}E_{k−1} ⊂ π^{∗}(E_{k}/E_{k−1}) is isomorphic to,

O_{ℙ(E)}(−1)⊗(π^{∗}π_{∗}[g^{∗}S(k, V)⊗O_{ℙ(E)}(−1)])^{∨} ⊂ π^{∗}E⊗(π^{∗}π_{∗}[g^{∗}S(k, V)⊗O_{ℙ(E)}(−1)])^{∨},

compatibly with the isomorphism to π^{∗}ι^{∗}E′, this follows from the definition of h. □

Corollary 2.3. Denote by ι : Flag(k − 1, k, k + 1; V) → M the inverse morphism of ι′. There are canonical isomorphisms,

ι^{∗}T_{ev} ≅ [(S_{k+1}/S_{k})^{∨} ⊗ ((V ⊗ O_{F})/S_{k+1})] ⊕ [(S_{k}/S_{k−1}) ⊗ S_{k−1}^{∨}],
ι^{∗}ψ^{∨} ≅ (S_{k+1}/S_{k}) ⊗ (S_{k}/S_{k−1})^{∨}.

Proof. Because ev ∘ ι′ is the tautological projection, (ι′)^{∗}T_{ev} is the vertical tangent bundle of the projection. This projection is the fiber product of the relative Grassmannian Flag(1, (V ⊗O_{F})/S_{k}) and the relative Grassmannian Flag(k−1, S_{k}). The first isomorphism follows from the well-known computation of the vertical tangent bundle of a Grassmannian bundle. The second isomorphism follows from the isomorphisms (ι′)^{∗}L = L′ = S_{k}/S_{k−1} and (ι′)^{∗}(E/L) = E′/L′ = S_{k+1}/S_{k}. □

3. Pointed lines on isotropic Grassmannians

Assume that char(κ) is not 2. This subsection gives the analogues of Proposition 2.2 and Corollary 2.3 in the isotropic case. Let V be an n-dimensional vector space with a symmetric or skew-symmetric nondegenerate pairing. Let X be the Grassmannian of isotropic k-planes in V. The later sections prove existence of a very twisting family of pointed lines. The proof breaks up into several cases.

I. This is the case when k is odd and n ≥ max(4, 2k + 2).

II. This is the case when k is even and n ≥ 2k + 2.

III. This is the case when n = 2k and the pairing is symmetric. The case when

n = 2k − 1 and the pairing is symmetric reduces to this case. IV. This is the case when n = 2k and the pairing is skew-symmetric.

Definition 3.1. Let B be a κ-scheme. A symmetric pairing over B, resp. a skew-symmetric pairing on B, is a triple (E, L, β) of a locally free O_{B}-module E of finite, constant rank, an invertible O_{B}-module L, and an isomorphism of O_{B}- modules, β : E → E^{∨} ⊗ L such that β equals β^{†} ⊗ Id_{L}, resp. β equals −β^{†} ⊗ Id_{L}. If the invertible sheaf L equals O_{B}, the pair is written (E, β).

Notation 3.2. Let (E, β) be a symmetric pairing over B or a skew-symmetric pairing over B. For every increasing sequence of integers \underline{k} = (k_{1} < ⋯ < k_{r}), denote by Flag\underline{k}(E, β) the bundle over B parametrizing \underline{k}-flags of isotropic locally direct summands of E. Denote by π : Flag\underline{k}(E, β) → B the projection, and denote by S_{k_{1}}(E, β) ⊂ ⋯ ⊂ S_{k_{r}}(E, β) ⊂ π^{∗}E the universal \underline{k}-flag of isotropic locally direct summands.

Let k and n be positive integers with n ≥ 2k. Let W be a κ-vector space of dimension n. Let (W, β) be a symmetric or skew-symmetric pairing. Denote by X the isotropic flag variety,

X = Flag_{k}(W, β).

By the universal property of Flag(k; W), there exists a unique morphism e : X → Flag(k; W) pulling back the universal locally direct summand S(k, W) of W to the universal isotropic locally direct summand S_{k}(W, β). The morphism e is a closed immersion. Except when β is odd and n = 2k or 2k − 1, the invertible sheaf O_{X}(1) is defined to be the pullback by e of the Plücker invertible sheaf O(1) on Flag(k; W). In these cases, the induced morphism \overline{M}_{0,1}(e, 1) : \overline{M}_{0,1}(X, 1) → \overline{M}_{0,1}(Flag(k; W), 1) is a closed immersion. Thus, using Proposition 2.2, there is a canonical closed immersion of \overline{M}_{0,1}(X, 1) in Flag(k − 1, k, k + 1; W). The cases when n = 2k or 2k − 1 are a bit more complicated.

Symmetric case, n ≥ 2k + 2. Denote by \overline{M} the scheme \overline{M}_{0,1}(X, 1). Denote by M the isotropic flag variety,

M = Flag_{k−1,k,k+1}(W, β).

Denote by S_{k−1} ⊂ S_{k} ⊂ S_{k+1} ⊂ W ⊗_{κ} O_{M} the universal isotropic flag. By the universal property of Flag(k − 1, k, k + 1; W), there is a unique morphism e′ : M → Flag(k − 1, k, k + 1; W) pulling back the universal flag to S_{k−1} ⊂ S_{k} ⊂ S_{k+1} ⊂ W ⊗_{κ} O_{M}. The morphism e′ is a closed immersion.

Define E′ to be the rank 2 locally free sheaf, S_{k+1}/S_{k−1}, and define L′ to be the invertible sheaf, S_{k}/S_{k−1}. Denote by π′ : ℙ(E′) → M the associated ℙ^{1}-bundle. There is a unique section σ′ : M → ℙ(E′) such that the pullback of the rank 1 locally direct summand O_{ℙ(E′)}(−1) of (π′)^{∗}E′ equals L′. On ℙ(E′) there is a rank k locally direct summand S′_{k} of (π′)^{∗}S_{k+1} defined as the preimage of the rank 1 locally direct summand O_{ℙ(E′)}(−1) of (π′)^{∗}(S_{k+1}/S_{k−1}). Because (π′)^{∗}S_{k+1} is a rank k+1 locally direct summand of W ⊗_{κ} O_{ℙ(E′)}, S′_{k} is a rank k locally direct summand of W ⊗_{κ} O_{ℙ(E′)}. Because (π′)^{∗}S_{k+1} is isotropic, also S′_{k} is isotropic. By the universal property of Flag_{k}(W, β), there exists a unique morphism g′ : ℙ(E′) → Flag_{k}(W, β) such that (g′)^{∗}S_{k} equals S′_{k} as a locally direct summand of W ⊗_{κ} O_{ℙ(E′)}.

Proposition 3.3. Assume n is at least 2k +2. The datum (π′, σ′, g′) is a family of pointed lines in Flag_{k}(W, β) parametrized by M. The associated morphism ι : M → \overline{M} is an isomorphism, compatible with the closed immersions into Flag(k −1, k, k + 1; W). Moreover, ev ∘ ι equals the tautological projection Flag_{k−1,k,k+1}(W, β) → Flag_{k}(W, β).

Proof. The proof that (π′, σ′, g′) is a family of pointed lines is identical to the argument in the proof of Proposition 2.2. The morphism ι is clearly compatible with the closed immersions into Flag(k − 1, k, k + 1; W). So ι is a closed immersion. Since both M and \overline{M} are smooth, to prove ι is an isomorphism, it suffices to prove that ι is surjective. The fact about ev follows from the analogous fact in Proposition 2.2.

Let [E_{k−1} ⊂ E_{k} ⊂ E_{k+1} ⊂ W] be any flag in Flag(k − 1, k, k + 1; W) contained in the image of \overline{M}. Because the pointed line in Flag(k; W) associated to this flag is contained in Flag_{k}(W, β), for every rank k subspace S′_{k} ⊂ E_{k+1} containing E_{k−1}, S′_{k} is isotropic. In particular, since every vector in E_{k+1} is contained in such a subspace, every vector in E_{k+1} is isotropic. By the polarization identity for symmetric bilinear pairings (which holds because char(k) is not two!), the subspace E_{k+1} is isotropic. Thus [E_{k−1} ⊂ E_{k} ⊂ E_{k+1} ⊂ W] is contained in the image of M. □

On M there is a rank n − k − 1 locally direct summand S_{k+1}^{⊥} of W ⊗_{κ} O_{M} defined as the annihilator of S_{k+1} under β. Since S_{k+1} is isotropic, by definition S_{k+1} is a locally direct summand of S_{k+1}^{⊥}.

Corollary 3.4. Assume n is at least 2k + 2. The pullbacks under ι of T_{ev} and ψ^{∨} admit canonical isomorphisms,

ι^{∗}T_{ev} ≅ [(S_{k+1}/S_{k})^{∨} ⊗ (S_{k+1}^{⊥}/S_{k+1})] ⊕ [(S_{k}/S_{k−1}) ⊗ S_{k−1}^{∨}],
ι^{∗}ψ^{∨} ≅ (S_{k+1}/S_{k}) ⊗ (S_{k}/S_{k−1})^{∨}.

Proof. The projection Flag_{k−1,k,k+1}(W, β) → Flag_{k}(W, β) is the fiber product of the relative isotropic Grassmannian, Flag_{1}(S_{k}^{⊥}/S_{k}, β̃), and the relative classical Grassmannian, Flag(k−1, S_{k}). The proof of the corollary is almost identical the the proof of Corollary 2.3. The one new element is the well-known isomorphism of the vertical tangent bundle of Flag_{1}(E, β) → B with S_{1}(E, β)^{∨}⊗(S_{1}(E, β)^{⊥}/S_{1}(E, β)), using Notation 3.2. □

Remark 3.5. If k = 1, then S_{k−1} is the zero sheaf. If n = 2k +2, then S_{k+1}^{⊥} equals S_{k+1} so that S_{k+1}^{⊥}/S_{k+1} is the zero sheaf.

Symmetric case, n = 2k + 1, 2k + 2. If n equals 2k + 1 or 2k + 2, then the Picard group is ℤ, but the invertible sheaf giving the Plücker embedding, the Plücker invertible sheaf, is not a generator of the Picard group. First consider the case when n = 2k + 1. Let (1, β_{1}) be a symmetric pairing such that dim(1) equals 1. Define (W′, β′) to be the orthogonal direct sum of (W, β) and (1, β_{1}). This has rank n′ = n + 1. Denote k + 1 by k′. Denote by M′ the isotropic flag variety, M′ = Flag_{k′}(W′, β′). Because every isotropic subspace of W has dimension ≤ k, no k′-dimensional isotropic subspace of W′ is contained in W. Therefore the following morphism of O_{M′}-modules is surjective,

S_{k′} → W′ ⊗_{κ} O_{M′} ⟶^{pr_{1}} 1 ⊗_{κ} O_{M′}.

Denote the kernel by K. This is a rank k locally direct summand of W ⊗_{κ} O_{M′}. Because S_{k′} is isotropic for β′, K is isotropic for β. By the universal property of Flag_{k}(W, β), there exists a unique morphism e : M′ → M, pulling back the universal isotropic flag S_{k} ⊂ W ⊗_{κ} O_{M} to K ⊂ W ⊗_{κ} O_{M′}.

Lemma 3.6. If n = 2k + 1, the morphism e : Flag_{k+1}(W′, β′) → Flag_{k}(W, β) is an étale, finite morphism of degree 2 identifying Flag_{k+1}(W′, β′) with a disjoint union of 2 copies of Flag_{k}(W, β).

Proof. The first part is just the fact that a symmetric, bilinear pairing on a rank 2 vector space has precisely 2 isotropic lines. That this cover is trivial can be checked directly. It also follows from the fact that Flag_{k}(W, β) is separably rationally connected, in fact separably unirational, together with a corollary of Kollár: a separably rationally connected variety has trivial étale fundamental group, cf. [Deb03, Cor. 3.6]. □

Of course dim(W′) = n + 1 = 2(k + 1) = 2k′. Thus the case n = 2k + 1 is reduced to the case n′ = 2k′.

Next consider the case when n = 2k. Let X be one of the two connected components of Flag_{k}(V, β). As above, there is an embedding of X in Flag(k, V), and thus a Plücker invertible sheaf on X. Because X is smooth and rational, there is no torsion in the Picard group of X. Therefore there exists a minimal ample invertible sheaf some power of which equals the Plücker invertible sheaf.

Notation 3.7. Assume n equals 2k. Denote by O_{X}(1) the unique minimal ample invertible sheaf on Flag_{k}(W, β) some power of which equals the Plücker invertible sheaf.

Lemma 3.8. The Picard group of X is generated by O_{X}(1) and the Plücker invertible sheaf is isomorphic to O_{X}(2). In particular, a smooth rational curve in X is a line with respect to O_{X}(1) iff it has degree 2 with respect to the Plücker invertible sheaf.

Proof. Write X as SO_{2k}/P = Spin_{2k}/P′. There is a natural isomorphism of the Picard group of X and the character group of P′. Choose an isomorphism W ≅ V ⊕ V^{∨} sending β to the canonical symmetric bilinear pairing on V ⊕ V^{∨}. Then the stabilizer group P of the isotropic flag V ⊂ V ⊕ V^{∨} is the group of all maps,

( \begin{matrix}U^{−1} & B \\ 0 & U^{†} \end{matrix} ) ,

where U is in GL(V) and B : V^{∨} → V is any skew-symmetric map. The group P′ is a connected extension of P by μ_{2}. Because char(κ) is not 2, there is no nontrivial μ_{2}-extension of the additive group of skew-symmetric matrices. Therefore P′ is the basechange by P → GL(V) of a connected μ_{2}-extension G′ of GL(V). There is precisely one such, namely the basechange by det : GL(V) → G_{m} of the unique connected μ_{2} extension of G_{m}, (∗)^{2} : G_{m} → G_{m}. The character group of P′ equals the character group of G′, which is a free Abelian group containing the character group of GL(V) as an index 2 subgroup. The Plücker invertible sheaf corresponds to the character det : GL(V) → G_{m}. Since this character is twice the generator of the character group of G′, the Plücker invertible sheaf is isomorphic to the square of the generator of the Picard group of X. □

Denote by \overline{M} the the space \overline{M}_{0,1}(X, 1). Denote by M the isotropic flag variety,

M = Flag_{k−2,k}(W, β).

Denote by S_{k−2} ⊂ S_{k} ⊂ W ⊗_{κ} O_{M} the universal isotropic flag. Denote by S_{k−2}^{⊥} the rank k + 2 locally direct summand of W ⊗_{κ} O_{M} defined as the annihilator of S_{k−2}. Because S_{k}, resp. S_{k−2}, is isotropic, it is a locally direct summand of S_{k−2}^{⊥}. Denote by F the rank 4 locally free sheaf, S_{k−2}^{⊥}/S_{k−2}. Associated to β there is a symmetric pairing (F, β_{F}) on M. Denote by C′ the isotropic flag variety,

C′ = Flag_{2}(E, β̃).

Denote by π : C′ → M the projection. Denote by G the rank 2, locally direct summand S_{k}/S_{k−2} of F. Because S_{k} is isotropic for β, G is isotropic for β_{F}. By the universal property of the isotropic flag variety, there exists a unique section σ : M → C′ such that the pullback of the universal flag S_{2}(F, β_{F}) ⊂ π^{∗}F equals G ⊂ F. On C′ there is a rank k locally direct summand S′_{k} of π^{∗}S_{k−2}^{⊥} defined as the preimage of the rank 2 locally direct summand S_{2}(E, β_{F}) of π^{∗}F = π^{∗}(S_{k−2}^{⊥}/S_{k−2}). Because S_{2}(F, β_{F}) is isotropic for β_{F}, S′_{k} is isotropic for β. By the universal property of Flag_{k}(W, β), there exists a unique morphism g : C′ → Flag_{k}(W, β) such that g^{∗}S_{k} equals S′_{k} as a locally direct summand of W ⊗_{κ} O_{C′}.

Proposition 3.9. Assume n equals 2k. The morphism π′ : C′ → M is a proper, smooth morphism, and every fiber is a disjoint union of 2 copies of ℙ^{1}. There is a unique open and closed subscheme C containing the image of σ such that π : C → M is a ℙ^{1}-bundle. The datum (π : C → M, σ, g) is a family of pointed lines in Flag_{k}(W, β) parametrized by M. The associated morphism ι : M → \overline{M}_{0,1}(Flag_{k}(W, β), 1) is an isomorphism, and ev ∘ ι is the tautological projection Flag_{k−2,k}(W, β) → Flag_{k}(W, β).

Proof. The first part follows from the fact that for n = 4, Flag_{2}(W, β) is a disjoint union of two copies of ℙ^{1}. Each ℙ^{1} has degree 2 with respect to the Plücker embedding, thus it has degree 1 with respect to O_{X}(1). Therefore (π, σ, g) is a family of pointed lines in X. As in the proofs of Proposition 2.2 and Proposition 3.3, to prove ι is an isomorphism, it suffices to prove it is bijective on points.

Let C be a smooth conic in Flag(k, W). By [Buc03, Lemma 1], there is a rank k − 2 subspace S_{k−2} of W such that every point of C parametrizes a subspace S_{k} containing S_{k−2}. If C is contained in Flag_{k}(W, β), then S_{k} is isotropic, hence also S_{k−2} is isotropic. Because S_{k} is isotropic and contains S_{k−2}, S_{k} is contained in the rank k + 2 annihilator, S_{k−2}^{⊥}. Denote by F the quotient space S_{k−2}^{⊥}/S_{k−2}. Associated to β there is a symmetric bilinear pairing β_{F} on F. A rank k subspace S_{k} of W containing S_{k−2} is isotropic for β iff the subspace S_{k}/S_{k−2} of F is isotropic for β_{F}. Therefore C is also a smooth conic in Flag_{2}(F, β_{F}). As above, Flag_{2}(F, β_{F}) is a disjoint union of two smooth conics, i.e., C is one of the two connected components of Flag_{2}(F, β_{F}). Also, since the subspaces of F parametrized by Flag_{2}(F, β_{F}) collectively span F and have common intersection (0), S_{k−2} is the common intersection in W of the spaces S_{k} for every point [S_{k}] of C. Therefore the space of conics in Flag_{k}(W, β) is the bijective image of the étale double cover Flag_{2}(F, β_{F}) of Flag_{k−2}(W, β). Therefore ι is bijective. □

Corollary 3.10. Assume n equals 2k. The pullbacks under ι of T_{ev} and ψ^{∨} admit canonical isomorphisms,

ι^{∗}T_{ev} ≅ (S_{k}/S_{k−2}) ⊗ S_{k−2}^{∨},

ι^{∗}ψ^{∨} ≅ ⋀^{2}(S_{k}/S_{k−2})^{∨}.

Proof. The projection Flag_{k−2,k}(W, β) → Flag_{k}(W, β) is the relative Grassmannian Flag(k − 2, S_{k}). So the first isomorphism follows from the well-known isomorphism of the vertical tangent bundle of a relative Grassmannian. By construction, π : C → M is one connected component of the relative isotropic Grassmannian Flag_{2}(F, β_{F}), where F := S_{k−2}^{⊥}/S_{k−2}. The vertical tangent bundle of Flag(2, F) equals S(2, F)^{∨} ⊗ π^{∗}F/S(2, F). The restriction of this sheaf to Flag_{2}(F, β_{F}) is S_{2}(F, β_{F})^{∨} ⊗ S_{2}(F, β_{F})^{∨}. By [HT84, p. 474], the normal bundle of the regular embedding Flag_{2}(F, β_{F}) → Flag(2, F) equals Sym^{2}(S_{2}(F, β_{F}))^{∨}. Therefore the vertical tangent bundle of Flag_{2}(F, β_{F}) is the kernel of the map S_{2}(F, β_{F})^{∨} ⊗ S_{2}(F, β_{F})^{∨} → Sym^{2}(S_{2}(F, β_{F}))^{∨}, i.e., ⋀^{2} S_{2}(F, β_{F})^{∨}. The bundle ψ^{∨} is the pullback of the vertical tangent bundle by the section σ. Since σ pulls back S_{2}(F, β_{F}) to G = S_{k}/S_{k−2}, the bundle ψ^{∨} is canonically isomorphic to ⋀^{2}(S_{k}/S_{k−2})^{∨}. □

Skew-symmetric case, n ≥ 2k. Assume (W, β) is a skew-symmetric pairing of dimension n. Assume n is at least 2k. There is a natural embedding of the isotropic Grassmannian in the classical Grassmannian, Flag_{k}(W, β) ⊂ Flag(k, W). Denote by O_{X}(1) the pullback of O_{ℙ(⋀^{k} W)}(1) under the Plücker embedding.

Denote by \overline{M} the space \overline{M}_{0,1}(X, 1). Denote by M_{pre} the isotropic flag variety,

M = Flag_{k−1,k}(W, β).

Denote by S_{k−1,pre} ⊂ S_{k,pre} ⊂ W ⊗_{κ} O_{M_{pre}} the universal isotropic flag. On M_{pre} there is a rank n − k + 1 locally direct summand S_{k−1,pre}^{⊥} of W ⊗_{κ} O_{M_{pre}} defined as the annihilator of S_{k−1,pre} under β. Since S_{k,pre} and S_{k−1,pre} are isotropic, each one is a locally direct summand of S_{k−1,pre}. Denote by G the rank n + 1 − 2k, locally free sheaf S_{k−1,pre}^{⊥}/S_{k,pre}.

Denote by ρ : M → M_{pre} the projective bundle,

M = Flag(1, G) = ℙ(G).

Denote by S_{k−1}, resp. S_{k}, the locally direct summand of W ⊗_{κ} O_{M} obtained by pulling back S_{k−1,pre}, resp. S_{k,pre}. Each one is an isotropic locally direct summand of W ⊗_{κ} O_{M}. Also the annihilator S_{k−1}^{⊥} of S_{k−1} with respect to β equals ρ^{∗}S_{k−1,pre}^{⊥} as subsheaves of W ⊗_{κ} O_{M}. There is a rank k + 1 locally direct summand R_{k+1} of S_{k−1}^{⊥} defined as the preimage of the rank 1 locally direct summand O_{P P (G)}(−1) of π^{∗}G = S_{k−1}^{⊥}/S_{k}.

Altogether, this defines a flag of locally direct summands of W ⊗_{κ} O_{M},

S_{k−1} ⊂ S_{k} ⊂ R_{k+1} ⊂ W ⊗_{κ} O_{M}.

The first two terms are isotropic. The term R_{k+1} is not necessarily isotropic, but it is contained in S_{k−1}^{⊥}. The following lemma is straightforward.

Lemma 3.11. The flag S_{k−1} ⊂ S_{k} ⊂ R_{k+1} ⊂ W ⊗_{κ} O_{M} is the universal (k − 1, k, k + 1)-flag of locally direct summands of W such that S_{k} is isotropic and R_{k+1} is contained in S_{k−1}^{⊥}.

Remark 3.12. There is a natural action of Sp(W, β) on M. There are 2 orbits. The closed orbit is the projective homogeneous space Flag_{k−1,k,k+1}(W, β). The complement of the closed orbit is a non-projective homogeneous space. If 1 < k < n/2, the automorphism group of M equals Sp(W, β), thus M is not a homogeneous space for any group.

Define E′ to be the rank 2 locally free sheaf, R_{k+1}/S_{k−1}, and define L′ to be the invertible sheaf, S_{k}/S_{k−1}. Denote by π : ℙ(E′) → M the associated ℙ^{1}-bundle. There is a unique section σ′ : M → ℙ(E′) such that the pullback of the rank 1 locally direct summand O_{ℙ(E′)}(−1) of π^{∗}E′ equals L′.

On ℙ(E′) there is a rank k locally direct summand S′_{k} of π^{∗}R_{k+1} defined as the preimage of the rank 1 locally direct summand O_{ℙ(E′)}(−1) of π^{∗}E′ = π^{∗}(R_{k+1}/S_{k−1}). Because π^{∗}R_{k+1} is a rank k+1 locally direct summand of W ⊗_{κ} O_{ℙ(E′)}, S′_{k} is a rank k locally direct summand of W ⊗_{κ} O_{ℙ(E)}. Denote by F the quotient S_{k−1}^{⊥}/S_{k−1}. Associated to β there is a skew-symmetric pairing β_{F} for F. The locally direct summand O_{ℙ(E′)}(−1) of π^{∗}F is isotropic for β_{F} because every rank 1 locally direct summand of a skew-symmetric pairing is isotropic. Because S′_{k}/π^{∗}S_{k−1}) is isotropic for β_{F}, S′_{k} is isotropic for β. By the universal property of Flag_{k}(W, β), there exists a unique morphism g′ : ℙ(E′) → Flag_{k}(W, β) such that (g′)^{∗}S_{k} equals S′_{k} as a locally direct summand of W ⊗_{κ} O_{ℙ(E′)}.

Proposition 3.13. Assume n is at least 2k. The datum (π′, σ′, g′) is a family of pointed lines in Flag_{k}(W, β) parametrized by M. The associated morphism ι : M → \overline{M} is an isomorphism.

Proof. As in the proof of Proposition 3.3, the first statement follows from Proposition 2.2, and the second statement reduces to surjectivity of ι. Let [S_{k−1} ⊂ S_{k} ⊂ R_{k} ⊂ W] be a flag parametrizing a line in Flag(k, W) contained in Flag_{k}(W, β). Then S_{k} is isotropic, and hence also S_{k−1} is isotropic. Every vector in R_{k} is contained in a subspace S′_{k} containing S_{k−1}. Because S′_{k} is isotropic, S′_{k} is contained in S_{k−1}^{⊥}. Therefore R_{k} is contained in S_{k−1}^{⊥}. By Lemma 3.11, [S_{k−1} ⊂ S_{k} ⊂ R_{k} ⊂ W] is contained in the image of ι. □

Corollary 3.14. Assume n is at least 2k. The pullback under ι of T_{ev} admits a short exact sequence,

0 → [(R_{k+1}/S_{k})^{∨} ⊗ (S_{k−1}^{⊥}/R_{k+1})] → ι^{∗}T_{ev} → [(S_{k}/S_{k−1}) ⊗ S_{k−1}^{∨}] → 0.

And the pullback of ψ^{∨} under ι admits an isomorphism,

ι^{∗}ψ^{∨} ≅ (R_{k+1}/S_{k}) ⊗ (S_{k}/S_{k−1})^{∨}.

Proof. This is very similar to the proof of Corollary 3.4. □

4. Maps of vector bundles on the projective line

Using the propositions of Subsections 2 and 3, a very twisting family of pointed lines on a classical or isotropic Grassmannian is equivalent to a flag of locally direct summands of W ⊗_{κ} O_{ℙ^{2}} such that the associated locally free sheaf ζ^{∗}T_{ev} is ample and the associated invertible sheaf ζ^{∗}ψ^{∨} has nonnegative degree. However, in the isotropic case, the flags are difficult to construct directly. This subsection contains the proof of a fact about maps of vector bundles on the projective line which will be useful for constructing flags.

Let a ≤ b be nonnegative integers. Define H to be the rank ab(b + 1 − a) vector space,

H = Hom_{O_{ℙ^{1}}} (O_{ℙ^{1}}^{⊕a}, O_{ℙ^{1}}(b − a)^{⊕b}).

There is an open subset of H parametrizing maps whose cokernel is locally free of rank b − a. There is an open subset of this subset parametrizing maps whose cokernel is isomorphic to O_{ℙ^{1}}(b)^{⊕(b−a)}. Denote by H^{o} this open subset of H.

Proposition 4.1. The open subset H^{o} is not empty.

Proof. If a is zero, this is vacuous. Thus assume a ≥ 1. Let U be a rank 2 vector space and identify ℙ^{1} with ℙ(U). The homogeneous coordinate ring of ℙ^{1} is S := Sym^{•}(U^{∨}). Denote the graded pieces by S_{k}, i.e., S_{k} = Sym^{k}(U^{∨}). By convention, define S_{−1} to be the zero vector space. Denote by A the associative, unital κ-algebra of linear maps from S to S, A = Hom(S, S). This has a natural structure of left S-module by (p · L)(q) = pL(q) for every p, q in S and every L in A. Denote by Diff the subalgebra of Hom(S, S) of differential operators on S. This is a left S-submodule of A.

Denote by d the unique κ-derivation,

d : S → S ⊗_{κ} U^{∨},

such that d|_{S_{1}} : U^{∨} → S ⊗_{κ} U^{∨} factors as

U^{∨} ⟶^{Id} κ ⊗_{κ} U^{∨} = S_{0} ⊗_{κ} U^{∨} ⊂ S ⊗_{κ} U^{∨}.

The derivation d identifies U as a linear subspace of Diff. Define Diff_{a−1} to be the linear subspace of Diff generated by U^{⊗(a−1)}. This, of course, is just an isomorphic copy of Sym^{a−1}(U). In particular it has rank a. Also every element in Diff_{a−1} has order a − 1.

There is a canonical map c : A → (⋀^{2} U) ⊗_{κ} A defined as follows. Choose an ordered basis e_{0}, e_{1} for U, and denote by T_{0}, T_{1} the dual ordered basis for U^{∨}. For every linear map L : S → S, define c(L) : S → (⋀^{2} U) ⊗ S to be,

c(L)(p) = (e_{0} ∧ e_{1}) ⊗ (T_{1}L(T_{0} · p) − T_{0}L(T_{1} · p)).

It is straightforward to check this is independent of the choice of basis. The map c is a morphism of left S-modules (where ⋀^{2} U is given the trivial S-module structure). More importantly, if L is a differential operator of order k + 1, then c(L) is a differential operator of order k. Define c^{l} : A → (⋀^{2} U)^{⊗l} ⊗_{κ} A to be the lfold composition of c in the obvious way. Then Ker(c^{l}) contains the subspace of differential operators of order ≤ l − 1.

For every integer k, define Q_{k} to be the quotient Hom(S_{k}, S) of A, and denote by π_{k} : A → Q_{k} the quotient map. By convention, define Q_{−1} to be the zero vector space. The space Q_{k} has a natural left S-module structure by (q · L)(p) = qL(p) and π_{k} is a morphism of left S-modules. Make Q_{k} a graded S-module by defining (Q_{k})_{l} = Hom(S_{k}, S_{k+l}) for every integer l. The associated sheaf on ℙ(U) is S_{k}^{∨} ⊗_{κ} O_{ℙ(U)}(k). For every integer 0 ≤ l ≤ k + 1, there is a unique degree 2l map of graded S-modules, C_{l} : Q_{k} → (⋀^{2} U)^{⊗l} ⊗_{κ} Q_{k−l}, such that the following diagram commutes,

A ⟶^{c^{l}} (⋀^{2} U)^{⊗l} ⊗_{κ} A
π_{k}↓  ↓^{Id⊗π_{k−l}}
Q_{k} ⟶^{C_{l}} (⋀^{2} U)^{⊗l} ⊗_{κ} Q_{k−l}.

This induces a map of associated sheaves,

C̃_{l} : S_{k}^{∨} ⊗_{κ} O_{ℙ(U)}(k) → [(⋀^{2} U)^{⊗l} ⊗_{κ} S_{k−l}^{∨}] ⊗_{κ} O_{ℙ(U)}(k + l).

The composite map Diff_{a−1} ↪ A ⟶^{π_{b−1}} Q_{b−1} has image in the subspace (Q_{b−1})_{−(a−1)}, and so induces a map of associated sheaves,

ϕ_{a,b} : Diff_{a−1} ⊗_{κ} O_{ℙ(U)} → S_{b−1}^{∨} ⊗_{κ} O_{ℙ(U)}(b − a).

Twisting the map C̃_{a} appropriately gives a map of associated sheaves,

ψ_{a,b} : S_{b−1}^{∨} ⊗_{κ} [(⋀^{2} U)^{⊗l} ⊗ S_{b−a−1}^{∨}] ⊗_{κ} O_{ℙ(U)}(b).

Since dim(Diff_{a−1}) equals a, dim(S_{b−1}^{∨}) equals b, and dim((⋀^{2} U)^{⊗l}⊗S_{b−a−1}^{∨}) equals b − a, the proposition is implied by the following.

Claim 4.2. The following sequence of sheaves on O_{ℙ(U)} is exact,

0 → Diff_{a−1}⊗_{κ}O_{ℙ(U)} ⟶^{ϕ_{a,b}} S_{b−1}^{∨}⊗_{κ}O_{ℙ(U)}(b−a) ⟶^{ψ_{a,b}} [(⋀^{2} U)^{∨}⊗S_{a+b−1}^{∨}]⊗_{κ}O_{ℙ(U)}(b) → 0.

First of all, because Diff_{a−1} is contained in the kernel of c^{a}, ψ_{a,b} ∘ ϕ_{a,b} is the zero map. There is a natural action of GL(U) on ℙ(U). Each vector bundle in the claim has a natural GL(U)-linearization and each of ϕ_{a,b} and ψ_{a,b} is GL(U)-equivariant. Thus to prove the sequence is exact, it suffices to prove it is exact at one point of ℙ(U). With respect to the bases e_{0}, e_{1} of U and the dual basis T_{0}, T_{1}, denote by ∂_{0}, ∂_{1} the elements of Diff_{1} corresponding to e_{0}, e_{1}. Then an ordered basis for Diff_{a−1} consists of,

\frac{1}{(a − 1)! · 0!}∂_{0}^{a−1}, …, \frac{1}{(a − k)! · (k − 1)!}∂_{0}^{a−k}∂_{1}^{k−1}, …, \frac{1}{0! · (a − 1)!}∂_{1}^{a−1}.

An ordered basis for S_{b−1} consists of,

T_{0}^{b−1}, …, T_{0}^{b−j}T_{1}^{j−1}, ⋯, T_{1}^{b−1},

and this gives a dual ordered basis for S_{b−1}^{∨}. Similarly, an ordered basis for S_{b−a−1} consists of,

T_{0}^{a+b−1}, …, T_{0}^{a+b−1−i}T_{1}^{i−1}, …, T_{1}^{a+b−1},

and, tensored with (e_{0}∧e_{1})^{⊗l}, this gives a dual ordered basis for (⋀^{2} U)^{⊗l}⊗S_{a+b−1}^{∨}. With respect to these ordered bases, the entries of the matrix of ϕ_{a,b} equal,

(ϕ_{a,b})_{j,k} = \binom{b − j}{a − k}\binom{j − 1}{k − 1}T_{0}^{b−1+k−j}T_{1}^{j−k},

and the entries of the matrix of ψ_{a,b} equal,

(ψ_{a,b})_{i,j} = (−1)^{j−i}\binom{a}{j − i}T_{0}^{j−i}T_{1}^{a+i−j}.

Plugging in T_{0} = 1, T_{1} = 0 gives the matrices,

ϕ_{a,b}|_{[1,0]} = ( \frac{D_{a,a}}{0_{b−a,a}} ) ,

and

ψ_{a,b}|_{[1,0]} = (−1)^{a} · ( 0_{a,b−a} I_{b−a,b−a} ) ,

where 0_{k,l} is the zero k × l matrix, I_{k,k} is the k × k identity matrix, and D_{a,a} is the invertible a × a diagonal matrix with entries (D_{a,a})_{j,j} = \binom{b−j}{a−j}. These matrices visibly give a short exact sequence of κ-vector spaces. □

Because the entries of ϕ_{a,b} and ψ_{a,b} are monomials, the very twisting families constructed later are more likely to be orbit curves.

5. Application to isotropic subspaces

This subsection applies Proposition 4.1 to construct some useful isotropic subspaces of W ⊗_{κ} O_{ℙ^{1}}.

Hypothesis 5.1. Let a and b be positive integers. Assume that b is at least 2a.

Let W_{b,+} be an b-dimensional κ-vector space. Denote the dual vector space by W_{b,−}. By Proposition 4.1, there exists a map,

ϕ_{a,b,+} : O_{ℙ^{1}}(−(b − a))^{⊕a} → W_{b,+} ⊗_{κ} O_{ℙ^{1}},

whose cokernel is isomorphic to O_{ℙ^{1}}(a)^{⊕(b−a)}. Thus the annihilator in W_{b,−} ⊗_{κ} O_{ℙ^{1}} of the image of ϕ_{a,b,+} is the image of a map,

ψ_{a,b,+}^{†} : O_{ℙ^{1}}(−a)^{⊕(b−a)} → W_{b,−} ⊗_{κ} O_{ℙ^{1}}.

By hypothesis, b − a is at least a. Thus, by Proposition 4.1, there exists a map,

ϕ_{a,b−a} : O_{ℙ^{1}}(−(b − a))^{⊕a} → O_{ℙ^{1}}(−a)^{⊕(b−a)},

whose cokernel is isomorphic to O_{ℙ^{1}}^{⊕(b−2a)}. Define ϕ_{a,b,−} to be the composite map,

ψ_{a,b,+}^{†} ∘ ϕ_{a,b−a} : O_{ℙ^{1}}(−(b − a))^{⊕a} → W_{b,−} ⊗_{κ} O_{ℙ^{1}}.

Define W_{2b} to be W_{b,+} ⊕ W_{b,−}. Thus (W_{2b})^{∨} is canonically isomorphic to W_{b,−} ⊕ W_{b,+}. In the symmetric case, define β_{2b} : W_{2b} → W_{2b}^{∨} to be the linear map (W_{b,+} ⊕ W_{b,−}) → (W_{b,−} ⊕ W_{b,+}) with matrix,

β_{2b} = ( \begin{matrix}0 & Id_{W_{b,−}} \\ Id_{W_{b,+}} & 0 \end{matrix} ) .

In the skew-symmetric case, define β_{2b} to be the linear map with matrix,

β_{2b} = ( \begin{matrix}0 & −Id_{W_{b,−}} \\ Id_{W_{b,+}} & 0 \end{matrix} ) .

Define E_{2a,2b} to be the image in W_{2b} ⊗_{κ} O_{ℙ^{1}} of the sheaf map,

ϕ_{a,b,+}⊕ϕ_{a,b,−} : O_{ℙ^{1}}(−(b−a))^{⊕a}⊕O_{ℙ^{1}}(−(b−a))^{⊕a} → (W_{b,+}⊗_{κ}O_{ℙ^{1}})⊕(W_{b,−}⊗_{κ}O_{ℙ^{1}}).

Denote by E_{2a,2b}^{⊥} the annihilator of E_{2a,2b} for β_{2b}.

Lemma 5.2. The subsheaf E_{2a,2b} is a rank 2a locally direct summand of W_{2b}⊗_{κ}O_{ℙ^{1}} isotropic for β_{2b}. The quotient (E_{2a,2b})^{⊥}/E_{2a,2b} is isomorphic to O_{ℙ^{1}}^{⊕(2b−4a)}. And, E_{2a,2b}^{∨} is an ample vector bundle on ℙ^{1}.

Proof. Denote by F the rank 2b−4a) locally free sheaf E_{2a,2b}^{⊥}/E_{2a,2b}. Associated to β, there is a symmetric, resp. skew-symmetric, pairing β_{F} for F. By construction, G := Image(ψ_{a,b,+}^{†})/Image(ϕ_{a,b,−}) is an rank b − 2a locally direct summand of F, isotropic for β_{F}. Therefore F is isomorphic to G ⊕ G^{∨}. By construction, G is isomorphic to O_{ℙ^{1}}^{⊕(b−2a)}. Therefore F is isomorphic to O_{ℙ^{1}}^{⊕(2b−4a)}.

By construction, E_{2a,2b}^{∨} is isomorphic to O_{ℙ^{1}}(b−a)^{⊕2a}. By hypothesis, a is positive, and b − 2a is nonnegative, so also b − a = (b − 2a) + a ≥ a > 0. Thus O_{ℙ^{1}}(b − a) is ample. Since 2a > 0, O_{ℙ^{1}}(b − a)^{⊕2a} is ample. □

6. The classical Grassmannian

Let n > 2 be an integer and let k be an integer 0 < k < n. Let V be an ndimensional κ-vector space, and denote by (X, O_{X}(1)) the Grassmannian Flag(k, V) and the Plücker invertible sheaf. Replacing Flag(k, V) by the isomorphic scheme Flag(n−k, V^{∨}) if necessary, assume k ≤ n/2. Of course Flag(k, V) equals SL_{n}/P_{k}, where P_{k} is the maximal parabolic group corresponding to the k^{th} node of the Dynkin diagram A_{n−1}.

By Proposition 2.2, a morphism ζ : ℙ^{1} → \overline{M}_{0,1}(Flag(k, V), 1) is equivalent to a (k − 1, k, k + 1)-flag of locally direct summands,

E_{k−1} ⊂ E_{k} ⊂ E_{k+1} ⊂ V ⊗_{κ} O_{ℙ^{1}}.

By Corollary 2.3, the morphism ζ is very twisting iff (i) the bundle,

[(E_{k+1}/E_{k})^{∨} ⊗ ((V ⊗ O_{ℙ^{1}})/E_{k+1})] ⊕ [(E_{k}/E_{k−1}) ⊗ E_{k−1}^{∨}],

is ample, and (ii) the bundle,

(E_{k+1}/E_{k}) ⊗ (E_{k}/E_{k−1})^{∨},

has nonnegative degree.

Proposition 6.1. There exists a very twisting morphism ζ : ℙ^{1} → \overline{M}_{0,1}(X, 1).

Proof. It is equivalent to prove there exists a flag E_{k−1} ⊂ E_{k} ⊂ E_{k+1} ⊂ V ⊗ O_{ℙ^{1}} satisfying Conditions (i)–(ii). Let T_{0}, T_{1} denote homogeneous coordinates on ℙ^{1}.

Let V′_{2k−2} be a rank 2k − 2 vector space, let E′_{k−1} be O_{ℙ^{1}}(−1)^{⊕(k−1)}, and let ϕ′_{k−1,2k−2} : E′_{k−1} → V′_{2k−2} ⊗_{κ} O_{ℙ^{1}} be a morphism whose cokernel is isomorphic to O_{ℙ^{1}}(1)^{⊕(k−1)}, as in Proposition 4.1. Let V′′_{n+1−2k} be a rank n + 1 − 2k vector space, let E′′_{1} be O_{ℙ^{1}}(−(n−2k)), and let ϕ′′_{1,n+1−2k} : E′′_{1} → V′′_{n+1−2k} ⊗_{κ} O_{ℙ^{1}} be a morphism whose cokernel is isomorphic to O_{ℙ^{1}}(1)^{n−2k}, as in Proposition 4.1. Finally, let V′′′_{1} be a rank 1 vector space, let E′′′_{1} be V′′′_{1} ⊗_{κ} O_{ℙ^{1}}, and let ϕ′′′_{1,1} be the identity map.

Define V to be the direct sum of V′_{2k−2}, V′′_{n+1−2k} and V′′′_{1}. Define E_{k+1} to be the image of ϕ′_{k−1,2k−1} ⊕ ϕ′′_{1,n+1−2k} ⊕ ϕ′′′_{1,1} in V ⊗ O_{ℙ^{1}}. The cokernel is isomorphic to O_{ℙ^{1}}(1)^{⊕(k−1)} ⊕ O_{ℙ^{1}}(1)^{⊕(n−2k)}, i.e., O_{ℙ^{1}}(1)^{⊕(n−k−1)}.

Define E_{k} to be the image of ϕ′_{k−1,2k−2} ⊕ ϕ′′_{1,n+1−2k}. The quotient E_{k+1}/E_{k} equals E′′′_{1} ≅ O_{ℙ^{1}}. In particular, (E_{k+1}/E_{k})^{∨} ⊗ ((V ⊗ O_{ℙ^{1}})/E_{k+1}) is isomorphic to O_{ℙ^{1}}(1)^{n−k−1}, which is ample. This is half of Condition (i). For E_{k−1}, there are two cases.

Case I: k = 1. In this case, define E_{k−1} to be (0). The quotient E_{k}/E_{k−1} equals E′′_{1} ≅ O_{ℙ^{1}}(−(n − 2k)). So deg(E_{k+1}/E_{k}) equals 0 and deg(E_{k}/E_{k−1}) equals −(n − 2k). Since n ≥ 2k, −(n − 2k) ≤ 0, i.e., Condition (ii) holds. Since k = 1, Condition (i) holds.

Case II: k > 1. Decompose E′_{k−1} as E′_{k−2,a} ⊕ E′_{1,b}, where E′_{k−2,a} is the first k − 2 summands and E′_{1,b} is the last summand. Define E′′′′_{1} to be O_{ℙ^{1}}(−(n + 1 − 2k)). Define ϕ′_{k−2,a} : E′_{k−2,a} → E′_{k−2,a} to be the identity map. And define ϕ′′′′_{1,2} : E′′′′_{1} → E′_{1,b} ⊕ E′′_{1}, i.e., O_{ℙ^{1}}(−(n + 1 − 2k)) → O_{ℙ^{1}}(−1) ⊕ O_{ℙ^{1}}(−(n − 2k)), to be the map with matrix,

ϕ′′′′_{1,2} = ( \begin{matrix}T_{0}^{n−2k} \\ T_{1} \end{matrix} )

Define E_{k−1} to be E′_{k−2,a} ⊕ E′′′′_{1}, and define ϕ_{k−1} : E_{k−1} → E_{k} to be ϕ′_{k−2,a} ⊕ ϕ′′′′_{1,2}. The map ϕ′′′′_{1,2} is injective with cokernel O_{ℙ^{1}}. Thus ϕ_{k−1} is injective and E_{k}/E_{k−1} equals O_{ℙ^{1}}. Since both E_{k+1}/E_{k} and E_{k}/E_{k−1} have degree 0, Condition (ii) holds. Also, E_{k−1}^{∨} ⊗ (E_{k}/E_{k−1}) equals O_{ℙ^{1}}(1)^{⊕(k−2)} ⊕ O_{ℙ^{1}}(n + 1 − 2k). Since n ≥ 2k, this is an ample bundle. Therefore Condition (i) holds. □

Claim 6.2. The very twisting family in the proof of Proposition 6.1 can be chosen to be an orbit curve.

Proof. For simplicity, assume k > 1; the case k = 1 is similar and easier. Choose ϕ′_{k−1,2k−2} to be the map with matrix,

ϕ′_{k−1,2k−2} = ( \begin{matrix}T_{0} & 0 & … & 0 \\ T_{1} & 0 & … & 0 \\ 0 & T_{0} & … & 0 \\ 0 & T_{1} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & T_{0} \\ 0 & 0 & … & T_{1} \end{matrix} )

Choose ϕ′′_{1,n+1−2k} to be the map with matrix,

ϕ′′_{1,n+1−2k} = ( T_{0}^{n−2k} ; ⋮ ; T_{0}^{n−2k−j}T_{1}^{j} ; ⋮ ; T_{1}^{n−2k} )

Let λ′ : G_{m} × V → V be the linear action compatible with the direct sum decomposition V′_{2k−2} ⊕ V′′_{n+1−2k} ⊕ V′′′_{1} given by the diagonal matrices D_{1}, D_{2},

D_{1} = ( \begin{matrix}1 & 0 & 0 & 0 & … & 0 & 0 \\ 0 & t & 0 & 0 & … & 0 & 0 \\ 0 & 0 & 1 & 0 & … & 0 & 0 \\ 0 & 0 & 0 & t & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & … & 1 & 0 \\ 0 & 0 & 0 & 0 & … & 0 & t \end{matrix} )

D_{2} = ( \begin{matrix}t & 0 & … & 0 \\ 0 & t^{2} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & t^{n+1−2k} \end{matrix} )

and D_{3} is the 1 × 1 matrix t^{c} for c = −[(k − 1) + (n + 2 − 2k)(n + 1 − 2k)/2]. Let E′_{k−1} be the subspace of V′_{2k−2} which is the image of the matrix,

( \begin{matrix}1 & 0 & … & 0 \\ 1 & 0 & … & 0 \\ 0 & 1 & … & 0 \\ 0 & 1 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 1 \\ 0 & 0 & … & 1 \end{matrix} ) .

Let E′′_{1} be the subspace of V′′_{n+1−2k} spanned by the vector,

( 1 ⋮ 1 ) .

And let E′′′_{1} equals V′′′_{1}. Define E_{k+1} = E′_{k−1}⊕E′′_{1}⊕E′′′_{1}. Define E_{k} to be E′_{k−1}⊕E′′_{1}.

Decompose V′_{2k−2} as V′_{2k−4,a} ⊕ V′_{2,b} where V′_{2k−4,a} is the first 2k − 4 summands, and V′_{2,b} is the last 2 summands. Define E′_{k−2,a} to be the subspace of V′_{2k−4,a} which is the image of the matrix,

( \begin{matrix}1 & 0 & … & 0 \\ 1 & 0 & … & 0 \\ 0 & 1 & … & 0 \\ 0 & 1 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 1 \\ 0 & 0 & … & 1 \end{matrix} ) .

Define E′′′′_{1} to be the subspace of V′_{2,b} ⊕ V′′_{n+1−2k} spanned by the vector,

( 1 ⋮ 1 1 ⋮ 1 ) .

Define E_{k−1} to be E′_{k−2,a} ⊕ E′′′′_{1}. This gives a flag of subbundle of V, E_{k−1} ⊂ E_{k} ⊂ E_{k+1} ⊂ V. Define P to be the maximal parabolic subgroup of SL(V) that is the stabilizer of the flag E_{k} ⊂ V and define P′ to be the stabilizer of the flag E_{k−1} ⊂ E_{k} ⊂ E_{k+1} ⊂ V. The 1-parameter subgroup λ : G_{m} → SL(V) is defined above. The rational curve ζ : ℙ^{1} → Flag(k − 1, k, k + 1, V) equals the orbit curve associated to λ and the flag [E_{k−1} ⊂ E_{k} ⊂ E_{k+1} ⊂ V]. □

7. Isotropic Grassmannians, Case I

Let (W, β) be a symmetric or skew-symmetric pairing of dimension n. If n = 2 or 3 there is no very twisting family of pointed lines to an isotropic Grassmannian of (W, β). Thus assume n ≥ 4. This subsection proves existence of a very twisting family of pointed lines on the Grassmannian of isotropic k-planes when k is odd and n ≥ 2k + 2.

Hypothesis 7.1. The pairing (W, β) is symmetric of dimension 2m or 2m + 1 or the pairing (W, β) is skew-symmetric of dimension 2m, and k = 2l + 1. Assume that l is nonnegative and m is at least max(2, 2l + 2).

The last inequality is equivalent to k is positive, n ≥ 4, and n ≥ 2k + 2.

Let (W′_{4}, β_{4}) and E′_{2,4} be as in Subsection 5 for a = 1 and b = 2. In particular, E′_{2,4} is isomorphic to O_{ℙ^{1}}(−1)^{⊕2}.

Case Ia: k = 1. Assume that k = 1. Let (W′′_{n−4}, β′′) be a symmetric pairing, resp. skew-symmetric pairing, of dimension n − 4. Define (W, β) to be the orthogonal direct sum of (W′_{4}, β′_{4}) and (W′′_{n−4}, β′′). Define E_{2} to be E′_{2,4} considered as a locally direct summand of W ⊗_{κ} O_{ℙ^{1}} via the embedding W′_{4} ⊗_{κ} O_{ℙ^{1}} ↪ W ⊗_{κ} O_{ℙ^{1}}. Define E_{1} to be a direct summand O_{ℙ^{1}}(−1) in E_{2}, and define E_{0} to be the zero sheaf.

Lemma 7.2. Assume k = 1 and n ≥ 4. The flag E_{0} ⊂ E_{1} ⊂ E_{2} ⊂ W ⊗_{κ} O_{ℙ^{1}} is a (0, 1, 2)-flag of isotropic locally direct summands for β. The cokernel E_{2}^{⊥}/E_{2} is isomorphic to W′′_{n−4} ⊗_{κ} O_{ℙ^{1}}. The cokernels E_{2}/E_{1} and E_{1}/E_{0} are isomorphic to O_{ℙ^{1}}(−1). And E_{0}^{∨} is the zero sheaf.

Proof. By construction, E_{2} is isotropic of rank 2. Since E_{1} is contained in E_{2}, it is isotropic. Of course (0) is isotropic. By construction, the annihilator E_{2}^{⊥} of E_{2} with respect to β equals the direct sum of the annihilator (E′_{2,4})^{⊥} of E′_{2,4} with respect to β′_{2,4} and W′′_{n−4} ⊗_{κ} O_{ℙ^{1}}. By Lemma 5.2, (E′_{2,4})^{⊥} equals E′_{2,4}. Therefore E_{2}^{⊥}/E_{2} equals W′′_{n−4} ⊗_{κ} O_{ℙ^{1}}. By definition of E_{1} and E_{0}, E_{2}/E_{1} ≅ E_{1}/E_{0} ≅ O_{ℙ^{1}}(−1). The dual of the zero sheaf is the zero sheaf. □

Proposition 7.3. Assume k = 1. In the skew-symmetric case, assume n ≥ 4. In the symmetric case, assume n ≥ 5. The morphism ζ : ℙ^{1} → M associated to the flag in Lemma 7.2 is a very twisting family of pointed lines on Flag_{1}(W, β).

Proof. By Corollary 3.4 and Corollary 3.14, in both the symmetric and skewsymmetric case ι^{∗}ψ^{∨} equals (E_{k+1}/E_{k}) ⊗ (E_{k}/E_{k−1})^{∨}. By Lemma 7.2, this equals O_{ℙ^{1}}, and so has nonnegative degree. This is (iii) of Definition 1.1. In the skewsymmetric case, by Corollary 3.14 and since E_{0}^{∨} is the zero sheaf, ι^{∗}T_{ev} is isomorphic to (E_{2}/E_{1})^{∨} ⊗ (E_{0}^{⊥}/E_{2}). Also since E_{0} is the zero sheaf, E_{0}^{⊥} equals W ⊗_{κ} O_{ℙ^{1}}. In particular, E_{0}^{⊥}/E_{2} has rank n − 2, which is positive by the hypothesis that n ≥ 4. Since W ⊗_{κ} O_{ℙ^{1}} is globally generated, the quotient E_{0}^{⊥}/E_{2} is globally generated of positive rank. The tensor product of an ample bundle and a globally generated, positive rank bundle is an ample bundle. Since (E_{2}/E_{1})^{∨} is ample, the tensor product (E_{2}/E_{1})^{∨} ⊗ (E_{0}^{⊥}/E_{2}) is ample.

The argument in the symmetric case is the same, except E_{0}^{⊥}/E_{2} is replaced by E_{2}^{⊥}/E_{2}, which is globally generated of rank n − 4 by Lemma 7.2. The rank n − 4 is positive by the hypothesis that n ≥ 5. □

An argument similar to the proof of Claim 6.2 proves the very twisting family can be chosen to be an orbit curve.

Case Ib: k > 1. Assume now that k > 1, i.e., l ≥ 1. Let (W′′_{2m−4}, β′′_{2m−4}) and E′′_{2l,2m−4,pre} be as in Subsection 5 for a = l and b = m − 2. Hypothesis 7.1 implies b ≥ 2a, i.e., Hypothesis 5.1 holds. By Lemma 5.2, (E′′_{2l,2m−4,pre})^{∨} is ample. Let f : ℙ^{1} → ℙ^{1} be any finite morphism such that f^{∗}[(E′′_{2l,2m−4,pre})^{∨}] ⊗ O_{ℙ^{1}}(−1) is ample. In every case except (l, m) = (1, 4), it suffices to take f to be the identity map. If (l, m) = (1, 4), it suffices to take f to be any finite morphism of degree ≥ 2. At any rate, define E′′_{2l,2m−4} to be f^{∗}(E′′_{2l,2m−4,pre}) considered as a subsheaf of f^{∗}(W′′_{2m−4} ⊗_{κ} O_{ℙ^{1}}) = W′′_{2m−4} ⊗_{κ} O_{ℙ^{1}}.

If n = 2m, define (W, β) to be the orthogonal direct sum of (W′_{4}, β′_{4}) and (W′′_{2m−4}, β′′_{2m−4}). If n = 2m + 1, which can only occur in the symmetric case, let (1, β_{1}) be a symmetric pairing of dimension 1, and define (W, β) to be the orthogonal direct sum of (W′_{4}, β′_{4}), (W′′_{2m−4}, β′′_{2m−4}) and (1, β_{1}). Define E_{2l+2} to be the direct sum E′_{2,4} and E′′_{2l,2m−4}. Define E_{2l+1} to be the direct sum of one direct summand O_{ℙ^{1}}(−1) of E′_{2,4} and E′′_{2l,2m−4}. Finally, define E_{2l} to be E′′_{2l,2m−4}.

Lemma 7.4. Assume l ≥ 1 and m ≥ 2l+2. The flag E_{2l} ⊂ E_{2l+1} ⊂ E_{2l+2} ⊂ W ⊗_{κ} O_{ℙ^{1}} is a (k−1, k, k+1)-flag of isotropic locally direct summands for β. The cokernel E_{2l+2}^{⊥}/E_{2l+2} is isomorphic to O_{ℙ^{1}}^{⊕(2m−4l−4)} if n = 2m, respectively O_{ℙ^{1}}^{⊕(2m−4l−3)} if n = 2m + 1. The cokernel E_{2l}^{⊥}/E_{2l+2} is isomorphic to O_{ℙ^{1}}(1)^{⊕2} ⊕ O_{ℙ^{1}}^{⊕(2m−4l−4)} if n = 2m, respectively O_{ℙ^{1}}(1)^{⊕2} ⊕ O_{ℙ^{1}}^{⊕(2m−4l−3)} if n = 2m + 1. The cokernels E_{2l+2}/E_{2l+1} and E_{2l+1}/E_{2l} are each isomorphic to O_{ℙ^{1}}(−1). And E_{2l}^{∨}⊗(E_{2l+1}/E_{2l}) is ample.

Proof. Since E′_{2,4} is isotropic for β′_{2,4} and E′′_{2l,2m−4} is isotropic for β′′_{2l,2m−4}, E_{2l+2} is isotropic for β. Since E_{2l+1} and E_{2l} are contained in E_{2l+2}, they are also isotropic for β. The annihilator E_{2l+2}^{⊥} of E_{2l+2} with respect to β is the direct sum of the annihilator (E′_{2,4})^{⊥} of E′_{2,4} with respect to β′_{2,4}, the annihilator (E′′_{2l,2m−4})^{⊥} of E′′_{2l,2m−4} with respect to β′′_{2l,2m−4}, and also O_{ℙ^{1}} if n = 2m + 1. Therefore E_{2l+2}^{⊥}/E_{2l+2} equals the direct sum of (E′_{2,4})^{⊥}/E′_{2,4}, (E′′_{2l,2m−4})^{⊥}/E′′_{2l,2m−4}, and also O_{ℙ^{1}} if n = 2m + 1. By Lemma 5.2, E′_{2,4} is its own annihilator, and (E′′_{2l,2m−4})^{⊥}/E′′_{2l,2m−4}

equals O_{ℙ^{1}}^{⊕(2m−4l−4)}. Therefore E_{2l+2}^{⊥}/E_{2l+2} equals O_{ℙ^{1}}^{⊕(2m−4l−4)} if n = 2m, and equals O_{ℙ^{1}}^{⊕(2m−4l−3)} if n = 2m + 1.

The computation of E_{2l}^{⊥}/E_{2l+2} is similar, except the summand (E′_{2,4})^{⊥}/E′_{2,4} is replaced by W′_{4} ⊗_{κ} O_{ℙ^{1}}/E′_{2,4}. Since E′_{2,4} equals its own annihilator, W′_{4} ⊗_{κ} O_{ℙ^{1}}/E′_{2,4} equals the dual of E′_{2,4}. Thus W′_{4} ⊗_{κ} O_{ℙ^{1}}/E′_{2,4} equals (O_{ℙ^{1}}(−1)^{⊕2})^{∨}, i.e., O_{ℙ^{1}}(1)^{⊕2}. Therefore E_{2l}^{⊥}/E_{2l+2} equals O_{ℙ^{1}}(1)^{⊕2} ⊕ O_{ℙ^{1}}^{⊕(2m−4l−4)} if n = 2m, and O_{ℙ^{1}}(1)^{⊕2} ⊕ O_{ℙ^{1}}^{⊕(2m−4l−3)} if n = 2m + 1.

By definition, E_{2l+2}/E_{2l+1} and E_{2l+1}/E_{2l} are isomorphic to O_{ℙ^{1}}(−1) and E_{2l}^{∨} ⊗ (E_{2l+1}/E_{2l}), i.e., E_{2l}^{∨} ⊗ O_{ℙ^{1}}(−1), is ample. □

Proposition 7.5. Assume l ≥ 1 and m ≥ 2l + 1. The morphism ζ : ℙ^{1} → M associated to the flag in Lemma 7.4 is a very twisting family of pointed lines on Flag_{k}(W, β).

Proof. This is very similar to the proof of Proposition 7.3. □

It seems likely ζ can be chosen to be an orbit curve. However, since the entries of the matrix for ϕ′′_{2l,2m−4} are typically not monomials, it is not certain.

8. Isotropic Grassmannians, Case II

Let (W, β) be a symmetric or skew-symmetric pairing of dimension n. This subsection proves existence of a very twisting family of pointed lines on the Grassmannian of isotropic k-planes when k is even and n ≥ 2k + 2.

Hypothesis 8.1. The pairing (W, β) is symmetric of dimension 2m or 2m + 1 or the pairing (W, β) is skew-symmetric of dimension 2m, and k = 2l. Assume that l is positive and m is at least 2l + 1.

The last inequality is equivalent to k is positive and n ≥ 2k + 2. In particular, observe that m is at least 3.

Define (W′_{6}, β′_{6}) as in Subsection 5 for b = 3. Choose an ordered basis e_{1}, e_{2}, e_{3} of W′_{3,+}. Denote the dual ordered basis of W′_{3,−} by x_{1}, x_{2}, x_{3}. Define E′_{3,6} to be the rank 3 locally free O_{ℙ^{1}}-module O_{ℙ^{1}}(−2)f ⊕ O_{ℙ^{1}}(−1)g_{1} ⊕ O_{ℙ^{1}}(−1)g_{2}, where the symbols f, g_{1}, g_{2} are simply place-holders. There is an isomorphism ϕ′_{3,6} of E′_{3,6} to an isotropic locally direct summand of W′_{6} ⊗_{κ} O_{ℙ^{1}}. The definition of ϕ′_{3,6} is different in the symmetric and skew-symmetric case.

Symmetric case. Define ϕ′_{3,6} : E′_{3,6} → W′_{6} ⊗_{κ} O_{ℙ^{1}} to be the unique O_{ℙ^{1}}-module homomorphism satisfying,

\begin{aligned}ϕ′_{3,6}(g) & = & T_{0}^{2}e_{1} + T_{0}T_{1}e_{2} + T_{1}^{2}x_{3} \\ ϕ′_{3,6}(f_{1}) & = & T_{0}e_{3} − T_{1}x_{2} \\ ϕ′_{3,6}(f_{2}) & = & T_{1}x_{1} − T_{0}x_{2} \end{aligned}

The proof of the following lemma is a straightforward computation.

Lemma 8.2. Let (W′_{6}, β′_{6}) be the symmetric pairing from above. The image of ϕ′_{3,6} is a rank 3 locally direct summand of W′_{6} ⊗_{κ} O_{ℙ^{1}} isotropic for β′_{6}. It equals its own annihilator with respect to β′_{6}.

Skew-symmetric case. Define ϕ′_{3,5} : E′_{3,6} → W′_{6} ⊗_{κ} O_{ℙ^{1}} to be the unique O_{ℙ^{1}}- module homomorphism satisfying,

ϕ′_{3,6}(g) = −T_{0}^{2}e_{1} + T_{0}T_{1}(e_{2} − x_{2}) + T_{1}^{2}x_{3}
ϕ′_{3,6}(f_{1}) =  T_{0}(e_{2} + x_{2}) + 2T_{1}x_{1}
ϕ′_{3,6}(f_{2}) =  T_{1}(e_{2} + x_{2}) + 2T_{0}e_{3}

The proof of the following lemma is a straightforward computation.

Lemma 8.3. Let (W′_{6}, β′_{6}) be the skew-symmetric pairing from above. The image of ϕ′_{3,6} is a rank 3 locally direct summand of W′_{6} ⊗_{κ} O_{ℙ^{1}} isotropic for β′_{6}. It equals its own annihilator with respect to β′_{6}.

Case IIa: k = 2. Assume that k = 2, i.e., l = 1. Then Hypothesis 8.1 is equivalent to n ≥ 6. Let (W′′_{n−6}, β′′) be a symmetric pairing, resp. skew-symmetric pairing, of dimension n − 6. Define (W, β) to be the orthogonal direct sum of (W′_{6}, β′_{6}) and (W′′_{n−6}, β′′). Define E_{3} to be E′_{3,6}. Define E_{2} to be direct summand O_{ℙ^{1}}(−2)g ⊕ O_{ℙ^{1}}(−1)f_{1}, and define E_{1} to be O_{ℙ^{1}}(−2)g.

Lemma 8.4. Assume k = 2 and n ≥ 6, either the symmetric or the skew-symmetric case. The flag E_{1} ⊂ E_{2} ⊂ E_{3} ⊂ W ⊗_{κ} O_{ℙ^{1}} is a (1, 2, 3)-flag of isotropic locally direct summands for β. The cokernel E_{3}^{⊥}/E_{3} is isomorphic to W′_{n−6} ⊗_{κ} O_{ℙ^{1}}. The cokernel E_{1}^{⊥}/E_{3} is isomorphic to O_{ℙ^{1}}(1)^{⊕2} ⊕ W′_{n−4} ⊗_{κ} O_{ℙ^{1}}. The cokernels E_{3}/E_{2} and E_{2}/E_{1} are isomorphic to O_{ℙ^{1}}(−1). And E_{1}^{∨} ⊗ (E_{2}/E_{1}) is isomorphic to the ample invertible sheaf O_{ℙ^{1}}(1).

Proof. By Lemma 8.2, resp. Lemma 8.3, E_{3} is isotropic of rank 3. Since E_{2} and E_{1} are contained in E_{3}, they are isotropic. By construction, the annihilator E_{3}^{⊥} of E_{3} with respect to β is the direct sum of the annihilator (E′_{3,6})^{⊥} of E′_{3,6} with respect to β′_{3,6} and W′′_{n−6} ⊗_{κ} O_{ℙ^{1}}. By Lemma 8.2, resp. Lemma 8.3, (E′_{3,6})^{⊥} equals E′_{3,6}. Therefore E_{3}^{⊥}/E_{3} equals W′′_{n−6} ⊗_{κ} O_{ℙ^{1}}. Similarly, E_{1}^{⊥} is the direct sum of the annihilator of O_{ℙ^{1}}(−2)g with respect to β′_{3,6} and W′′_{n−6} ⊗_{κ} O_{ℙ^{1}}. Since E′_{3,6} equals its own annihilator, (O_{ℙ^{1}}(−2)g)^{⊥}/E′_{3,6} equals the dual of E′_{3,6}/O_{ℙ^{1}}(−2)g. Thus (O_{ℙ^{1}}(−2)g)^{⊥}/E′_{3,6} equals (O_{ℙ^{1}}(−1)^{⊕2})^{∨}, i.e., O_{ℙ^{1}}(1)^{⊕2}. Therefore E_{1}^{⊥}/E_{3} equals O_{ℙ^{1}}(1)^{⊕2} ⊕ W′′_{n−6} ⊗_{κ} O_{ℙ^{1}}.

By the definition of E_{2} and E_{1}, E_{3}/E_{2} ≅ E_{2}/E_{1} ≅ O_{ℙ^{1}}(−1). Since E_{1} ≅ O_{ℙ^{1}}(−2), E_{1}^{∨} ⊗ (E_{2}/E_{1}) equals O_{ℙ^{1}}(2) ⊗ O_{ℙ^{1}}(−1), i.e., O_{ℙ^{1}}(1). □

Proposition 8.5. Assume k = 2 and n ≥ 6, either the symmetric or the skewsymmetric case. The morphism ζ : ℙ^{1} → M associated to the flag in Lemma 8.4 is a very twisting family of pointed lines on Flag_{2}(W, β).

Proof. This is very similar to the proof of Proposition 7.3. Of course now the term E_{1}^{∨} ⊗ (E_{2}/E_{2}) in ζ^{∗}T_{ev} is nonzero. But by the last part of Lemma 8.4, this is ample. □

An argument similar to the proof of Claim 6.2 proves the very twisting family can be chosen to be an orbit curve.

Case IIb: k > 2. Assume now that k > 2, i.e., l ≥ 2. In both the symmetric case and the skew-symmetric case, let (W′′_{2m−6}, β′′_{2m−6}) and E′′_{2l−2,2m−6,pre} be as in Subsection 5 for a = l − 1 and b = m − 3. Because l ≥ 2, a is positive. By Hypothesis 8.1, b ≥ 2a. In particular, b is positive and Hypothesis 5.1 holds. By Lemma 5.2, (E′′_{2l−2,2m−6}, pre)^{∨} is ample. Let f : ℙ^{1} → ℙ^{1} be any finite morphism such that f^{∗}[(E′′_{2l−2,2m−6,pre})^{∨}] ⊗ O_{ℙ^{1}}(−1) is ample. In every case except (l, m) = (2, 5), it suffices to take f to be the identity map. If (l, m) = (2, 5), it suffices to take f to be any finite morphism of degree ≥ 2. At any rate, define E′′_{2l−2,2m−6} to be f^{∗}(E′′_{2l−2,2m−6,pre}) considered as a subsheaf of f^{∗}(W′′_{2m−6}⊗_{κ}O_{ℙ^{1}}) = W′′_{2m−6}⊗_{κ}O_{ℙ^{1}}.

If n = 2m, define (W, β) to be the orthogonal direct sum of (W′_{6}, β′_{6}) and (W′′_{2m−6}, β′′_{2m−6}). If n = 2m + 1, which can only occur in the symmetric case, let (1, β_{1}) be a symmetric pairing of dimension 1, and define (W, β) to be the orthogonal direct sum of (W′_{6}, β′_{6}), (W′′_{2m−6}, β′′_{2m−6}) and (1, β_{1}). Define E_{2l+1} to be the direct sum E′_{3,6} and E′′_{2l−2,2m−6}. Define E_{2l} to be the direct sum of O_{ℙ^{1}}(−2)g ⊕ O_{ℙ^{1}}(−1)f_{1} and E′′_{2l−2,2m−6}. Finally, define E_{2l−1} to be the direct sum of O_{ℙ^{1}}(−2)g and E′′_{2l−2,2m−6}.

Lemma 8.6. Assume l ≥ 2 and m ≥ 2l+1. The flag E_{2l−1} ⊂ E_{2l} ⊂ E_{2l+1} ⊂ W ⊗_{κ} O_{ℙ^{1}} is a (k−1, k, k+1)-flag of isotropic locally direct summands for β. The cokernel E_{2l+1}^{⊥}/E_{2l+1} is isomorphic to O_{ℙ^{1}}^{⊕(2m−4l−2)} if n = 2m, respectively O_{ℙ^{1}}^{⊕(2m−4l−1)} if n = 2m + 1. The cokernel E_{2l−1}^{⊥}/E_{2l+1} is isomorphic to O_{ℙ^{1}}(1)^{⊕2} ⊕ O_{ℙ^{1}}^{⊕(2m−4l−2)} if n = 2m, respectively O_{ℙ^{1}}(1)^{⊕2} ⊕ O_{ℙ^{1}}^{⊕(2m−4l−1)} if n = 2m + 1. The cokernels E_{2l+1}/E_{2l} and E_{2l}/E_{2l−1} are each isomorphic to O_{ℙ^{1}}(−1). And E_{2l−1}^{∨}⊗(E_{2l}/E_{2l−1}) is ample.

Proof. Since E′_{3,6} is isotropic for β′_{3,6} and E′′_{2l−2,2m−6} is isotropic for β′′_{2l−2,2m−6}, E_{2l+1} is isotropic for β. Since E_{2l} and E2l − 1 are contained in E_{2l+1}, they are also isotropic for β. The annihilator E_{2l+1}^{⊥} of E_{2l+1} with respect to β is the direct sum of the annihilator (E′_{3,6})^{⊥} of E′_{3,6} with respect to β′_{3,6}, the annihilator (E′′_{2l−2,2m−6})^{⊥} of E′′_{2l−2,2m−6} with respect to β′′_{2l−2,2m−6}, and also O_{ℙ^{1}} if n = 2m + 1. Therefore E_{2l+1}^{⊥}/E_{2l+2} equals the direct sum of (E′_{3,6})^{⊥}/E′_{3,6}, (E′′_{2l−2,2m−6})^{⊥}/E′′_{2l−2,2m−6}, and also O_{ℙ^{1}} if n = 2m + 1. By Lemma 8.2, resp. Lemma 8.3, E′_{3,6} is its own annihilator. By Lemma 5.2, (E′′_{2l−2,2m−6})^{⊥}/E′′_{2l−2,2m−6} equals O_{ℙ^{1}}^{⊕(2m−4l−2)}. Therefore

E_{2l+1}^{⊥}/E_{2l+1} equals O_{ℙ^{1}}^{⊕(2m−4l−2)} if n = 2m, and equals O_{ℙ^{1}}^{⊕(2m−4l−1)} if n = 2m + 1.

The computation of E_{2l−1}^{⊥}/E_{2l+1} is similar, except the summand (E′_{3,6})^{⊥}/E′_{3,6} is replaced by (O_{ℙ^{1}}(−2)g)^{⊥}/E′_{3,6}. Since E′_{3,6} equals its own annihilator, (O_{ℙ^{1}}(−2)g)^{⊥}/E′_{3,6} equals the dual of E′_{3,6}/O_{ℙ^{1}}(−2)g. Thus (O_{ℙ^{1}}(−2)g)^{⊥}/E′_{3,6} equals (O_{ℙ^{1}}(−1)^{⊕2})^{∨}, i.e., O_{ℙ^{1}}(1)^{⊕2}. Therefore E_{2l−1}^{⊥}/E_{2l+1} equals O_{ℙ^{1}}(1)^{⊕2} ⊕ O_{ℙ^{1}}^{⊕(2m−4l−2)} if n = 2m, and O_{ℙ^{1}}(1)^{⊕2} ⊕ O_{ℙ^{1}}^{⊕(2m−4l−1)} if n = 2m + 1.

By definition, E_{2l+1}/E_{2l} and E_{2l}/E_{2l−1} are isomorphic to O_{ℙ^{1}}(−1). Since (E′′_{2l−2,2m−6})^{∨}⊗ O_{ℙ^{1}}(−1) is ample, also E_{2l−1}^{∨} ⊗ (E_{2l}/E_{2l−1}), which equals the direct sum of O_{ℙ^{1}}(1) and (E′′_{2l−2,2m−6})^{∨} ⊗ O_{ℙ^{1}}(−1), is ample. □

Proposition 8.7. Assume l ≥ 1 and m ≥ 2l + 1. The morphism ζ : ℙ^{1} → M associated to the flag in Lemma 8.6 is a very twisting family of pointed lines on Flag_{k}(W, β).

Proof. The proof is very similar to the proof of Proposition 8.5. □

It seems likely ζ can be chosen to be an orbit curve. However, since the entries of the matrix for ϕ′′_{2l,2m−4} are typically not monomials, it is not certain.

9. Isotropic Grassmannians, Case III

Let (W, β) be a symmetric pairing of dimension n = 2k. This subsection proves existence of a very twisting family of pointed lines on the Grassmannian of isotropic k-planes when k ≥ 3. There is no very twisting family if k = 1 or k = 2.

Case IIIa, k even. Let l ≥ 2 be an integer, and let k equal 2l. Let n equal 2k, i.e., 4l. Let (W′_{4}, β′_{4}) and E′_{2,4} be as in Subsection 5 for a = 1 and b = 2. In particular, E′_{2,4} is isomorphic to O_{ℙ^{1}}(−1)^{⊕2}.

Let (W′′_{4l−4}, β′′_{4l−4}) and E′′_{2l−2,4l−4,pre} be as in Subsection 5 for a = l−1 and b = 2l−2. Since l ≥ 2, a is positive. And, of course, b equals 2a. Thus Hypothesis 5.1 holds. By Lemma 5.2, (E′′_{2l−2,4l−4,pre})^{∨} is ample. Let f : ℙ^{1} → ℙ^{1} be any finite morphism such that f^{∗}[(E′′_{2l−2,4l−4,pre})^{∨}] ⊗ O_{ℙ^{1}}(−1) is ample. In every case except l = 2, it suffices to take f to be the identity map. If l = 2, it suffices to take f to be any finite morphism of degree ≥ 2. At any rate, define E′′_{2l−2,4l−4} to be f^{∗}(E′′_{2l−2,4l−4,pre}) considered as a subsheaf of f^{∗}(W′′_{4l−4} ⊗_{κ} O_{ℙ^{1}}) = W′′_{4l−4} ⊗_{κ} O_{ℙ^{1}}.

Define (W, β) to be the orthogonal direct sum of (W′_{4}, β′_{4}) and (W′′_{4l−4}, β′′_{4l−4}). Define E_{2l} to be the direct sum of E′_{2,4} and E′′_{2l−2,4l−4}. And define E_{2l−2} to be E′′_{2l−2,4l−4}.

Lemma 9.1. Assume l ≥ 2. Let k equal 2l and let n equal 4l. The flag E_{2l−2} ⊂ E_{2l} ⊂ W ⊗_{κ} O_{ℙ^{1}} is a (k − 2, k)-flag of isotropic locally direct summands for β. The cokernel E_{2l}/E_{2l−2} is isomorphic to O_{ℙ^{1}}(−1)^{⊕2}. The determinant ⋀^{2}(E_{2l}/E_{2l−2})^{∨} is isomorphic to O_{ℙ^{1}}(2). And (E_{2l}/E_{2l−2}) ⊗ E_{2l−2}^{∨} is ample.

Proof. The proof is very similar to the proof of Lemma 7.4. The novel feature is the computation of ⋀^{2}(E_{2l}/E_{2l−2})^{∨}, which is obviously O_{ℙ^{1}}(2) since E_{2l}/E_{2l−2} equals O_{ℙ^{1}}(−1)^{⊕2}. □

Proposition 9.2. Assume l ≥ 2. Let k equal 2l and let n equal 4l. The morphism ζ : ℙ^{1} → M associated to the flag in Lemma 9.1 is a very twisting family of pointed lines on Flag_{k}(W, β).

Proof. The proof is very similar to the proof of Proposition 7.3. Instead of Corollary 3.4, use Corollary 3.10. □

An argument similar to the proof of Claim 6.2 proves the very twisting family can be chosen to be an orbit curve.

Case IIIb, k odd. Let l ≥ 1 be an integer, let k equal 2l + 1, and let n equal 2k = 4l + 2. For (n, k) = (2, 1), there is no very twisting family of lines. Let (W′_{6}, β′_{6}) and E′_{3,6} be as in Subsection 8 for the symmetric pairing. In particular, E′_{3,6} is isomorphic to O_{ℙ^{1}}(−2)g ⊕ O_{ℙ^{1}}(−1)f_{1} ⊕ O_{ℙ^{1}}(−1)f_{2}. If l = 1, i.e., k = 3 and n = 6, define (W, β) to be (W′_{6}, β′_{6}), define E′_{k} to be E′_{3,6} and define E′_{k−2} to be O_{ℙ^{1}}(−2)g.

Next assume l ≥ 2. Let (W′′_{4l−4}, β′′_{4l−4}) and E′′_{2l−2,4l−4,pre} be as in Subsection 5 for a = l − 1 and b = 2l − 2. Since l ≥ 2, a is positive. And, of course, b = 2a. Thus Hypothesis 5.1 holds. By Lemma 5.2, (E′′_{2l−2,4l−4,pre})^{∨} is ample. Let f : ℙ^{1} → ℙ^{1} be any finite morphism such that f^{∗}[(E′′_{2l−2,4l−4,pre})^{∨}]⊗O_{ℙ^{1}}(−1) is ample. In every case except l = 2, it suffices to take f to be the identity map. If l = 2, it suffices to take f to be any finite morphism of degree ≥ 2. At any rate, define E′′_{2l−2,4l−4} to be f^{∗}(E′′_{2l−2,4l−4,pre}) considered as a subsheaf of f^{∗}(W′′_{4l−4} ⊗_{κ} O_{ℙ^{1}}) = W′′_{4l−4} ⊗_{κ} O_{ℙ^{1}}.

Define (W, β) to be the orthogonal direct sum of (W′_{6}, β′_{6}) and (W′′_{4l−4}, β′′_{4l−4}). Define E_{2l+1} to be the direct sum of E′_{3,6} and E′′_{2l−2,4l−4}. And define E_{2l−1} to be the direct sum of O_{ℙ^{1}}(−2)g and E′′_{2l−2,4l−4}.

Lemma 9.3. Assume l ≥ 1. Let k equal 2l + 1 and let n equal 4l + 2. The flag E_{2l−1} ⊂ E_{2l+1} ⊂ W ⊗_{κ} O_{ℙ^{1}} is a (k − 2, k)-flag of isotropic locally direct summands for β. The cokernel E_{2l+1}/E_{2l−1} is isomorphic to O_{ℙ^{1}}(−1)^{⊕2}. The determinant ⋀^{2}(E_{2l+1}/E_{2l−1})^{∨} is isomorphic to O_{ℙ^{1}}(2). And (E_{2l+1}/E_{2l−1}) ⊗ E_{2l−1}^{∨} is ample.

Proof. The proof is very similar to the proof of Lemma 9.1. □

Proposition 9.4. Assume l ≥ 1. Let k equal 2l + 1 and let n equal 4l + 2. The morphism ζ : ℙ^{1} → M associated to the flag in Lemma 9.1 is a very twisting family of pointed lines on Flag_{k}(W, β).

Proof. The proof is very similar to the proof of Proposition 9.2. □

An argument similar to the proof of Claim 6.2 proves the very twisting family can be chosen to be an orbit curve.

10. Isotropic Grassmannians, Case IV

Let (W, β) be a skew-symmetric pairing of dimension n = 2k. This subsection proves existence of a very twisting family of pointed lines on the Grassmannian of isotropic k-planes when k ≥ 2. For k = 1 there is no very twisting family of pointed lines.

Case IVa, k even. Let l ≥ 1 be an integer, and let k equal 2l. Let n equal 2k, i.e., 4l. Let (W′_{4}, β′_{2}) be (W′_{2,+} ⊕ W′_{2,−}, β′_{4}) as in Subsection 5 for b = 2. Choose an ordered basis e_{1}, e_{2} for W′_{2,+}, and let x_{1}, x_{2} be the dual ordered basis for W′_{2,−}. Define R′_{3} to be the image of the sheaf,

ϕ′_{3} : O_{ℙ^{1}}{a} ⊕ O_{ℙ^{1}}(−1){b_{+}, b_{−}} → W′_{4} ⊗ O_{ℙ^{1}},
ϕ′_{3}(a) = e_{2} − x_{2}, ϕ′_{3}(b_{+}) = T_{0}e_{1} + T_{1}e_{2}, ϕ′_{3}(b_{−}) = −T_{1}x_{1} + T_{0}x_{2}.

Define E′_{2} and E′_{1} to be the subsheaves of R′_{3} given by,

\begin{matrix}E′_{2} = ϕ′_{3}(O_{ℙ^{1}}(−1)\{b_{+}, b_{−}\}), \\ E′_{1} = ϕ′_{3}(O_{ℙ^{1}}(−2)\{T_{0}b_{+} − T_{1}b_{−}\}) = O_{ℙ^{1}}(−2)\{T_{0}^{2}e_{1} + T_{0}T_{1}(e_{2} − x_{2}) + T_{1}^{2}x_{2}\}. \end{matrix}

Lemma 10.1. Let (W′_{4}, β′_{4}) be the skew-symmetric pairing from above. The sheaf E′_{2} is a rank 2 isotropic locally direct summand of W′_{4} ⊗_{κ} O_{ℙ^{1}}. The subsheaf E′_{1} is a rank 1 isotropic locally direct summand. The annihilator of E′_{1} with respect to β′_{4} equals R′_{3}. The cokernels R′_{3}/E′_{2} and E′_{2}/E′_{1} are both isomorphic to O_{ℙ^{1}}. And (E′_{1})^{∨} is an ample invertible sheaf.

Proof. Since b_{+} maps into W′_{2,+} ⊗_{κ} O_{ℙ^{1}} and b_{−} maps into W′_{2,−} ⊗_{κ} O_{ℙ^{1}}, the images are mutually orthogonal. Therefore E′_{2} is isotropic. It is straightforward to compute that E′_{2} and E′_{1} are locally direct summands of rank 2, respectively rank 1. It is also straightforward to compute that R′_{3} is a rank 3 locally direct summand that annihilates E′_{1}. Thus it is all of the annihilator of E′_{1}. The cokernels R′_{3}/E′_{2} and E′_{2}/E′_{1} are invertible sheaves. Comparing the degrees of E′_{1}, E′_{2} and R′_{3}, the cokernels have degree 0, thus are isomorphic to O_{ℙ^{1}}. Of course (E′_{1})^{∨} is isomorphic to the ample invertible sheaf O_{ℙ^{1}}(2). □

If l = 1, define W′′_{0} to be the zero vector space, and define E′′_{0} to be the zero sheaf on ℙ^{1}. If l > 1, let (W′′_{4l−4}, β′′_{4l−4}) and E′′_{2l−2,4l−4} be as in Subsection 5 for a = l − 1 and b = 2l − 2. Since l > 1, a is positive. And, of course, b equals 2a. Thus Hypothesis 5.1 holds. By Lemma 5.2, (E′′_{2l−2,4l−4})^{∨} is ample.

Define (W, β) to be the orthogonal direct sum of (W′_{2}, β′_{2}) and (W′′_{4l−4}, β′′_{4l−4}), which is just (W′_{2}, β′_{2}) if l equals 1. Define R_{2l+1} to be the direct sum of R′_{3} and E′′_{2l−2,4l−4}. Define E_{2l} to be the direct sum of E′_{2} and E′′_{2l−2,4l−4}. And define E_{2l−1} to be the direct sum of E′_{1} and E′′_{2l−2,4l−4}.

Lemma 10.2. Assume l ≥ 2. Let k equal 2l and let n equal 4l. The flag E_{2l−1} ⊂ E_{2l} ⊂ R_{2l+1} ⊂ W ⊗_{κ} O_{ℙ^{1}} is a (k − 1, k, k + 1)-flag parametrized by a morphism ζ : ℙ^{1} → M. The annihilator of E_{2l−1} equals R_{2l+1}. The cokernels R_{k+1}/E_{k} and E_{k}/E_{k−1} are each isomorphic to O_{ℙ^{1}}. And (E_{k}/E_{k−1}) ⊗ E_{k−1}^{∨} is ample.

Proof. This follows by combining Lemma 10.1 with the method of proof of Lemma 7.2. The novelty is that R_{2l+1} is not isotropic. However, it does equal the annihilator of E_{2l−1} with respect to β_{2l}. □

Proposition 10.3. Assume l ≥ 2. Let k equal 2l and let n equal 4l. The morphism ζ : ℙ^{1} → M associated to the flag in Lemma 10.2 is a very twisting family of pointed lines on Flag_{k}(W, β).

Proof. This is very similar to the proof of Proposition 7.3. □

Case IVb, k odd. Let l ≥ 1 be an integer, let k equal 2l + 1, and let n equal 2k = 4l + 2. For (n, k) = (2, 1), there is no very twisting family of lines. Let (W′_{2}, β′_{2}) be as in 5 for b = 1. In particular, W′_{2} is the direct sum of W′_{1,+} and W′_{1,−}.

Let (W′′_{4l}, β′′_{4l}) and E′′_{2l,4l} be as in Subsection 5 for a = l and b = 2l. Since l ≥ 1, a is positive. And, of course, b equals 2a. Thus Hypothesis 5.1 holds.

Define (W, β) to be the orthogonal direct sum of (W′_{2}, β′_{2}) and (W′′_{4l}, β′′_{4l}). Define R_{2l+2} to be the direct sum of W′_{2} and E′′_{2l,4l}. Define E_{2l+1} to be the direct sum of W′_{1,+} and E′′_{2l,4l}. And define E_{2l} to be E′′_{2l,4l}.

Lemma 10.4. Assume l ≥ 1. Let k equal 2l + 1 and let n equal 4l + 2. The flag E_{2l} ⊂ E_{2l+1} ⊂ R_{2l+2} ⊂ W ⊗_{κ} O_{ℙ^{1}} is a (k − 1, k, k + 1)-flag parametrized by a morphism ζ : ℙ^{1} → M. The annihilator of E_{2l} equals R_{2l+2}. The cokernels R_{k+1}/E_{k} and E_{k}/E_{k−1} are each isomorphic to O_{ℙ^{1}}. And (E_{k}/E_{k−1}) ⊗ E_{k−1}^{∨} is ample.

Proof. This is similar to the proof of Lemma 10.2. □

Proposition 10.5. Assume l ≥ 1. Let k equal 2l + 1 and let n equal 4l + 2. The morphism ζ : ℙ^{1} → M associated to the flag in Lemma 10.4 is a very twisting family of pointed lines on Flag_{k}(W, β).

Proof. This is similar to the proof of Proposition 10.5. □

An argument similar to the proof of Claim 6.2 proves the very twisting family can be chosen to be an orbit curve.

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

E-mail address: dejong@math.mit.edu

Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

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