Verlinde bundles of families of hypersurfaces and their jumping lines

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Introduction
Let π : X → S be a proper family of schemes with a polarization L . For k ≥ 1, if the sheaf π * (L ⊗k ) is locally free, we call it the k-th Verlinde bundle of the family π .
For example (Iyer 2013), let C → T be a smooth projective family of curves of fixed genus. Consider the relative moduli space π : SU(r ) → T of semistable vector bundles of rank r and trivial determinant. This family is equipped with a polarization Θ, the determinant bundle. The Verlinde bundles π * (Θ k ) of this family are projectively flat (Hitchin 1990;Axelrod et al. 1991), and their rank is given by the Verlinde formula.
In this article, we study the example of the universal family π : X → |O P n (d)| of hypersurfaces of degree d in the complex projective space P n , with n > 1. This family comes equipped with the polarization L given by the pullback of O(1) along the projection map X → P n . For k ≥ 1, the sheaf π * L ⊗k is locally free, as can be seen by considering the structure sequence of an arbitrary hypersurface of degree d in P n . For k ≥ 1, we denote the k-th Verlinde bundle of the family π by V k . To better understand V k we study its splitting type when restricted to lines in |O(d)|. Let T ⊆ |O(d)| be a line. On T = P 1 , we define the vector bundle V k,T := V k | T . The splitting type of V k,T is the unique non-increasing tuple (b 1 , . . . , b r (k) . Sequence (2.1) puts constraints on the b i : they are all non-negative and they sum up to d (k) := deg(V k ). The set of such tuples (b i ) can be ordered by defining the With this definition, smaller types are more general: the vector bundle O(b i ) on P 1 specializes to O(b i ) in the sense of Shatz (1976) If d (k) ≤ r (k) , then the most generic possible type has thus the form (1, . . . , 1, 0, . . . , 0). We call this the generic splitting type. A computation shows We have the following result on the cohomology class of the set of jumping lines Furthermore, let b range over the integers with the property 0 1. If dim Z is odd or n = 2, we have (1.1) 2. If dim Z is even and n = 3, we have The computation is carried out by the method of undetermined coefficients, leading into various calculations in the Chow ring of the Grassmannian. The assumption n ≤ 3 is needed for a certain dimension estimation.

Attained splitting types
There exists a short exact sequence of vector bundles on |O(d)| (2.1) as can be seen by taking the pushforward of a twist of the structure sequence of X on P n × |O(d)|. The map M is given by multiplication by the section In particular, we have Proof Define E 1 := ker(pr 2 • ψ), which is a locally free sheaf on P 1 . By comparing determinants in the short exact sequence 0 → E 1 → E → O → 0 we see that E 1 is free, hence by an Ext 1 computation the sequence splits. The property im(ϕ) ⊆ E 1 follows from the definition.
Proposition 1 Let f 1 , f 2 ∈ |O(d)| span the line T ⊆ |O(d)| and let p be the number of zero entries in the splitting type of V k,T . We have To prove the other inequality, consider the induced sequence We obtain the matrix description of M| T from the matrix description of M| T as follows.
The property im(M| T ) ⊆ E 1 implies that after some row operations, the matrix A has a zero row. By the construction of M| T , the same row operations lead to the matrix B having a zero row, but this is a contradiction, since the map M| T is surjective.

Corollary 2 Let T ⊆ |O(d)| be a line spanned by the polynomials f
In particular, if d (k) ≤ r (k) but k > 2d then the generic type never occurs.
Proof By Corollary 1, the bundle V k,t has non-generic type if and only if there exist linearly independent g 1 , we may take g 1 , g 2 to be multiples of f 1 /h and f 2 /h, respectively.
Proof Assume that the type of V k at some line ( f 1 , f 2 ) is other than the two above. Then the type has at least two more zero entries than the general type. By Proposition 1, we have dim f 1 θ, f 2 θ | θ ≤ 2d (k) − 2, so we find g 1 , g 2 , g 1 , g 2 ∈ H 0 (P n , O(1)) and two linearly independent equations with both sets (g 1 , g 2 ), (g 1 , g 2 ) linearly independent. From the first equation it follows that f 1 = g 2 h and f 2 = −g 1 h, for some common factor h. Applying this to the second equation, we find g 1 g 2 = g 2 g 1 , hence g 1 = αg 1 and g 2 = αg 2 for some scalar α, a contradiction.

The cohomology class of the set of jumping lines
Definition 1 Let k ≥ 1 and (b i ) be a splitting type for V k . We define the set Z (b i ) of all points t ∈ Gr(1, |O(d)|) such that V k,t has splitting type (b i ). For the set of points t where V k,t has generic splitting type, we also write Z gen , and define the set of jumping lines Z := Gr(1, |O(d)|)\Z gen . Now let k = d + 1. By Corollary 3, Z is the subvariety given as the image of the finite, generically injective multiplication map sending the tuple ((sg 1 + tg 2 ) (s:t)∈P 1 , h) to the line (shg 1 + thg 2 ) (s:t)∈P 1 .
To perform calculations in the Chow ring A of Gr(1, |O(d)|), we follow the conventions found in Eisenbud and Harris (2016). We assume char(k) = 0 for simplicity.
The ring A is generated by the Schubert classes σ a,b of the cycles Σ a,b . The class Σ a,b has codimension a + b, and we use the convention σ a := σ a,0 .
Proof (of Theorem 1) We have dim Z = n + 1 + d−1+n n since Z is the image of the generically injective map ϕ.
Let Q ⊂ |O(d)| be the image of the multiplication map The map f is birational on its image, since a general point of Q has the form gh with h irreducible. The Chow group A codim Z is generated by the classes σ a ,b with and 0 else. Hence, multiplying the above equation with the complementary classes σ a,b and taking degrees gives Using Giambelli's formula σ a,b = σ a σ b −σ a+1 σ b−1 (Eisenbud and Harris 2016, Prop. 4.16), we reduce to computing deg( where H and H are general linear subspaces of |O(d)| of dimension N − a − 2 and N − b − 2, respectively.
To a point p = g p h p ∈ Q with g p ∈ |O(1)| and h p ∈ |O(d − 1)|, associate a closed reduced subscheme Λ p ⊂ Q containing p as follows. If h p is irreducible, let Λ p be the image of the linear embedding |O(1)|×{h p } → |O(d)| given by g → gh p .
If h p is reducible, define the subscheme Λ p as the union h im(|O(1)| × {h} → |O(d)|), where h ranges over the (up to multiplication by units) finitely many divisors of p of degree d − 1.
By the definition of Z , all lines T ∈ Z lie in Q. Furthermore, if T meets the point p, then T ⊆ Λ p . For H ⊆ |O(d)| a linear subspace of dimension N − a − 2, define Q := H ∩ Q. For general H , the subscheme Q is a smooth subvariety of dimension b − n + 1 such that for a general point p = gh of Q with h ∈ |O(d)|, the polynomial h is irreducible.
Next, we consider the case n = 2 or dim Z odd.

Claim For genereal H , for each point
The fibers of the induced map X → H have dimension at least one. Hence, to prove that the desired condition on H is an open condition, it suffices to prove dim(X ) ≤ dim(H ).
The fiber of the map X → Q over a point p consists of the union of finitely many closed subsets of the form X p = {H ∈ H : dim(H ∩ Λ p ) ≥ 1}, where Λ p P n ⊆ |O(d)| is one of the components of Λ p . The space X p is a Schubert cycle Next, let and Λ := |O(1)| × pr 2 f −1 (Q ).
By the choice of H , the map f −1 (Q ) → Q is birational and the map f −1 (Q ) → pr 2 f −1 (Q ) is even bijective. It follows that Λ and hence Λ have dimension b + 1.
The intersection of Λ with a general linear subspace H of dimension N − b − 2 is a finite set of points. For each point p ∈ Q , the linear subspace H intersects each component Λ p of Λ p in at most one point. For each point p ∈ H ∩ Λ there exists a unique p such that p ∈ Λ p .
The only line T ∈ Z meeting both p and H is the one through p and p . If the intersection H ∩ Λ p is empty, then there will be no line meeting p and H . Hence, deg([Z ] · σ a σ b ) is the number of intersection points of Λ with a general H .
Finally, the pre-image f −1 (Q ) = f −1 (H ) is smooth for a general H by Bertini's Theorem. If ζ is the class of a hyperplane section of |O(d)| we have f * (ζ ) = α + β, where α and β are classes of hyperplane sections of |O(1)| and |O(d)|, respectively. Since pr 2 and f have degree one, we compute Hence, by the push-pull formula: We then use Giambelli's formula to obtain Eq. 1.1. In case n = 3 and dim Z even, we show that for b = dim Z /2 we have deg([Z ] · σ b,b ) = 0. In this case, the hyperplanes H and H have the same dimension N − b − 2.
For p ∈ Q, the set Λ p is defined as before.
Claim for general H of dimension N − b − 2, we have dim(Λ p ∩ H ) = 1.
Proof (of Claim) Define as before the closed subset X ⊆ Q × H by The generic fiber of the projection map ϕ : X → H is one-dimensional, hence we have dim ϕ(X ) = dim(X ) − 1 = dim H . The last equation holds with n = 3 and 2b = dim Z . Hence for all H ∈ H we have dim(Λ p ∩ H ) ≥ 1.
On the other hand, the equality dim(Λ p ∩ H ) = 1 is attained by some, and hence by a general, H . Indeed, Define the closed subset X ⊆ Q × H by By a similar argument as before, one needs to show that dim(H ) − dim(X ) + 1 ≥ 0. The fiber X p is a Schubert cycle of codimension 3(dim Q − b + 1). Lastly, a computation shows dim(H ) − dim( X ) + 1 = codim( X p ) − dim(Q) + 1 = 1 2 (2 dim Q + 18 − 5n) ≥ 0. Now, define Λ as above. We have dim Λ = dim |O(1)| + dim pr 2 f −1 (Q ) = b. Since f is generically of degree one, we still have dim Λ = Λ, hence dim Λ + dim H = N − 2 < dim |O(d)|. It follows that a generic H does not meet any of the lines T ⊂ Z , hence σ b σ b · [Z ] = 0.