Verlinde bundles of families of hypersurfaces and their jumping lines

Verlinde bundles are vector bundles $V_k$ arising as the direct image $\pi_*(\mathcal L^{\otimes k})$ of polarizations of a proper family of schemes $\pi\colon \mathfrak X \to S$. We study the splitting behavior of Verlinde bundles in the case where $\pi$ is the universal family $\mathfrak X \to |\mathcal O(d)|$ of hypersurfaces of degree $d$ in $\mathbb P^n$ and calculate the cohomology class of the locus of jumping lines of the Verlinde bundles $V_{d+1}$ in the cases $n=2,3$.


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 ( [Iye13]), 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 ( [Hit90], [ADPW91]), 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, The 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 expression With this definition, smaller types are more general: the vector bundle , then the most generic possible type has thus the form (1, . . . , 1, 0, . . . , 0). We call this the generic splitting type. A computation shows that 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 (1.1) (ii) 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.

Aknowledgement
This work is a condensed version of my Master's thesis, supervised by Daniel Huybrechts. I would like to take the opportunity to thank him for his mentorship during the writing of the thesis, as well as for his help during the preparation of this article.

Attained splitting types
There exists a short exact sequence of vector bundles on |O(d)| 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 Lemma 2.1. Let E be a free O P 1 -module of finite rank, and let 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.
Proof. Let s and t denote the homogeneous coordinates of 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. Proof. By Corollary 2.3, the bundle V k,T has non-generic type if and only if there exist linearly independent g 1 , g 2 ∈ H 0 (P n , O(k − d)) such that g 1 f 1 + g 2 f 2 = 0. Let h := gcd(f 1 , f 2 ) and d ′ := deg h.
and 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 2.2, 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 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 3.1. 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 2.6, Z is the subvariety given as the image of the finite, generically injective multiplication map The map f is birational on its image, since a general point of Q has the form gh with h irreducible of degree d − 1. The Chow group A codim Z is generated by the classes ⌋ and 0 else. Hence, multiplying the above equation with the complementary classes σ a,b and taking degrees gives 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 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. The fiber of the map X → Q over a point p consists of the union of finitely many closed subsets of the form with H n+1 an (n+1)-dimensional subspace of H 0 (O(d)). The codimension of the cycle is 2(dim Q−b), hence also codim( and 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 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 ′ . Hence, by the push-pull formula: We then use Giambelli's formula to obtain Equation (1.1).
In case n = 3 and dim Z even, we show that In this case, the hyperplanes H and H ′ have the same dimension N − b − 2.