Universal degeneracy classes for vector bundles on $\mathbb{P}^1$ bundles

Abstract

Given a vector bundle on a P 1 bundle, the base is stratified by degeneracy loci measuring the spitting type of the vector bundle restricted to each fiber. The classes of these degeneracy loci in the Chow ring or cohomology ring of the base are natural invariants characterizing the degenerations of the vector bundle. When these degeneracy loci occur in the expected codimension, we find their classes. This yields universal formulas for degeneracy classes in terms of naturally arising vector bundles on the base. Our results hold over arbitrary fields of any characteristic.Example 1.1. Suppose E is a rank 2, degree 0 vector bundle on B×P 1 . On the open set B\Σ (−2,2) , the theorem on cohomology and base change shows that the pushforwards π * E and π * E(1) are locally free sheaves of ranks 2 and 4 respectively. There is a natural map between rank 4 vector bundles φ : π * E ⊗ H 0 (O(1)) → π * E(1), and Σ (−1,1) = {b ∈ B : rank φ b ≤ 3}.

A refined Brill-Noether theory over Hurwitz spaces

“…Theorem 5.1 (Thm. 1.1 of [16]). If E is a vector bundle on π : B × P 1 → B and codim Σ e (E) = u( e), then the class [Σ e (E)] is given by a universal formula in terms of Chern classes π * E(m) and π * E(m − 1) for suitably large m. Moreover, if this expected class is non-zero, then Σ e (E) is non-empty.…”

“…The above is therefore akin to the observation that the formula for the class of W r d (C) for general C in M g computed by Kempf-Kleiman-Laksov [13,14,15] depends only on d − g. Lemma 5.4 allows us to leverage the combinatorics of the partial ordering to deduce existence from calculations for certain special splitting types. Following [16] Proof of Theorem 1.2. We will show that a e is non-zero for all e. By the second half of Theorem 5.1, this will imply Σ e (C, f ) is non-empty whenever u( e) ≤ g. Then, Lemmas 2.1 and 3.5 show that Σ e (C, f ) has dimension g − u( e) and is the closure of Σ e (C, f ).…”

“…In this section, we exploit the combinatorial structure of splitting loci stratifications to deduce existence from a simple calculation. This relies on the existence of universal enumerative formulas for splitting loci, as determined by [16,Thm. 1.1].…”

“…Remark. Since it has the expected codimension, the class of Σ e (C, f ) is determined by the universal splitting degeneracy formulas of [16] (see Example 5.3).…”

“…Acknowledgements. This work was inspired by Geoffrey Smith, who introduced the notion of Brill-Noether splitting loci in a seminar at Stanford and asked if the author's results in [16] could be applied to show their existence. I am grateful for his insight.…”