Maximal Subbundles and Gromov-Witten Invariants

Abstract

Abstract. Let C be a nonsingular irreducible projective curve of genus g ≥ 2 defined over the complex numbers. Suppose that 1 ≤ n ′ ≤ n − 1 and. It is known that, for the general vector bundle E of rank n and degree d, the maximal degree of a subbundle of E of rank n ′ is d ′ and that there are finitely many such subbundles. We obtain a formula for the number of these maximal subbundles when (n ′ , d ′ ) = 1. For g = 2, n ′ = 2, we evaluate this formula explicitly. The numbers computed here are Gromov-Witten invariants in the sense of a recent paper of Ch. Okonek and A. Teleman (to appear in Commun. Math. Phys.) and our results answer a question raised in that paper.

Genus 2 curves and generalized theta divisors

“…The set T s is locally closed, irreducible of dimension dim T s = (r − 1) 2 + 1 + s(s − r + 2), see [LN02], [RT99]. If r ≥ 4 and s ≤ r − 3, then dim T s < dim U C (r − 1, r), which proves (1).…”

“…(3) Let [F ] ∈ U C (r − 1, r) be a general element and [S] ∈ M r−2 (F ), then S is semistable and we have an exact sequence 0 → S → F → Q → 0 with Q ∈ Pic 2 (C). Moreover S and Q are general in their moduli spaces as in [LN02]. This implies that Hom(F, Q) ≃ C. In fact, by taking the dual of the above sequence and tensoring with Q we obtain 0 → Q * ⊗ Q → F * ⊗ Q → S * ⊗ Q → 0 and, passing to cohomology we get…”

“…Then actually T r−2 = U C (r − 1, r) and a general [F ] ∈ U C (r − 1, r) has finitely many maximal subbundles of rank r − 2. See [LN02], [RT99] for a proof in the general case and [LN83] for r = 3, where actually the property actually holds for any [F ] ∈ U C (2, 3).…”

Moduli of rank 4 symplectic vector bundles over a curve of genus 2

“…By dimension count, irreducibility and projectivity of the spaces, it is surjective. Finally, we use results of Lange and Newstead [14] to calculate the degree of Φ.…”

“…If symplectic bundles in higher even genus are still general enough then, by Lange-Newstead [14], Prop. 2.4, the problem involves determining those elementary transformations 0 → F → E * → T → 0, where T is a torsion sheaf of length 2g − 2, which lift to the above W .…”