Varieties of maximal line subbundles

Abstract

The point of this note is to make an observation concerning the variety M(E) parametrizing line subbundles of maximal degree in a generic stable vector bundle E over an algebraic curve C. M(E) is smooth and projective and its dimension is known in terms of the rank and degree of E and the genus of C (see Section 1). Our observation (Theorem 3·1) is that it has exactly the Chern numbers of an étale cover of the symmetric product SδC where δ = dim M(E).This suggests looking for a natural map M(E) → SδC; however, it is not clear what such a map should be. Indeed, we exhibit an example in which M(E) is connected and deforms non-trivially with E, while there are only finitely many isomorphism classes of étale cover of the symmetric product. This shows that for a general deformation in the family M(E) cannot be such a cover (see Section 4).One may conjecture that M(E) is always connected. This would follow from ampleness of a certain Picard-type bundle on the Jacobian and there seems to be some evidence for expecting this, though we do not pursue this question here.Note that by forgetting the inclusion of a maximal line subbundle in E we get a natural map from M(E) to the Jacobian whose image W(E) is analogous to the classical (Brill–Noether) varieties of special line bundles. (In this sense M(E) is precisely a generalization of the symmetric products of C.) In Section 2 we give some results on W(E) which generalise standard Brill–Noether properties. These are due largely to Laumon, to whom the author is grateful for the reference [9].

Non-special scrolls with general moduli

“…Finally, in § 7. 4 we extend the result in [16] and [27,Example 3.2], by computing the genus of the curve parametrizing those divisors in Div 1,m S passing through d m − 1 general points of S. This paper has to be regarded as the continuation of a project initiated with [3]. In [4] we make further application of the ideas contained in [3] and in the present paper to the BrillNoether theory of sub-line bundles of rank-two vector bundles on curves.…”

Maximal Subbundles and Gromov-Witten Invariants

“…(due essentially to Laumon [10], see also [15]). If H 1 (E ′ * ⊗ E/E ′ ) = 0, then by Serre duality there exists a nonzero homomorphism E/E ′ −→ E ′ ⊗ K. We thus have a non-zero homomorphism…”

“…In the simplest case, when n ′ = 1, the number is known; in fact, [5,9,17] for n = 2, [15] and [14,Proposition 3.9] in general). Our object in this paper is to obtain a formula for m n ′ (E) in the case (n ′ , d ′ ) = 1.…”

Brill–Noether theory and non-special scrolls

“…[13,Theorem 3], [14] and [28], where W n (F) is denoted by W (F)). The map π n can be viewed as an analogue of the classical Abel-Jacobi map and M n (F) has to be viewed as an analogue of the symmetric product of the curve C. …”

“…In order to study the morphism (4.33) and the schemes W p n (F), for p ≥ 0, a basic ingredient is the following contraction map n (F). The maximal case n = n has been studied in [28]. We recall the results.…”