On coherent systems of type (n, d, n  +  1) on Petri curves

Abstract

We study coherent systems of type (n, d, n + 1) on a Petri curve X of genus g ≥ 2. We describe the geometry of the moduli space of such coherent systems for large values of the parameter α. We determine the top critical value of α and show that the corresponding "flip" has positive codimension. We investigate also the nonemptiness of the moduli space for smaller values of α, proving in many cases that the condition for non-emptiness is the same as for large α. We give some detailed results for g ≤ 5 and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known.

Higher rank BN-theory for curves of genus 6

“…This completes the proof of the proposition. (3,8,4). If E is not generated, a negative elementary transformation yields a bundle F with n F = 3, d F = 7 and h 0 (F ) = 4.…”

“…Note that B (3,9,6) = ∅ if C is trigonal. Part (i) and (ii) then follow from Proposition 7.4(i) and (ii).…”

“…It remains to consider E ∈ B (3,9,5). Certainly E has no subpencil, so it must possess a subbundle F of rank 2 with h 0 (F ) ≥ 3.…”

Tangent cones to generalised theta divisors and generic injectivity of the theta map

“…Before proceeding, we need to recall some background on coherent systems (see [BBN08,§2] for an overview and [BGMN03] for more detail): We recall that a coherent system of type (r, d , k) is a pair (W, Π) where W is a vector bundle of rank r and degree d over C, and Π ⊆ H 0 (C, W) is a subspace of dimension k. There is a stability condition for coherent systems depending on a real parameter α, and a moduli space G(α; r, d , k) for equivalence classes of α-semistable coherent systems of type (r, d , k).…”

Some examples of rank-2 Brill–Noether loci

“…Let C be a general curve of genus g defined over the complex numbers. The main focus of this paper is to study certain rank-2 Brill-Noether loci in the case g = 6 and, in particular, to show that B (2,10,4) is reducible (see below for the definitions); this is contrary to naïve expectations. We consider also similar situations in genus 5 and in higher genus and finish with some results on bundles computing the rank-2 Clifford index for low values of g. These examples are presented as a contribution to higher rank Brill-Noether theory, which is still far from fully understood even in rank 2.…”