On linear series and a conjecture of D. C. Butler

Abstract

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H 0 (L) of dimension n + 1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V ⊗ O C → L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

Higher rank BN-theory for curves of genus 6

“…However, there exist trigonal curves for which K C ⊗ T * 2 = T (p) for some p ∈ C and is not generated. In the latter case, there are no generated bundles in B (1,4,2).…”

“…[14] for a detailed account). Certainly T (p) ∈ B(1, 4, 2) for all p ∈ C and also K C ⊗T * 2 ∈ B (1,4,2). In fact, it is easy to see that these are the only possibilities.…”

“…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.…”

“…(iv) B(2, 11, 5) ≃ B (2,9,4) by Serre duality; similarly B(2, 12, 5) ≃ B (2, 8, 3). The result follows from Propositions 7.1(v) and 6.1.…”

On kernel bundles over reducible curves with a node

“…In the complete case, Butler, in his seminal work [But94], proved that on a smooth ir- Much works has been done in the direction of solving this conjecture. In particular it has been completely proved in [BBN15] in the case of line bundles. Moreover, many conditions for stability are given (see also [BBN08]).…”