Linear stability and stability of syzygy bundles

Abstract

Let C be a smooth irreducible projective curve and let (L, H 0 (C, L)) be a complete and generated linear series on C. Denote by ML the kernel of the evaluation map H 0 (C, L) ⊗ OC → L. The exact sequence 0 → ML → H 0 (C, L) ⊗ OC → L → 0 fits into a commutative diagram that we call the Butler's diagram. This diagram induces in a natural way a multiplication map on global sections mW :is a subspace and S ∨ is the dual of a subbundle S ⊂ ML. When the subbundle S is a stable bundle, we show that the map mW is surjective. When C is a Brill-Noether general curve, we use the surjectivity of mW to give another proof on the semistability of ML, moreover we fill up a gap of an incomplete argument by Butler: With the surjectivity of mW we give conditions to determinate the stability of ML, and such conditions implies the well known stability conditions for ML stated precisely by Butler. Finally we obtain the equivalence between the stability of ML and the linear stability of (L, H 0 (L)) on γ-gonal curves.The bundle M V,L is called Lazarsfeld-Mukai bundle. When V = H 0 (C, L), we will denote the bundle M H 0 (L),L by M L . The vector bundle M V,L and its dual M ∨ V,L have been studied from different points of view because of the rich geometry they encode. The study of the stability of M V,L is related with: the study of Brill-Noether varieties (see [4]), the Resolution Minimal Conjecture (see [6]), the stability of the tangent bundle of a projective space restricted to a curve; and the theta divisors of vector bundles on curves (see [7], [8]). Ein and Lazarsfeld used the stability of M V,L to prove the stability of the Picard bundle (see [5]). In ([10]), Paranjape and Ramanan proved that M K C is semistable, and David C. Butler showed that M L is stable for d > 2g, and it is semistable for d = 2g (see [2] and [10]).David Mumford introduced the concept of linear stability for projective varieties X ⊂ P n (see [9]). In some sense, this definition is a way to measure how X sits in P n . It was generalized for linear series (L, V ) over a curve C (see [8]). Linear stability of a generated linear series (L, V ) is a weaker condition than the stability for the vector bundle M V,L , that is, the stability of M V,L

On linear stability and syzygy stability for rank 2 linear series

“…The first and third named author proved in [CTL18] that Conjecture 2.6 does hold in the two opposite cases: when C is a hyperelliptic curve and when C is a Brill-Noether-Petri general curve.…”

“…In fact the first and third named authors showed in [CTL18] that the conjecture holds when C is a hyperelliptic curve or a Brill-Noether-Petri general curve.…”

Higher rank Brill-Noether theory and coherent systems: Open questions

“…In [MS12], this was shown to be true in many cases and some counterexamples were given. When V = H 0 (E), the question has been answered in the affirmative for Petri curves and hyperelliptic curves [CT18], but counter-examples are known for smooth plane curves of genus 7 [CMT20].…”

New examples of theta divisors for some syzygy bundles

“…Let L ∈ P ic d (C) be a globally generated line bundle over a general curve C of genus g with h 0 (L) = r+1. In ( [3], Corollary 4.3), the authors proved that the syzygy bundle M L is semistable (not stable) if and only if all the following three conditions hold:…”

“…Proof. We recall by ( [3], Corollary 4.3) that if M L is strictly semistable, then h 1 (L) = 0, r divides g and there exists an effective divisor Z of degree 1 + g r such that h 0 (L(−Z)) = r and L(−Z) is generated. The vector bundles M L and M L(−Z) fit in the following exact sequence…”