Rank two bundles with canonical determinant and four sections

Abstract

Let C be a smooth irreducible complex projective curve of genus g and let B k (2, K C ) be the Brill-Noether locus parametrizing classes of (semi)stable vector bundles E of rank two with canonical determinant over C with h 0 (C, E) ≥ k. We show that B 4 (2, K C ) has an irreducible component B of dimension 3g −13 on a general curve C of genus g ≥ 8. Moreover, we show that for the general element [E] of B, E fits into an exact sequence 0 → O C (D) → E → K C (−D) → 0 with D a general effective divisor of degree three, and the corresponding coboundary map ∂ : H 0 (C, K C (−D)) → H 1 (C, O C (D)) has cokernel of dimension three.

Some examples of rank-2 Brill–Noether loci

“…Mukai states that B(2, K C , 4) ≃ P 2 (see the table in [18, section 4]). Since β(1, 4, 2) = 1, it follows from (5.2) that all components of B (2,8,4) have dimension 2. In particular, B (2,8,4) has no component of dimension β (2,8,4).…”

“…The only way to achieve equality is with d L = 6 and h 0 (L) = 1 and then all sections of L * ⊗ K C must lift to E. On the other hand, if E ∈ B(2, 24, 8) \ B(2, K C , 8), then E is expressible in the form (2.2) with L, L ′ ∈ B (1,8,2), L ≃ L ′ . It is not clear whether there are any L, L ′ for which there exist non-trivial extensions of this form for which all sections of L ′ * ⊗ K C lift, but, if these do exist, E is necessarily stable.…”

An stratification of $B^4(2,K_C)$ over a general curve

“…there is an open, dense subset P(E) 0 ⊆ P(E) and a morphism π d,δ : P(E) 0 → U C (d). In [2] the authors showed that on a general curve C of genus g ≥ 8, there exists an irreducible component of W 3 , ∆ 3 ⊂ P(E) of dimension 3g − 13. Moreover, this component fills up an irreducible component B 3 ⊆ B 4 (2, K C ) of dimension 3g − 13.…”

“…The following construction is a generalization of the one given in [2], and this is an adapted argument due to Robert Lazarsfeld. Lemma 3.1.…”

“…We study the Brill-Noether locus B 4 (2, K C ) that parametrizes classes of stable rank two vector bundles with canonical determinant and four sections over a general irreducible complex projective curve C. In [2], the authors construct an irreducible component of B 4 (2, K C ) of the expected dimension by studying the space of extensions of line bundles following the spirit in [3]. In this work we generalize such construction over the general curve for the case of rank two bundles with four sections in the following sense: we exhibit a chain of irreducible closed subloci B 3 ⊃ • • • ⊃ B n inside B 4 (2, K C ) where B 3 is an irreducible component of expected dimension 3g − 13 (B 3 is the component constructed in [2]) and the dimension of any stratrum B n is 3g − 10 − n. In order to give such stratification we analize the space Ext 1 := Ext 1 (K C (−D n ), O C (D n )) which parametrizes extensions of the form…”