The Maximal Rank Conjecture and Rank Two Brill-Noether Theory

Abstract

We describe applications of Koszul cohomology to the BrillNoether theory of rank 2 vector bundles. Among other things, we show that in every genus g > 10, there exist curves invalidating Mercat's Conjecture for rank 2 bundles. On the other hand, we prove that Mercat's Conjecture holds for general curves of bounded genus, and its failure locus is a Koszul divisor in the moduli space of curves. We also formulate a conjecture concerning the minimality of Betti diagrams of suitably general curves, and point out its consequences to rank 2 Brill-Noether theory.

Vector Bundles of Rank 2 Computing Clifford Indices

“…An interesting by-product of this investigation is that, for r ≥ 4, we have γ ′ 2 < γ 1 , yielding further counterexamples to Mercat's conjecture in rank 2 (see proposition 2.7) to add to those already described in [3] and [10]. In particular Proposition 4.5 and Theorem 4.8 give the following theorem.…”

“…This has also been proved for g ≤ 16 in [3, Theorem 1.7] (it is a consequence of (2.1) and (7.1) for g ≤ 10, g = 12 and g = 14). It is conjectured in [3] that this holds for general curves of arbitrary genus. Lemma 7.3.…”

“…For such curves it is conjectured (see [3]) that γ ′ 2 = γ 1 and this is certainly true for g ≤ 16. We work out the possible bundles computing γ ′ 2 under this assumption (see Theorem 7.4).…”

“…be not injective can be restated in terms of Koszul cohomology and that there are close connections between the problems discussed here and the maximal rank conjecture (see [3]). Our results also have implications for the non-emptiness of higher rank Brill-Noether loci, but we have not developed this here because we have no "unexpected" results for general curves.…”

“…Examples of such curves are known for all genera g ≥ 11 (see [3], [10] and Theorem 1.1 above). In this case we show that all bundles computing γ ′ 2 are stable with h 0 (E) = 2 + s, s ≥ 2, and do not possess a line subbundle with h 0 ≥ 2 (we refer to such bundles as bundles of type PR).…”