Counterexamples to Mercat’s conjecture

Abstract

In this paper, we provide counterexamples to Mercat's conjecture on vector bundles on algebraic curves. For any n ≥ 4, we provide examples of curves lying on K3 surfaces and vector bundles on those curves which invalidate Mercat's conjecture in rank n.

25 open questions about vector bundles and their moduli

“…Even though many counterexamples are known, see for instance [116], the conjecture has been proved in some cases, [6], so still it remained unclear whether a weaker conjecture may describe well the situation of high genus. After the first counterexamples, the conjecture was solved in some low rank cases for curves with very low gonality (i.e.…”

An effective restriction theorem via wall-crossing and Mercat's conjecture

“…the bundle E C,A | C is slope-stable. Corollary 1.2 gives a better lower bound for g. There are also some other results which use different techniques, such as taking evaluation map on the curve instead of the surface to find counterexample for (M 3 ) [LMN12], or restricting the bundle E C,A to a curve of higher degree to show existence of a counterexample for (M n ) when n > 3 [Sen16]. But, we show that for any smooth curve C ∈ |H| with genus g, the Mercat's conjecture M n fails for 4 ≤ n < √ g and M 3 fails where g = 9 or g ≥ 11.…”