Higher rank Clifford indices of curves on a K3 surface

Abstract

Let (X, H) be a polarized K3 surface with Pic(X) = ZH, and let C ∈ |H| be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when g ≥ r 2 ≥ 4, the rank r Clifford index of C can be computed by the restriction of Lazarsfeld-Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree d(≥ 5) smooth plane curve is d − 4, which is the same as the Clifford index of the curve.

Walls and asymptotics for Bridgeland stability conditions on 3-folds

“…Proof. Chasing through the seven triangles listed in display (14) with A 11 = A 01 = 0, one concludes that à C A 1 , which is the same as H −1 (A) H −1 β (A) by the isomorphisms in (15). The vanishing of A 01 also implies that A 0 A 00 by the sequences in display (12), hence H 0 (A) H 0 β (A).…”

“…Bridgeland's notion of stability conditions on triangulated categories, introduced in [9] and [10], provides a new set of tools to study moduli spaces of sheaves on smooth projective varieties. Such tools have been successfully applied by many authors first to the study of sheaves on surfaces, for example, [1,2,4,5,13,14,15,21,39], and more recently on three-folds (especially P 3 ), see for instance [16,25,34,36]. One way to study moduli spaces of sheaves using Bridgeland stability spaces is to restrict attention to the so called geometric stability conditions parametrized by (a subset of) the upper half plane H. Once we know that the moduli space of Bridgeland stable objects is asymptotically given by the Gieseker semistable moduli space along an unbounded path we can try to locate all the points where the moduli space changes along this path (these isolated points are called walls) and compute the change to the moduli space.…”

On stability conditions for the quintic threefold

“…It is pity that none of the results mentioned above fit in our situation since we need the sharp bounds at some critical slopes µ = 5, 10, 30 and 35. Based on the idea in [Fey17], together with Feyzbakhsh, we develop our own methods to estimate the Clifford type bound for curves supported on K3 and Fano surfaces via stability conditions in [FL18]. Especially for this case, we think C 2,2,5 as a curve on a degree four del Pezzo surface.…”

“…Especially for this case, we think C 2,2,5 as a curve on a degree four del Pezzo surface. More introductions about the technical details in 4 can be found in [Fey17,FL18].…”

Stability and Applications

“…In this section, we present ideas of Bayer from [Bay18] to give a new proof of Lazarsfeld's theorem by using wallcrossing in Bridgeland stability. These techniques also lead to new results: for instance, there are applications to Mukai's program on reconstructing a K3 surface from a curve ([ABS14, Fey17]) and to higher rank Clifford indices of curves ( [FL18]).…”