Higher rank Clifford indices of curves on a K3 surface

Abstract

Let (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ 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 $$g\ge r^2\ge 4$$ 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 $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

Stability condition on Calabi–Yau threefold of complete intersection of quadratic and quartic hypersurfaces

“…Most of this can be found in [Bri08]. A review of it can be found in [FL21]. Let (𝑆, 𝐻) be a polarized K3 surface with a Picard group generated by H. We use the same heart of bounded t-structure Coh 𝛽,𝐻 (𝑆) as above and define the central charge as…”

“…Let Γ(𝑥) = 𝐻 2 2 𝑥 2 − 𝛾(𝑥) and Γ + to be the region above Γ. Then Bridgeland showed that (𝛼, 𝛽) with 𝛼 > Γ(𝛽) defines a stability condition 𝜎 𝛼,𝛽 = (Coh 𝛽,𝐻 (𝑋), 𝑍 𝛼,𝛽 ) on K3 surfaces [Bri08,FL21] by using Theorem 2.9. The slope is defined to be − (𝑍 𝛼,𝛽 ) (𝑍 𝛼,𝛽 ) .…”

The unreasonable effectiveness of wall-crossing in algebraic geometry

“…Riemann-Roch directly implies a stronger Bogomolov-Gieseker inequality on S 2,2 , and shifting the logic of Section 3.2 back to forward gear implies the desired Clifford bounds. These Clifford bound arguments yield new results even for planar curves [29].…”

On the Bogomolov-Gieseker inequality in positive characteristic

“…Among others, one of the important consequences of the BG inequality is the existence of Bridgeland stability conditions on surfaces [AB13,Bri07,Bri08]. The theory of Bridgeland stability conditions on surfaces has been applied to various classical problems in algebraic geometry such as birational geometry [ABCH13, BMW14, CH14, CH15, CH16, CHW17, LZ16, LZ18], Brill-Noether problem [Bay18,BL17], higher rank Clifford indices [FL18,Li19,Kos20a], and so on.…”