[Paper Review] Higher rank Clifford indices of curves on a K3 surface

Soheyla Feyzbakhsh, Chunyi Li|arXiv (Cornell University)|Oct 25, 2018

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper establishes a sharp upper bound on the dimension of global sections of semistable vector bundles on curves lying on K3 surfaces with Picard group ℤH, enabling computation of higher rank Clifford indices. It proves that for genus $ g \geq r^2 \geq 4 $, the rank $ r $ Clifford index of such a curve equals $ \frac{2}{r}(g-1) - \frac{2}{r}\left\lfloor \frac{g}{r}\right\rfloor $, computed via restrictions of Lazarsfeld-Mukai bundles, and verifies the first part of Mercat's conjecture for smooth plane curves by showing $ \mathrm{Cliff}_r(C) = l - 4 $ for degree $ l \geq 5 $ curves.

ABSTRACT

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

Motivation & Objective

• To establish a new, stronger upper bound on the dimension of global sections of semistable vector bundles on curves lying on K3 surfaces with Picard group ℤH.
• To compute the higher rank Clifford index $ \mathrm{Cliff}_r(C) $ for such curves when $ g \geq r^2 \geq 4 $, generalizing Green and Lazarsfeld's result to higher rank bundles.
• To verify the first part of Mercat's conjecture for smooth plane curves by showing $ \mathrm{Cliff}_r(C) = l - 4 $ for degree $ l \geq 5 $ curves.
• To extend the method to other surfaces like the projective plane and del Pezzo surfaces, using Bridgeland stability conditions and Bogomolov-Gieseker-type inequalities.

Proposed method

• Utilizes Bridgeland stability conditions on the derived category $ D^b(X) $ of coherent sheaves on a surface $ X $, focusing on a two-dimensional subspace of stability conditions.
• Applies the Harder-Narasimhan polygon method to bound $ h^0(C,E) $ via the central charge and slope behavior as $ Z(\mathcal{O}_X) \to 0 $, leveraging wall-crossing techniques.
• Employs the construction of Lazarsfeld-Mukai bundles $ E_{C,A} $ on $ X $ via the exact sequence $ 0 \to E_{C,A}^\vee \to H^0(C,A) \otimes \mathcal{O}_X \to A \to 0 $, whose restrictions compute the Clifford index.
• Derives bounds on $ h^0(C,E) $ using convexity of the Harder-Narasimhan polygon and geometric constraints on slopes and Chern characters.
• Applies the method to the projective plane by analyzing the Harder-Narasimhan polygon within a triangle defined by Chern character bounds, leading to piecewise upper bounds on $ h^0(C,E) $.
• Uses the resulting bounds to compute $ \mathrm{Cliff}_r(C) $ and proves it equals $ l - 4 $ for smooth plane curves of degree $ l \geq 5 $.

Experimental results

Research questions

• RQ1 What is the sharp upper bound on $ h^0(C,E) $ for semistable vector bundles $ E $ of rank $ r $ and degree $ d \leq r(g-1) $ on a smooth curve $ C \subset X $, where $ X $ is a K3 surface with $ \mathrm{Pic}(X) = \mathbb{Z}H $?
• RQ2 Can the higher rank Clifford index $ \mathrm{Cliff}_r(C) $ be computed via restrictions of Lazarsfeld-Mukai bundles on $ X $ when $ g \geq r^2 \geq 4 $?
• RQ3 Does the first part of Mercat's conjecture hold for smooth plane curves of degree $ l \geq 5 $, i.e., is $ \mathrm{Cliff}_r(C) = l - 4 $?
• RQ4 How does the method based on Bridgeland stability and Harder-Narasimhan polygons extend to surfaces like the projective plane or del Pezzo surfaces?

Key findings

• For a smooth curve $ C \subset X $ on a K3 surface with $ \mathrm{Pic}(X) = \mathbb{Z}H $, the bound $ h^0(C,E) < r + \frac{g}{4r(g-1)^2}d^2 + \frac{r}{g} $ holds for semistable $ E $ with $ d \leq r(g-1) $, which is stronger than the classical higher rank Clifford theorem.
• When $ g \geq r^2 \geq 4 $, the rank $ r $ Clifford index is $ \mathrm{Cliff}_r(C) = \frac{2}{r}(g-1) - \frac{2}{r}\left\lfloor \frac{g}{r}\right\rfloor $, computed via restrictions of Lazarsfeld-Mukai bundles.
• For smooth plane curves of degree $ l \geq 5 $, the rank $ r $ Clifford index is $ \mathrm{Cliff}_r(C) = l - 4 $, confirming the first part of Mercat's conjecture.
• The upper bound on $ h^0(C,E) $ for plane curves is piecewise: $ r + \left(\frac{3}{2l} + \frac{d}{2rl^2}\right)d $ if $ d \geq rl $, and $ \max\{3r + d - rl, r + \frac{rl + r}{rl^2 - d}d\} $ if $ r(l-1) \leq d < rl $.
• The bound on $ h^0(C,E) $ is sharp enough to show that $ \mathrm{Cliff}_r(C) = l - 4 $, with equality achieved by $ E = \mathcal{O}_C(1)^{\oplus r} $, and no bundle with $ h^0(C,E) \geq 2r $ yields a smaller Clifford index.
• The method generalizes beyond K3 surfaces: it applies to the projective plane, where the same Clifford index result holds, and suggests applicability to other surfaces with strong Bogomolov-Gieseker inequalities.