[Paper Review] Wall-Crossing implies Brill-Noether. Applications of stability conditions on surfaces

Arend Bayer|arXiv (Cornell University)|Apr 27, 2016

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper demonstrates that wall-crossing in Bridgeland stability conditions on K3 surfaces provides a powerful framework to reprove Lazarsfeld's Brill-Noether theorem, showing that the Brill-Noether variety $W^r_d(C)$ on a smooth curve $C$ in $|H|$ has expected dimension for all such curves, not just generic ones. The method leverages moduli spaces of stable objects and Harder-Narasimhan filtrations to control Ext groups and establish smoothness, offering a derived category-based alternative to classical vector bundle techniques.

ABSTRACT

Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces. We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces. The intended reader is an algebraic geometer with a limited working knowledge of derived categories. This article is based on the author's talk at the AMS Summer Institute on Algebraic Geometry in Utah, July 2015.

Motivation & Objective

• To reprove Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces using wall-crossing in Bridgeland stability conditions.
• To demonstrate that stability conditions provide a systematic method to study birational geometry of moduli spaces of sheaves.
• To show that the dimension of the Brill-Noether locus $W^r_d(C)$ is expected for every curve $C \in |H|$, not only for generic curves.
• To connect derived category techniques with classical algebraic geometry problems, such as projective embeddings and stability of vector bundles.
• To survey recent applications of stability conditions on surfaces, including flips, contractions, and nef cone computations.

Proposed method

• Construct Bridgeland stability conditions $\sigma_{\alpha,\beta}$ on the derived category $D^b(X)$ of a K3 surface $X$ with polarization $H$.
• Use the moduli space $M_\sigma(\mathbf{v})$ of $\sigma$-stable objects with fixed Chern character $\mathbf{v}$ as a birational model of the Hilbert scheme or sheaf moduli space.
• Apply wall-crossing to track destabilizing objects and control Harder-Narasimhan filtrations of twisted objects $E(-3)$.
• Establish smoothness of moduli spaces by proving $\mathrm{Ext}^2(F,F) = 0$, using Serre duality and phase control in the stability condition.
• Leverage the Positivity Lemma and S-equivalence of objects to identify nef divisors and describe effective and nef cones.
• Use the stability of Mukai-Lazarsfeld bundles restricted to curves to derive new counterexamples to Mercat’s conjecture.

Experimental results

Research questions

• RQ1 Can wall-crossing in Bridgeland stability conditions be used to reprove classical results like Lazarsfeld’s Brill-Noether theorem?
• RQ2 Does the derived category framework allow for stronger conclusions than classical methods, such as uniform dimension results across all curves in |H|?
• RQ3 How do stability conditions control the birational geometry of moduli spaces of sheaves on surfaces?
• RQ4 What is the relationship between the location of destabilizing walls and invariants like Castelnuovo-Mumford regularity?
• RQ5 Can stability conditions be used to determine the nef cone of Hilbert schemes or sheaf moduli spaces on rational surfaces?

Key findings

• The Brill-Noether variety $W^r_d(C)$ has expected dimension for every smooth curve $C \in |H|$, not just generic ones, under Assumption (*).
• The moduli space $M_\sigma(\mathbf{v})$ of $\sigma$-stable objects is smooth, as $\mathrm{Ext}^2(F,F) = 0$ is shown via phase control in the Harder-Narasimhan filtration of $F(-3)$.
• Wall-crossing induces the contraction from Gieseker-moduli spaces to Uhlenbeck spaces and realizes Thaddeus flips via sequences of walls.
• For surfaces of Picard rank one and large $n$, the nef cone of $\mathrm{Hilb}^n(X)$ can be determined using the Positivity Lemma and minimal degree curves.
• The restriction of the Mukai-Lazarsfeld bundle $M_L$ to any curve in $|H|$ is slope-stable, leading to new counterexamples to Mercat’s conjecture.
• The wall where an ideal sheaf in $\mathrm{Hilb}^n(\mathbb{P}^2)$ destabilizes is closely related to its Castelnuovo-Mumford regularity.