Wall-crossing implies Brill-Noether: Applications of stability conditions on surfaces

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.

Brill–Noether theory for curves on generic abelian surfaces

“…of semistable objects that is part of the Jordan-Hölder filtration of F . However, unlike in the case of K3 surfaces treated in [Bay16], we may have h 0 (F ′ ) = 0; in fact, this may be necessary for the extension F to exist, as otherwise Ext 1 (F ′ , O X ) might be too small.…”

“…We refer also to [MS16] or [Bay16] for more details and a sketch of the proof. Up to an action of the universal cover of GL + 2 (R), the above theorem in fact describes an entire component of Stab(X), but that fact will be irrelevant for us.…”

Mukai’s program (reconstructing a K3 surface from a curve) via wall-crossing

“…There exists a region in the space of stability conditions where the Brill-Noether behaviour of stable objects is completely controlled by the nearby Brill-Noether wall. This wall destabilises objects with non-zero global sections, and arguments similar to [Bay16] show that the Brill-Noether loci are mostly of expected dimension. Our first key result, Proposition 3.4, gives an extension to unstable objects: it gives a bound on the number of global sections in terms of their mass, i.e.…”

“…For any coherent sheaf, there exists a chamber which is called Gieseker chamber, where the notion of Bridgeland stability coincides with the old notion of Gieseker stability. Unlike the case of push-forward of line bundles considered in [Bay16], the Brill-Noether wall is not adjacent to the Gieseker chamber for the push-forward of semistable vector bundles F of higher ranks on the curve C. However, the wall that bounds the Gieseker chamber provides an extremal polygon which contains the Harder-Narasimhan polygon, see e.g. Lemma 4.4.…”

On stability conditions for the quintic threefold

“…Instead, we need to estimate the hom(O X , E) for Brill-Noether stable objects. The original idea for this estimation via stability conditions, as far as the author knows, first appears in [Bay16] which reproves the Brill-Noether generality of certain curves on K3 surfaces as that in [Laz86]. The estimation for hom(O X , E) necessarily relies on a stronger Bogomolov-Gieseker type inequality for the second Chern character of slope stable objects, which is the statement of Theorem 5.5.…”