Bridgeland Stability With a View Towards Classical Algebraic Geometry

The theory of Bridgeland stability was originally introduced to study question about mirror symmetry. Since then it has found many unexpected applications to more classical subjects in algebraic geometry such as birational geometry of moduli spaces, Brill-Noether theory, and space curves. This mini-course will give an introduction to the theory of stability conditions on surfaces and threefolds, and show how they can be applied to obtain various interesting results.

1. Lecture 1
The first lecture will start with some motivation. We will look at classical results about space curves (see [Har87]), and talk about Lazarfeld’s Brill-Noether theorem for curves on K3 surfaces from [Laz86]. In the second hour, we will recall basic facts about derived categories that will be crucial in the following lectures ([Huy06]). In particular, we will talk about t-structures which are fundamental in the theory of stability conditions ([BBD82]).

2. Lecture 2
In the second lecture, we will introduce Mumford’s stability of vector bundles on curves and how Bridgeland stability is a direct generalization of this idea. The original definition of Bridgeland stability is found in [Bri07]. We will proceed to explain how they can be constructed on surfaces according to [AB13, Bri08]. Moreover, we will discuss a conjectural construction on threefolds due to [BMT14]. For the exposition we will follow [MS17].

3. Lecture 3
In this lecture, we will explain how some of the results explained in lecture 1 can be proved or extended with the use of stability conditions. The references for this are [Bay18, MS18].

4. Lecture 4
Bayer and Macrì introduced a nef divisor on moduli spaces of Bridgeland-semistable objects in [BM14]. We will explain this construction and how it can be used to study the birational geometry of some moduli spaces. The first examples can be found in the last section of [ABCH13] for Hilbert schemes of points on P2. We will also discuss a result about nef cones from [BHL+16].

[AB13] D. Arcara and A. Bertram. Bridgeland-stable moduli spaces for K-trivial surfaces. J. Eur. Math. Soc. (JEMS), 15(1):1–38, 2013. With an appendix by Max Lieblich.
[ABCH13] D. Arcara, A. Bertram, I. Coskun, and J. Huizenga. The minimal model program for the Hilbert scheme of points on P2 and Bridgeland stability. Adv. Math., 235:580–626, 2013.
[Bay18] A. Bayer. Wall-crossing implies Brill-Noether: applications of stability conditions on surfaces. In Algebraic geometry: Salt Lake City 2015, volume 97 of Proc. Sympos. Pure Math., pages 3–27. Amer. Math. Soc., Providence, RI, 2018.
[BBD82] A. A. Beılinson, J. Bernstein, and P. Deligne. Faisceaux pervers. In Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Ast´erisque, pages 5–171. Soc. Math. France, Paris, 1982.
[BHL+16] B. Bolognese, J. Huizenga, Y. Lin, E. Riedl, B. Schmidt, M. Woolf, and X. Zhao. Nef cones of Hilbert schemes of points on surfaces. Algebra Number Theory, 10(4):907–930, 2016.
[BM14] A. Bayer and E. Macrì. Projectivity and birational geometry of Bridgeland moduli spaces. J. Amer. Math. Soc., 27(3):707–752, 2014.
[BMT14 ] A. Bayer, E. Macrì, and Y. Toda. Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities. J. Algebraic Geom., 23(1):117–163, 2014.
[Bri07] T. Bridgeland. Stability conditions on triangulated categories. Ann. of Math. (2), 166(2):317–345, 2007.
[Bri08] T. Bridgeland. Stability conditions on K3 surfaces. Duke Math. J., 141(2):241–291, 2008.
[Har87] R. Hartshorne. On the classification of algebraic space curves. II. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 145–164. Amer. Math. Soc., Providence, RI, 1987.
[Huy06] D. Huybrechts. Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2006.
[Laz86] R. Lazarsfeld. Brill-Noether-Petri without degenerations. J. Differential Geom., 23(3):299–307, 1986.
[MS17] E. Macr`ı and B. Schmidt. Lectures on Bridgeland stability. In Moduli of curves, volume 21 of Lect. Notes
Unione Mat. Ital., pages 139–211. Springer, Cham, 2017.
[MS18] E. Macr`ı and B. Schmidt. Derived categories and the genus of space curves, 2018. arXiv:1801.02709.