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Ian Morrison and David Swinarski
GIT constructions of moduli spaces with applications to their birational geometry

We plan a series of four lectures. Three will cover the notions from geometric invariant theory (GIT) needed to apply it to construct moduli spaces and review applications of such constructions, both classic and recent, to the birational geometry of these spaces. One will deal with related computer assisted calculations using Macaulay2 and other tools. The talks will not assume any familiarity with GIT, but it will be helpful to have read at least through section 3 of the paper of Laza cited below for some basic facts that will only be quoted.

The first talk will review the basics of GIT and explain, using the model of the moduli spaces Mg of stable curves of genus g, how it is used to construct these moduli spaces as quotients of Hilbert schemes of pluricanonical models.

Subsequent talks will focus on methods for checking the GIT stability of Hilbert points, which is the main technical problem in carrying out such constructions. We'll briefly review classical and more recent asymptotic criteria in which the auxiliary degree m used to linearize the action on the Hilbert scheme can be chosen very large. The log minimal model program (LMMP) for Mg, which we will review in the final talk, has focused attention on linearizations of small fixed degree} m, particularly of canonical and bicanonical Hilbert schemes. Here checking stability requires new ideas. The basic setup for doing this using state polytopes and their dual Gröbner fans will be the main focus of the second talk.

In low genera, stability of special curves with suitable automorphism groups can be established for fixed m by symbolic computations. The third talk will deal practical issues of using software including gfan, Macaulay2, magma, and polymake to find suitable examples and analyze their stability, confirming predictions of the LMMP. We plan two short exercise sessions to allow particpants to get a bit of hands-on experience with making these calculations.

Carrying out the next stages of the LMMP calls for finding families of examples in all genera that can be analyzed synthetically. In the last talk, we will review a recent breakthough by Alper, Fedorchuk and Smyth that yields examples of Hilbert semistable canonical and bicanonical curves for all g and all $m \ge 2$. The talk will conclude by opening the black box of the LMMP. We will first explain how it predicts which GIT quotients arise as log minimal models, what loci are contracted or flipped and what singularities appear at each stage, then review recent constructions for small g.

Some good surveys of the material in the theoretical lectures are the articles GIT and moduli with a twist by Radu Laza, "Constructions of Moduli Spaces of Stable Curves and Maps" (pp. 315-369 Geometry of Riemann surfaces and their moduli spaces, International Press, Somerville MA 2009) and this article by the first speaker. Our paper Groebner techniques for low degree Hilbert stability (pp. 34-56, Experimental Mathematics 20, Issue 1, 2011) and "GIT Constructions of Log Canonical Models of Mg by Jarod Alper and Donghoon Hyeon. Our paper "Gröbner techniques for low degree Hilbert stability" (Experimental Mathematics 20, Issue 1, pp. 34-56 (2011) is a good introduction to the calculations and the work of Alper, Fedorchuk and Smyth appears in "Finite Hilbert stability of (bi)canonical curves"