Compact moduli of plane curves

We construct a compactification M_d of the moduli space of plane curves of degree d. We regard a plane curve C as a surface-divisor pair (P^2,C) and define M_d as a moduli space of pairs (X,D) where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack M_d is smooth and the degenerate surfaces X can be described explicitly.Comment: 46 pages. Final version to be published in Duke Mathematical Journa

The moduli space of curves is rigid

We prove that the moduli stack of stable curves of genus g with n marked points is rigid, i.e., has no infinitesimal deformations. This confirms the first case of a principle proposed by Kapranov. It can also be viewed as a version of Mostow rigidity for the mapping class group.Comment: 11 pages. v2: Proof rewritten to avoid use of log structures. Example of nonrigid moduli space of surfaces adde

Homological mirror symmetry for log Calabi-Yau surfaces

Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w: M \to \mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b \text{Coh}(Y)$, and the wrapped Fukaya category $\mathcal{W} (M)$ is isomorphic to $D^b \text{Coh}(Y \backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D$. We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y,D)$, notably in work of Auroux-Katzarkov-Orlov and Abouzaid.Comment: Comments welcome

Mirror symmetry for log Calabi-Yau surfaces I

We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga's 1981 cusp conjecture.Comment: 144 pages, 3 figures, Second version significantly shorter, 109 pages. The first version has a lot of material (particularly in the introduction and material on cyclic quotient singularities) which does not appear in the new version. Download version 1 if this material is desired. Third and final version, small changes from Version 2, to appear in Publ. IHE

Canonical singularities of orders over surfaces

We classify the possible ramification data and etale local structure of orders over surfaces with canonical singularities.Comment: This contains major revisions, primarily to help introduce the reader to the minimal model program for orders on surface