Paul Hacking

Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces

AbstractWe give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification

The homology of tropical varieties

M_{0,n}\subset\overline{M}_{0,n}

Moduli of Surfaces with an Anti-Canonical Cycle

.

Compact Moduli Spaces of Surfaces and Exceptional Vector Bundles

Given a closed subvarietyX of an algebraic torusT, the associated tropical variety is a polyhedral fan in the space of 1-parameter subgroups of the torus which describes the behaviour of the subvariety at infinity. We show that the link of the origin has only top rational homology if a genericity condition is satisfied.

Canonical Bases for Cluster Algebras

We prove a global Torelli theorem for pairs

Mirror symmetry for log Calabi-Yau surfaces I

(Y,D)

Birational Geometry of Cluster Algebras

where

Theta functions on varieties with effective anti-canonical class

Y

The moduli space of curves is rigid

is a smooth projective rational surface and

Degenerations of del Pezzo surfaces I

D\in |-K_{Y}|