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Canonical bases for cluster algebras
In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonicalExpand
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Mori dream spaces and GIT.
The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We callExpand
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Intersection theory of moduli space of stable N-pointed curves of genus zero
We give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the Chow ring
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Quotients by Groupoids
We show that if a flat group scheme acts properly, with finite stabilizers, on an algebraic space, then a quotient exists as a separated algebraic space. More generally we show any flat groupid forExpand
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Mirror symmetry for log Calabi-Yau surfaces I
We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family isExpand
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In this paper we study the ample cone of the moduli space M g,n of stable n-pointed curves of genus g. Our motivating conjecture is that a divisor on M g,n is ample iff it has positive intersectionExpand
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Rational curves on quasi-projective surfaces
Introduction and statement of results Glossary of notation and conventions Gorenstein del Pezzo surfaces Bug-eyed covers Log deformation theory Criteria for log uniruledness Reduction toExpand
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Log abundance theorem for threefolds
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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 Expand
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Birational geometry of cluster algebras
We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the LaurentExpand
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