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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
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
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
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
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
Basepoint freeness for nef and big line bundles in positive characteristic
A necessary and sufficient condition is given for semi-ampleness of a numerically effective (nef) and big line bundle in positive characteristic. One application is to the geometry of the universalExpand
Towards the ample cone of \overline{}_{,}
In this paper we study the ample cone of the moduli space $\mgn$ of stable $n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on $\mgn$ is ample iff it has positiveExpand
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
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