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Riemann Surfaces

This course serves as a follow-up of Math 506. The students are assumed to know complex analysis and have some basic knowledge in algebra and topology. The courses are divided into two parts: The… Expand

Canonical bases for cluster algebras

- M. Gross, P. Hacking, S. Keel, M. Kontsevich
- Mathematics
- 5 November 2014

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 canonical… Expand

Mirror symmetry for log Calabi-Yau surfaces I

- M. Gross, P. Hacking, S. Keel
- Mathematics
- 24 June 2011

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 is… Expand

Compact moduli of plane curves

- P. Hacking
- Mathematics
- 22 October 2003

We construct a compactication Md of the moduli space of plane curves of degreed. We regard a plane curveC P 2 as a surface-divisor pair (P 2 ;C) and dene Md as a moduli space of pairs (X;D) where X… Expand

Smoothable del Pezzo surfaces with quotient singularities

- P. Hacking, Yuri G. Prokhorov
- Mathematics
- Compositio Mathematica
- 11 August 2008

Abstract We classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a ℚ-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in the… Expand

Birational geometry of cluster algebras

- M. Gross, P. Hacking, S. Keel
- Mathematics
- 10 September 2013

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 Laurent… Expand

IRAS Faint Source Catalogue, version 2.0.

- M. Moshir, G. Copan, +8 authors T. Chester
- Physics
- 1990

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

- P. Hacking, S. Keel, J. Tevelev
- Mathematics
- 18 February 2007

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

Moduli of surfaces with an anti-canonical cycle

- M. Gross, P. Hacking, S. Keel
- Mathematics
- Compositio Mathematica
- 27 November 2012

Abstract We prove a global Torelli theorem for pairs $(Y,D)$ where $Y$ is a smooth projective rational surface and $D\in |-K_{Y}|$ is a cycle of rational curves, as conjectured by Friedman in 1984.… Expand

The homology of tropical varieties

- P. Hacking
- Mathematics
- 12 November 2007

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… Expand

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