Related papers: On toric varieties and modular forms
We describe a very large class of conjectural relations in the tautological ring of the moduli space $\bar{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points, extending and generalizing the Faber-Zagier relations. These notes…
The automorphism group of a projective bundle P(E) over a simplicial toric variety is described when the bundle E is a direct sum of line bundles. Applications to study of moduli of complete intersections on toric varieties, including…
In this article we review some recent developments in heterotic compactifications. In particular we review an ``inherently toric'' description of certain sheaves, called equivariant sheaves, that has recently been discussed in the physics…
Lectures on the construction on the moduli space of principal bundles, given in the Mini-School on Moduli Spaces at the Banach center (Warsaw) 26-30 April 2005.
Recent advances in computational techniques for $K$-theory allow us to describe the $K$-theory of toric varieties in terms of the $K$-theory of fields and simple cohomological data.
An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is…
Summary of main work 1999-2012
This is a review article on modular categories, extending an invited talk given at the workshop "Categorical (co)algebraic methods in quantum informatics and linguistics", Oxford, October 29-31, 2010. To appear in C. Heunen, M. Sadrzadeh,…
We establish a natural and geometric 1-1 correspondence between projective toric varieties of dimension $n$ and horofunction compactifications of $\mathbb{R}^n$ with respect to rational polyhedral norms. For this purpose, we explain a…
These notes contain a brief introduction to the construction of toric Calabi--Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and…
These lecture notes are meant to serve as an introduction to some geometric constructions and techniques (in particular the ones of toric geometry) often employed by the physicist working on string theory compactifications. The emphasis is…
We prove a conjecture of Shokurov which characterises toric varieties using log pairs.
This paper surveys some applications of moduli theory to issues concerning the distribution of rational points on algebraic varieties. It will appear on the proceedings of the Fano Conference.
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation…
These notes are based on three lectures given at the 2013 CIME/CIRM summer school. The purpose of this series of lectures is to introduce the notion of a toric fibration and to give its geometrical and combinatorial characterizations.…
We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen…
Relationships between moduli spaces of curves and sheaves on 3-folds are presented starting with the Gromov-Witten/Donaldson-Thomas correspondence proposed more than 20 years ago with D. Maulik, N. Nekrasov, and A. Okounkov. The descendent…
We construct the moduli space for equivalence classes of n-pointed tropical curves of genus g, together with its compactification given by weighted tropical curves, and establish some of its basic topological properties. We compare it to…
We explore the positive geometry of statistical models in the setting of toric varieties. Our focus lies on models for discrete data that are parameterized in terms of Cox coordinates. We develop a geometric theory for computations in…