Related papers: Recent developments in toric geometry
These lecture notes are based on an introductory course given by the author at the summer school "Noncommutative Algebraic Geometry" at MSRI in June 2012. The emphasis throughout is on examples to illustrate the many different facets of…
Our paper develops a theory of Poisson slices and a uniform approach to their partial compactifications. The theory in question is loosely comparable to that of symplectic cross-sections in real symplectic geometry.
These are the notes from a survey talk given at Arbeitstagung 2001 covering the author's work with Lev Borisov and Sorin Popescu on toric varieties, modular forms, and equations of modular curves.
We present a construction of noncommutative double mirrors to complete intersections in toric varieties. This construction unifies existing sporadic examples and explains the underlying combinatorial and physical reasons for their…
Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction…
The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in…
These notes are based on a series of lectures given by the author at the Max Planck Institute for Mathematics in the Sciences in Leipzig. Addressed topics include affine and projective toric varieties, abstract normal toric varieties from…
Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the…
In this paper we will survey some recent developments in the last decade or so on variation of Geometric Invariant Theory and its applications to Birational Geometry such as the weak Factorization Theorems of nonsingular projective…
We give a survey of results on the geometry of complex algebraic Q-acyclic surfaces, so-called 'Q-homology planes', including some recent results.
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent r{\^o}le. This selection is largely…
In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my…
We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms…
This paper studies two related subjects. One is some combinatorics arising from linear projections of polytopes and fans of cones. The other is quotient varieties of toric varieties. The relation is that projections of polytopes are related…
Survey article on the geometry of spherical varieties. Invited survey for Transformation Groups.
We present two examples in toric geometry concerning the relationship between toric and quasitoric manifolds, and provide the sufficient conditions on the base polytope and characteristic map so that the resulting quasitoric manifold is…
We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical…
In this appendix, we summarize known results on the geometry of Severi varieties on toric surfaces - the varieties parameterizing integral curves of a given geometric genus in a given linear system. Till the last decade, Severi varieties…
This is an expository article describing recent results of the authors and David Nadler on microlocalization, the Fukaya category, and coherent sheaves on toric varieties. The original papers are arXiv:math/0604379, arXiv:math/0612399 and…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.