# Introduction to toric varieties

@inproceedings{Brasselet2004IntroductionTT, title={Introduction to toric varieties}, author={Jean-Paul Brasselet}, year={2004} }

The course given during the School and Workshop “The Geometry and Topology of Singularities”, 8-26 January 2007, Cuernavaca, Mexico is based on a previous course given during the 23o Coloquio Brasileiro de Matematica (Rio de Janeiro, July 2001). It is an elementary introduction to the theory of toric varieties. This introduction does not pretend to originality but to provide examples and motivation for the study of toric varieties. The theory of toric varieties plays a prominent role in various…

## Figures from this paper

## 1,215 Citations

TORIC FANO VARIETIES AND THEIR POLYTOPES

My research is chiefly concerned with classifying toric Fano varieties with certain classes of singularities. For the purposes of this brief talk we shall restrict our attention to smooth varieties.…

Toric Hyperkähler Varieties Tamás Hausel

Extending work of Bielawski-Dancer [3] and Konno [12], we develop a theory of toric hyperkähler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a…

Quiver representations in toric geometry

- Mathematics
- 2008

This article is based on my lecture notes from summer schools at the Universities of Utah (June 2007) and Warwick (September 2007). We provide an introduction to explicit methods in the study of…

Resolutions and Moduli for Equivariant Sheaves over Toric Varieties

- Mathematics
- 2003

In this thesis the combinatorial framework of toric geometry is extended to equivariant sheaves over toric varieties. The central questions are how to extract combinatorial information from the so…

Toric Topology

- Mathematics
- 2012

Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra. It has quickly grown up into a very…

Toric Hyperkahler Varieties

- Mathematics
- 2002

Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study…

Arithmetic geometry of toric varieties. metrics, measures and heights

- Mathematics
- 2014

We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and…

Toric quotients and flips

- 2008

As my contribution to these proceedings, I will discuss the geometric invariant theory quotients of toric varieties. Specifically, I will show that quotients of the same problem with respect to…

SPACES OF ARCS IN BIRATIONAL GEOMETRY

- 2006

These lecture notes have been prepared for the Summer school on ”Moduli spaces and arcs in algebraic geometry”, Cologne, August 2007. The goal is to explain the relevance of spaces of arcs to…

Ju n 19 95 Residues in Toric Varieties June 22 , 1995

Introduction Toric residues provide a tool for the study of certain homogeneous ideals of the homogeneous coordinate ring of a toric variety—such as those appearing in the description of the Hodge…

## References

SHOWING 1-10 OF 17 REFERENCES

Introduction to Toric Varieties.

- Mathematics
- 1993

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic…

Topological methods in algebraic geometry

- Mathematics
- 1966

Introduction Chapter 1: Preparatory material 1. Multiplicative sequences 2. Sheaves 3. Fibre bundles 4. Characteristic classes Chapter 2: The cobordism ring 5. Pontrjagin numbers 6. The ring…

THE GEOMETRY OF TORIC VARIETIES

- Mathematics
- 1978

ContentsIntroductionChapter I. Affine toric varieties § 1. Cones, lattices, and semigroups § 2. The definition of an affine toric variety § 3. Properties of toric varieties § 4. Differential forms on…

Picard Groups of compact toric Varieties and combinatorial Classes of Fans

- Mathematics
- 1993

We consider the question what can be said about the rank of the Picard group Pic Xσ of a compact toric variety Xσ if we know only the combinatorial type of the associated fan σ. We establish upper…

The Topology of Torus Actions on Symplectic Manifolds

- Mathematics
- 1991

This is an extended second edition of "The Topology of Torus Actions on Symplectic Manifolds" published in this series in 1991. The material and references have been updated. Symplectic manifolds and…

On simple polytopes

- Mathematics
- 1993

SummaryLetP be a simpled-polytope ind-dimensional euclidean space
$$\mathbb{E}^d $$
, and let Π(P) be the subalgebra of the polytope algebra Π generated by the classes of summands ofP. It is shown…

The number of faces of a simplicial convex polytope

- Mathematics
- 1980

Let P be a simplicial convex d-polytope with fi = fi(P) faces of dimension i. The vector f(P) = (f. , fi ,..., fdel) is called the f-vector of P. In 1971 McMullen [6; 7, p. 1791 conjectured that a…

Sufficiency of McMullen's conditions for $f$-vectors of simplicial polytopes

- Mathematics
- 1980

For convex d-polytope P let ft{P) equal the number of faces of P of dimension i, 0 < i < d 1. f(P) = (f0(P)9 . . . , fd^QP)) is called the f vector of P An important combinatorial problem is the…

Using Algebraic Geometry

- Mathematics
- 1998

Introduction.- Solving Polynomial Equations.- Resultants.- Computation in Local Rings.- Modules.- Free Resolutions.- Polytopes, Resultants, and Equations.- Integer Programming, Combinatorics, and…

Eventails et varietes toriques

- Mathematics
- 1980

© Séminaire sur les singularités des surfaces (École Polytechnique), 1976-1977, tous droits réservés. L’accès aux archives du séminaire sur les singularités des surfaces implique l’accord avec les…