Topic

# Toric variety

About: Toric variety is a(n) research topic. Over the lifetime, 2630 publication(s) have been published within this topic receiving 65604 citation(s). The topic is also known as: torus embedding.

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Book
12 Jul 1993
TL;DR: In this article, a mini-course is presented to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications, concluding with Stanley's theorem characterizing the number of simplicies in each dimension in a convex simplicial polytope.
Abstract: 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 geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

3,330 citations

Journal ArticleDOI

2,757 citations

Book
22 Dec 2012
TL;DR: In this article, the authors introduce the notion of Tangent Spaces to Grassmannians and describe the relationship between them and regular functions and maps. But they do not discuss their application in the context of dimension computations.
Abstract: 1: Affine and Projective Varieties. 2: Regular Functions and Maps. 3: Cones, Projections, and More About Products. 4: Families and Parameter Spaces. 5: Ideals of Varieties, Irreducible Decomposition. 6: Grassmannians and Related Varieties. 7: Rational Functions and Rational Maps. 8: More Examples. 9: Determinantal Varieties. 10: Algebraic Groups. 11: Definitions of Dimension and Elementary Examples. 12: More Dimension Computations. 13: Hilbert Functions and Polynomials. 14: Smoothness and Tangent Spaces. 15: Gauss Maps, Tangential and Dual Varieties. 16: Tangent Spaces to Grassmannians. 17: Further Topics Involving Smoothness and Tangent Spaces. 18: Degree. 19: Further Examples and Applications of Degree. 20: Singular Points and Tangent Cones. 21: Parameter Spaces and Moduli Spaces. 22: Quadrics.

1,346 citations

BookDOI
31 Jan 1993

1,297 citations

Book
01 Jan 1999
TL;DR: The quintic threefold Toric geometry Mirror symmetry constructions Hodge theory and Yukawa couplings Moduli spaces Gromov-Witten invariants Quantum cohomology Localization Quantum differential equations The mirror theorem Conclusion Singular varieties Physical theories Bibliography Index as mentioned in this paper
Abstract: Introduction The quintic threefold Toric geometry Mirror symmetry constructions Hodge theory and Yukawa couplings Moduli spaces Gromov-Witten invariants Quantum cohomology Localization Quantum differential equations The mirror theorem Conclusion Singular varieties Physical theories Bibliography Index.

1,068 citations

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##### Performance
###### Metrics
No. of papers in the topic in previous years
YearPapers
20222
2021104
2020105
201999
201893
2017102