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Torus (Geometry) --- Toric varieties --- Topology
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Combinatorial geometry --- Geometry, Algebraic --- Toric varieties
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Toric varieties. --- Embeddings, Torus --- Torus embeddings --- Varieties, Toric --- Algebraic varieties --- Mathematics --- Algebra --- Geometry --- Toric varieties --- Geometrie algebrique --- Colloque --- Varietes algebriques
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Toric varieties --- Arakelov theory --- Variétés toriques --- Arakelov, Théorie d' --- Variétés toriques. --- Arakelov, Théorie d'.
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Topological groups. Lie groups --- Functions, Zeta. --- Toric varieties --- Fonctions zêta --- Variétés toriques --- Toric varieties. --- 51 <082.1> --- Mathematics--Series --- Fonctions zêta --- Variétés toriques --- Functions, Zeta --- Embeddings, Torus --- Torus embeddings --- Varieties, Toric --- Algebraic varieties --- Zeta functions --- Arithmetical algebraic geometry --- Géométrie algébrique arithmétique
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Algebraic geometry --- Toric varieties. --- Calabi-Yau manifolds --- Noncommutative algebras. --- Geometry, Algebraic. --- Variétés toriques --- Variétés de Calabi-Yau --- Algèbres non commutatives --- Géométrie algébrique --- Calabi-Yau manifolds. --- 51 <082.1> --- Mathematics--Series --- Variétés toriques --- Variétés de Calabi-Yau --- Algèbres non commutatives --- Géométrie algébrique --- Geometry, Algebraic --- Noncommutative algebras --- Toric varieties --- Embeddings, Torus --- Torus embeddings --- Varieties, Toric --- Algebraic varieties --- Algebras, Noncommutative --- Non-commutative algebras --- Algebra --- Geometry --- Manifolds (Mathematics)
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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 "ed without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.
Algebraic geometry --- Differential geometry. Global analysis --- 512.7 --- Algebraic geometry. Commutative rings and algebras --- Toric varieties. --- 512.7 Algebraic geometry. Commutative rings and algebras --- Toric varieties --- Embeddings, Torus --- Torus embeddings --- Varieties, Toric --- Algebraic varieties --- Addition. --- Affine plane. --- Affine space. --- Affine variety. --- Alexander Grothendieck. --- Alexander duality. --- Algebraic curve. --- Algebraic group. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Big O notation. --- Characteristic class. --- Chern class. --- Chow group. --- Codimension. --- Cohomology. --- Combinatorics. --- Commutative property. --- Complete intersection. --- Convex polytope. --- Convex set. --- Coprime integers. --- Cotangent space. --- Dedekind sum. --- Dimension (vector space). --- Dimension. --- Direct proof. --- Discrete valuation ring. --- Discrete valuation. --- Disjoint union. --- Divisor (algebraic geometry). --- Divisor. --- Dual basis. --- Dual space. --- Equation. --- Equivalence class. --- Equivariant K-theory. --- Euler characteristic. --- Exact sequence. --- Explicit formula. --- Facet (geometry). --- Fundamental group. --- Graded ring. --- Grassmannian. --- H-vector. --- Hirzebruch surface. --- Hodge theory. --- Homogeneous coordinates. --- Homomorphism. --- Hypersurface. --- Intersection theory. --- Invertible matrix. --- Invertible sheaf. --- Isoperimetric inequality. --- Lattice (group). --- Leray spectral sequence. --- Limit point. --- Line bundle. --- Line segment. --- Linear subspace. --- Local ring. --- Mathematical induction. --- Mixed volume. --- Moduli space. --- Moment map. --- Monotonic function. --- Natural number. --- Newton polygon. --- Open set. --- Picard group. --- Pick's theorem. --- Polytope. --- Projective space. --- Quadric. --- Quotient space (topology). --- Regular sequence. --- Relative interior. --- Resolution of singularities. --- Restriction (mathematics). --- Resultant. --- Riemann–Roch theorem. --- Serre duality. --- Sign (mathematics). --- Simplex. --- Simplicial complex. --- Simultaneous equations. --- Spectral sequence. --- Subgroup. --- Subset. --- Summation. --- Surjective function. --- Tangent bundle. --- Theorem. --- Topology. --- Toric variety. --- Unit disk. --- Vector space. --- Weil conjecture. --- Zariski topology.
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Mathematical statistics --- Geometry, Algebraic --- Algebraic geometry -- Instructional exposition (textbooks, tutorial papers, etc.) --- Algebraic geometry -- Real algebraic and real analytic geometry -- Semialgebraic sets and related spaces. --- Algebraic geometry -- Special varieties -- Determinantal varieties. --- Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra. --- Algebraic geometry -- Tropical geometry -- Tropical geometry. --- Biology and other natural sciences -- Genetics and population dynamics -- Problems related to evolution. --- Commutative algebra -- Computational aspects and applications -- GrÃjabner bases; other bases for ideals and modules (e.g., Janet and border bases) --- Convex and discrete geometry -- Polytopes and polyhedra -- Lattice polytopes (including relations with commutative algebra and algebraic geometry) --- Operations research, mathematical programming -- Mathematical programming -- Integer programming. --- Probability theory and stochastic processes -- Markov processes -- Markov chains (discrete-time Markov processes on discrete state spaces) --- Statistics -- Instructional exposition (textbooks, tutorial papers, etc.) --- Statistics -- Multivariate analysis -- Contingency tables. --- Statistics -- Parametric inference -- Hypothesis testing. --- Commutative algebra -- Computational aspects and applications -- Solving polynomial systems; resultants.
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