Leseprobe
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Introduction to Toric Varieties. (AM-131), Volume 131
William Fulton
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
Kundenbewertungen
Open set, Mathematical induction, Pick's theorem, Dedekind sum, Divisor, Projective space, Line bundle, Graded ring, Resolution of singularities, Alexander Grothendieck, Commutative property, Natural number, Moment map, Surjective function, Fundamental group, Summation, Subgroup, Hodge theory, Convex polytope, Topology, Equivariant K-theory, Line segment, Lattice (group), Monotonic function, Leray spectral sequence, Polytope, Discrete valuation ring, Regular sequence, Affine space, Big O notation, Tangent bundle, Dimension (vector space), Addition, Serre duality, Quadric, Unit disk, Dimension, Invertible sheaf, Toric variety, Zariski topology, Intersection theory, Divisor (algebraic geometry), Simplicial complex, Hirzebruch surface, Alexander duality, Complete intersection, Limit point, Weil conjecture, Moduli space, Algebraic curve, Equation, Affine variety, Automorphism, Linear subspace, Spectral sequence, Equivalence class, Picard group, Euler characteristic, Explicit formula, Quotient space (topology), Isoperimetric inequality, Subset, Codimension, Grassmannian, Cohomology, H-vector, Homomorphism, Atiyah–Singer index theorem, Betti number, Riemann–Roch theorem