Introduction to Toric Varieties (eBook)

William Fulton (Autor)

eBook Download: PDF

2016
Princeton University Press (Verlag)
978-1-4008-8252-6 (ISBN)

Infos

FultonWilliam:

William Fulton is Professor of Mathematics at the University of Chicago.William Fulton is Professor of Mathematics at the University of Chicago.

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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 quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

Produkt-Details

• Erscheint lt. Verlag: 2.3.2016
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Graphentheorie
• Schlagworte: Addition • affine plane • Affine space • affine variety • Alexander duality • Alexander Grothendieck • 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 • Dimension (vector space) • Direct proof • discrete valuation • Discrete valuation ring • Disjoint union • Divisor • Divisor (algebraic geometry) • Dual basis • dual space • Equation • equivalence class • Equivariant K-theory • Euler characteristic • exact sequence • Explicit formula • Facet (geometry) • fundamental group • graded ring • Grassmannian • Hirzebruch surface • Hodge Theory • Homogeneous Coordinates • Homomorphism • h-vector • hypersurface • Intersection theory • Invertible matrix • Invertible sheaf • Isoperimetric inequality • Lattice (group) • Leray spectral sequence • Limit point • Linear subspace • line bundle • Line segment • 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
• ISBN-10: 1-4008-8252-4 / 1400882524
• ISBN-13: 978-1-4008-8252-6 / 9781400882526