Showing papers on "Toric variety published in 1988"

Convex bodies and algebraic geometry : an introduction to the theory of toric varieties

Abstract: 1. Fans and Toric Varieties.- 1.1 Strongly Convex Rational Polyhedral Cones and Fans.- 1.2 Toric Varieties.- 1.3 Orbit Decomposition, Manifolds with Corners and the Fundamental Group.- 1.4 Nonsingularity and Compactness.- 1.5 Equivariant Holomorphic Maps.- 1.6 Low Dimensional Toric Singularities and Finite Continued Fractions.- 1.7 Birational Geometry of Toric Varieties.- 2. Integral Convex Polytopes and Toric Projective Varieties.- 2.1 Equivariant Line Bundles, Invariant Cartier Divisors and Support Functions.- 2.2 Cohomology of Compact Toric Varieties.- 2.3 Equivariant Holomorphic Maps to Projective Spaces.- 2.4 Toric Projective Varieties.- 2.5 Mori's Theory and Toric Projective Varieties.- 3. Toric Varieties and Holomorphic Differential Forms.- 3.1 Differential Forms with Logarithmic Poles.- 3.2 Ishida's Complexes.- 3.3 Compact Toric Varieties and Holomorphic Differential Forms.- 3.4 Automorphism Groups of Toric Varieties and the Cremona Groups.- 4. Applications.- 4.1 Periodic Continued Fractions and Two-Dimensional Toric Varieties..- 4.2 Cusp Singularities.- 4.3 Compact Quotients of Toric Varieties.- Appendix. Geometry of Convex Sets.- A.1 Convex Polyhedral Cones.- A.2 Convex Polyhedra.- A.3 Support Functions.- A.4 The Mixed Volume of Compact Convex Sets.- A.5 Morphology for Convex Polytopes.- References.

On the classification of toric fano varieties

Abstract: Toric Fano varieties are algebraic varieties associated with a special class of convex polytopes inR?'. We extend results of V. E. Voskresenskij and A. A. Klyachko about the classification of such varieties using a purely combinatorial method of proof.

An embedding theorem for smooth projective toric varieties

Abstract: Since about 1970 an interesting connection between the theory of convex bodies and algebraic geometry has developed. It is based on the discovery of so-called toric varieties. We summarize the definition (see I-1-], [4-1, [8,1). Let E be a cell complex consisting of convex cones in R\", each being the positive linear hull of finitely many simple lattice vectors (e 7/\"). If a ~ E, let ~ be the dual cone ofa. All Laurent polynomials E a r , aj ~ C, z j := z /1 . . . z / \" , j l . . . . . j , ~ Z, only finitely many a~ ¢ 0, for which (jx . . . . . j ,) e 6 c~ Z\" form a ring R~. The (maximal) prime ideals of R~, called the spectrum Spec R~ of R~ define an affine algebraic variety X~. Using the complex structure of 22 the varieties X~ can be glued together in a natural way; the result is called a torie variety X~. Each X~ is open and dense in X~ as in the example of affine charts of a projective space. X~ is compact if and only if 22 covers R\". A basic result of toric geometry is the following: X~ is projective, that is, isomorphically embeddable in a projective space if and only if E can be obtained by projecting the faces of a convex polytope. A compact projective X~ can also be characterized by a polytope A obtained as intersection of translated cones: