BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
نویسندگان
چکیده
Algebraic Geometry is a deep subject of remarkable power that originated in the study of concrete objects, as it is concerned with the geometry of solutions to polynomial equations. It is this interplay between theoretical abstraction and tangible examples that gives the subject relevance and a reach into other areas of mathematics and its applications. There is no better subdiscipline of algebraic geometry than that of toric varieties for observing the application of its ideas and methods on a wide variety of explicit geometric objects. While toric varieties have been around as long as algebraic geometry—the familiar Cartesian plane and the projective plane are both toric varieties as is the parabola y = x—the subject of toric varieties began with the foundational paper of Demazure [5]. Toric varieties arose as a byproduct of his main interest, groups of rational automorphisms of k (k a field) that contain a split torus, (k). These groups turn out to be automorphism groups of smooth toric varieties, which Demazure defined and classified as schemes over Z. These are obtained by adding certain points at infinity to a split torus, and the classification involved objects from geometric combinatorics called éventails (fans en anglais) encoding the added points. Fans are certain collections of cones in Z, with each cone corresponding to an orbit of (k) in the toric variety, and this correspondence is contravariant. For example, the Cartesian k and projective P(k) planes are compactifications of the torus (k) encoded by the following fans.
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BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
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