Toric variety

About: Toric variety is a research topic. Over the lifetime, 2630 publications have been published within this topic receiving 65604 citations. The topic is also known as: torus embedding.

Theory of multi-fans

Abstract: We introduce the notion of a multi-fan. It is a generalization of that of a fan in the theory of toric variety in algebraic geometry. Roughly speaking a toric variety is an algebraic variety with an action of algebraic torus of the same dimension as that of the variety, and a fan is a combinatorial object associated with the toric variety. Algebro-geometric properties of the toric variety can be described in terms of the associated fan. We develop a combinatorial theory of multi-fans and define ``topological invariants'' of a multi-fan. A smooth manifold with an action of a compact torus of half the dimension of the manifold and with some orientation data is called a torus manifold. We associate a multi-fan with a torus manifold, and apply the combinatorial theory to describe topological invariants of the torus manifold. A similar theory is also given for torus orbifolds. As a related subject a generalization of the Ehrhart polynomial concerning the number of lattice points in a convex polytope is discussed.

Basic properties of convex polytopes

Abstract: Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial optimization. In this chapter we try to give a short introduction, provide a sketch of “what polytopes look like” and “how they behave,” with many explicit examples, and briefly state some main results (where further details are in the subsequent chapters of this Handbook). We concentrate on two main topics:

Recent developments in toric geometry

Abstract: This paper will appear in the Proceedings of the 1995 Santa Cruz Summer Institute. The paper is a survey of recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic geometry, combinatorics and mirror symmetry.

Mordell-Weil torsion and the global structure of gauge groups in F-theory

Abstract: We study the global structure of the gauge group G of F-theory compactified on an elliptic fibration Y. The global properties of G are encoded in the torsion subgroup of the Mordell-Weil group of rational sections of Y. Generalising the Shioda map to torsional sections we construct a specific integer divisor class on Y as a fractional linear combination of the resolution divisors associated with the Cartan subalgebra of G. This divisor class can be interpreted as an element of the refined coweight lattice of the gauge group. As a result, the spectrum of admissible matter representations is strongly constrained and the gauge group is non-simply connected. We exemplify our results by a detailed analysis of the general elliptic fibration with Mordell-Weil group ℤ2 and ℤ3 as well as a further specialization to ℤ ⊕ ℤ2. Our analysis exploits the representation of these fibrations as hypersurfaces in toric geometry.

Vanishing theorems on toric varieties

Abstract: We use Cox's description for sheaves on toric varieties and results about local cohomology with respect to monomial ideals to give a characteristic-free approach to vanishing results on toric varieties. As an application, we give a proof of a strong version of Fujita's Conjecture in the case of toric varieties. We also prove that every sheaf on a toric variety corresponds to a module over the homogeneous coordinate ring, generalizing Cox's result for the simplicial case.