Special linear Systems on Toric Varieties (original) (raw)
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The thesis provides an introduction into the theory of affine and abstract toric varieties. In the first chapter, tools from algebraic geometry indispensable for the comprehension of the topic are introduced. Many properties of convex polyhedral cones and affine toric varieties are proven and discussed in detail as is the deep connection between the two objects. The second chapter establishes the notion of an abstract variety and translates obtained results to this more general setting, giving birth to the theory of abstract toric varieties and the closely associated theory of fans. Finally, an algorithmic approach to the resolution of singularities on toric surfaces and its relation to continued fractions is revealed.
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We study the subvariety of singular sections, the discriminant, of a base point free linear system |L| on a smooth toric variety X. On one hand we describe pairs (X, L) for which the discriminant is of low dimension. Precisely, we collect some bounds on this dimension and classify those pairs whose dimension differs the bound less than or equal to two. On the other hand we study the degree of the discriminant for some relevant families on polarized toric varieties, describing their minimal values and the region of the ample cone where this minimal is attained.
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Some remarks on the symplectic and Kähler geometry of toric varieties
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Let M be a projective toric manifold. We prove two results concerning respectively Kähler-Einstein submanifolds of M and symplectic embeddings of the standard euclidean ball in M. Both results use the well-known fact that M contains an open dense subset biholomorphic to C n. Theorem 1.2. Let (M, ω) a toric manifold endowed with an integral toric Kähler form and let ∆ ⊆ R n be the image of the moment map for the torus action. Then, there exists a number c(∆) (explicitely computable from the polytope, see Corollary 3.5) such that any ball of radius r > c(∆), symplectically embedded into (M, ω), must intersect the divisor M \ C n. These two results are proved and discussed respectively in Section 2 (Theorem 2.6) and Section 3 (Corollary 3.5). The paper ends with an Appendix where, for the reader's convenience, we give an exposition (as self-contained as possible) of the classical facts about toric manifolds we need in Sections 2 and 3. 2. Kähler-Einstein submanifolds of Toric manifolds Let us briefly recall Calabi's work on Kähler immersions and diastasis function [7].
Characterizations of toric varieties via polarized endomorphisms
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Let X be a normal projective variety and f : X → X a non-isomorphic polarized endomorphism. We give two characterizations for X to be a toric variety. First we show that if X is Q-factorial and G-almost homogeneous for some linear algebraic group G such that f is G-equivariant, then X is a toric variety. Next we give a geometric characterization: if X is of Fano type and smooth in codimension 2 and if there is an f −1-invariant reduced divisor D such that f | X\D is quasietale and K X + D is Q-Cartier, then X admits a quasi-étale cover X such that X is a toric variety and f lifts to X. In particular, if X is further assumed to be smooth, then X is a toric variety.
The signature of a toric variety
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We identify a combinatorial quantity (the alternating sum of the h-vector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by Charney and Davis in their work, which in particular showed that its non-negativity is closely related to a conjecture of Hopf on the Euler characteristic of a non-positively curved manifold.