Special linear Systems on Toric Varieties (original) (raw)
A note on affine toric varieties
Linear Algebra and Its Applications, 2000
Let k be an arbitrary field and a toric set in the affine space A n k given parametrically by monomials. Using linear algebra we give necessary and sufficient conditions for to be an affine toric variety, and show some applications.
Toric Varieties and Their Applications
Toric Varieties and Their Applications, 2022
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.
These notes survey some basic results in toric varieties over a field F , with examples and applications.
Spaces of non-resultant systems of bounded multiplicity determined by a toric variety
Topology and its Applications
The space of non-resultant systems of bounded multiplicity for a toric variety X is a generalization of the space of rational curves on it. In our earlier work [24] we proved a homotopy stability theorem and determined explicitly the homotopy type of this space for the case X = CP m. In this paper we consider the case of a general non-singular toric variety and prove a homotopy stability theorem generalising the one for CP m .
Strong rational connectedness of Toric Varieties
Mathematical Research Letters, 2011
In this paper, we establish that, for any given finitely many distinct points P 1 , . . . , P r and a closed subvariety S of codimension ≥2 in a complete toric variety X over an algebraically closed field of characteristic 0, there exists a rational curve f : P 1 → X passing through P 1 , . . . , P r , disjoint from S \ {P 1 , . . . , P r } (see Main Theorem). As a corollary we obtain that the smooth loci of complete toric varieties are strongly rationally connected.
Discriminants of toric varieties
Communications in Algebra, 2021
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.
Families of pointed toric varieties and degenerations
Mathematische Zeitschrift
The Losev–Manin moduli space parametrizes pointed chains of projective lines. In this paper we study a possible generalization to families of pointed degenerate toric varieties. Geometric properties of these families, such as flatness and reducedness of the fibers, are explored via a combinatorial characterization. We show that such families are described by a specific type of polytope fibration which generalizes the twisted Cayley sums, originally introduced to characterize elementary extremal contractions of fiber type associated to projective {\mathbb {Q}}$$ Q -factorial toric varieties with positive dual defect. The case of a one-dimensional simplex can be viewed as an alternative construction of the permutohedra.
Some remarks on the symplectic and Kähler geometry of toric varieties
Annali di Matematica Pura ed Applicata (1923 -), 2015
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
Mathematische Zeitschrift
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
Duke Mathematical Journal, 2002
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.
Higher order selfdual toric varieties
Annali di Matematica Pura ed Applicata (1923 -)
The notion of higher order dual varieties of a projective variety, introduced in Piene [Singularities, part 2, (Arcata, Calif., 1981), Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, 1983], is a natural generalization of the classical notion of projective duality. In this paper, we present geometric and combinatorial characterizations of those equivariant projective toric embeddings that satisfy higher order selfduality. We also give several examples and general constructions. In particular, we highlight the relation with Cayley-Bacharach questions and with Cayley configurations.
Classification of normal toric varieties
2014
Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro's theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations.
Toric varieties from cyclic matrix groups
arXiv: Algebraic Geometry, 2020
We study cyclic groups and semigroups of matrices from an algebraic geometric viewpoint. We present and expand some existing results and highlight their connection with toric geometry.
Lecture Notes in Mathematics, 2014
P is isomorphic to a Cayley sum Cayley(R 0 , . . . , R t ) π,Y where t + 1 = codeg(P) with k > n 2 . (c) µ(L P ) = τ(L P ) (n + 3)/2. Conventions. We assume basic knowledge of toric geometry and refer to [EW, FU, ODA] for the necessary background on toric varieties. Throughout this paper, we work over the field of complex numbers C. Toric varieties, X, are always normal and thus defined by a fan Σ X ⊂ N. By Σ X (n) we will denote the collection of n-dimensional cones of Σ X . Let P ⊂ R n be a lattice polytope of dimension n. Consider the graded semigroup Π P generated by ({1} × P) ∩ (N × Z n ). The polarized variety (Pro j(C[Π P ]), O(1)) is a toric variety associated to the polytope P. It will be sometime denoted by (X P , L P ). Notice that the toric variety X P is defined by the (inner) normal fan of P. Viceversa the symbol P (X,L) will denote the lattice polytope associated to a polarized toric variety (X, L). The symbol ∆ n denotes the smooth (unimodular) simplex of dimension n. Recall that an n-dimensional polytope is simple if through every vertex pass exactly n edges. A lattice polytope is smooth if it is simple and the primitive vectors of the edges through every vertex form a lattice basis. Smooth polytopes are associates to smooth projective varieties. Simple polytopes are associated to Q-factorial projective varieties.
Cycle-Level Intersection Theory for Toric Varieties
Canadian Journal of Mathematics, 2004
This paper addresses the problem of constructing a cycle-level intersection theory for toric varieties. We show that by making one global choice, we can determine a cycle representative for the intersection of an equivariant Cartier divisor with an invariant cycle on a toric variety. For a toric variety defined by a fan inN, the choice consists of giving an inner product or a complete flag forMQ= ℚ ⊗ Hom(N, ℤ), or more generally giving for each cone σ in the fan a linear subspace ofMQcomplementary to σ⊥, satisfying certain compatibility conditions. We show that these intersection cycles have properties analogous to the usual intersections modulo rational equivalence. IfXis simplicial (for instance, ifXis non-singular), we obtain a commutative ring structure to the invariant cycles ofXwith rational coefficients. This ring structure determines cycles representing certain characteristic classes of the toric variety. We also discuss how to define intersection cycles that require no choi...
On the geometry of toric arrangements
Transform Groups, 2005
We introduce toric arrangements, essentially finite families of codimension 1 subtori of a torus or of their cosets, as a periodic generalization of hyperplane arrangements, compute cohomology of the complement of such an arrangement and apply the theory to give a geometric interpretation of the formulas of Brion--Szenes--Vergne counting the number of integral points in polytopes.
arXiv (Cornell University), 2022
In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety X with respect to the irrelevant ideal of X. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on X. In particular, we prove that toric Sylvester forms yield bases of some graded components of I sat /I, where I denotes an ideal generated by n + 1 generic forms, n is the dimension of X and I sat the saturation of I with respect to the irrelevant ideal of the Cox ring of X. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we give a new formula for computing toric residues of the product of two forms.