Contributions to Algebraic Geometry (original) (raw)
These notes survey some basic results in toric varieties over a field F , with examples and 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.
Invariants of hypersurfaces and logarithmic differential forms
EMS Series of Congress Reports
This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.
A Theory of Divisors for Algebraic Curves
2007
The purpose of this paper is twofold. We first prove a series of results, concerned with the notion of Zariski multiplicity, mainly for non-singular algebraic curves. These results are required in [6], where, following Severi, we introduced the notion of the "branch" of an algebraic curve. Secondly, we use results from [6], in order to develop a refined theory of g r n on an algebraic curve. This refinement depends critically on replacing the notion of a point with that of a "branch". We are then able to construct a theory of divisors, generalising the corresponding theory in the special case when the algebraic curve is non-singular, which is birationally invariant.
Singularities of theta divisors and the birational geometry of irregular varieties
Journal of the American Mathematical Society, 1997
The purpose of this paper is to show how the generic vanishing theorems of and can be used to settle a number of questions and conjectures raised in , Chapter 17, concerning the geometry of irregular complex projective varieties. Specifically, we focus on three sorts of results. First, we establish a well known conjecture characterizing principally polarized abelian varieties whose theta divisors are singular in codimension one. Secondly, we study the holomorphic Euler characteristic of varieties of general type having maximal Albanese dimension: we verify a conjecture of Kollár for subvarieties of abelian varieties, but show that it fails in general. Finally, we give a surprisingly simple new proof of a fundamental theorem of Kawamata [Ka] on the Albanese mapping of varieties of Kodaira dimension zero.
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...
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.
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.
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.
Computing with toric varieties
Journal of Symbolic Computation, 2007
A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any number of dimensions (written in both MAGMA and GAP). Among other things, the package implements the desingularization procedure, constructs some error-correcting codes associated with toric varieties, and computes the Riemann-Roch space of a divisor on a toric variety.
The Bott Formula for Toric Varieties
1999
The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf OmegaPp(D)\Omega_{\P}^p(D)OmegaPp(D) of p-th differential forms of Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope
Algebraic geometry (Selected Topics)
This topic contains lecture notes from one-semester course for graduate students which took place in Mathematical institute SANU in the spring semester 1994/95. It includes basic notions of algebraic geometry such as plane algebraic curves, Weil and Cartier divisors, dimension, sheaves and Czech cohomology, topological and arithmetical genus, linear systems, Riemann-Roch theorem for curves.
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.
Ju n 20 09 BIG DIVISORS ON A PROJECTIVE SYMMETRIC VARIETY
2009
We describe the big cone of a projective symmetric variety. Moreover, we give a necessary and sufficient combinatorial criterion for the bigness of a nef divisor (linearly equivalent to a G-stable divisor) on a projective symmetric variety. When the variety is toroidal, such criterion has an explicitly geometrical interpretation. Finally, we describe the spherical closure of a symmetric subgroup. keywords: Symmetric varieties, Big divisors. Mathematics Subject Classification 2000: 14L30, 14C20, (14M17) Brion give a description of the Picard group of a spherical variety in [Br89]. He also finds necessary and sufficient conditions for the ampleness and global generation of a line bundle. From these conditions follows that a line bundle is nef if and only if it is globally generated. It is natural to ask what are the conditions on a line bundle to be big. It is known that the effective cone is closed, polyhedral and, if the variety is Q-factorial, generated by the classes of the B-stab...
arXiv (Cornell University), 2023
For an affine T -variety X with the action of a torus T , this paper provides a combinatorial description of X with respect to the action of a subtorus T ′ ⊂ T in terms of a T /T ′ -invariant pp-divisor. We also describe the corresponding GIT fan.
An asymptotic description of the Noether-Lefschetz components in toric varieties
arXiv: Algebraic Geometry, 2019
We extend the definition of Noether-Leschetz components to quasi-smooth hypersurfaces in a projective simplicial toric variety mathbbPSigma2k+1\mathbb P_{\Sigma}^{2k+1}mathbbPSigma2k+1, and prove that asymptoticaly the components whose codimension is bounded from above by a suitable effective constant correspond to hypersurfaces containing a small degree kkk-dimensional subvariety. As a corollary we get an asymptotic characterization of the components with small codimension, generalizing the work of Otwinowska for mathbbPSigma2k+1=mathbbP2k+1\mathbb P_{\Sigma}^{2k+1}=\mathbb P^{2k+1}mathbbPSigma2k+1=mathbbP2k+1 and Green and Voisin for mathbbPSigma2k+1=mathbbP3\mathbb P_{\Sigma}^{2k+1}=\mathbb P^3mathbbPSigma2k+1=mathbbP3. Some tools that are developed in the paper are a generalization of Macaulay's theorem for Fano, irreducible normal varieties with rational singularities, satisfying a suitable additional condition, and an extension of the notion of Gorenstein ideal for normal varieties with finitely generated Cox ring.