Stable base loci, movable curves, and small modifications, for toric varieties

We show that the dual of the cone of divisors on a complete -factorial toric variety X whose stable base loci have dimension less than k is generated by curves on small modifications of X that move in families sweeping out the birational transforms of k-dimensional subvarieties of X. We give an example showing that it does not suffice to consider curves on X itself. Supported by a Graduate Research Fellowship from the NSF     

Local Euler obstructions of toric varieties

We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones.     

One-parameter toric deformations of cyclic quotient singularities

In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions introduced by Altmann [Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (2) (1995) 151-184.]. In particular, we show how to induce each deformation from a versal family, describe exactly to which reduced versal base space components each such deformation maps, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions.     

Varieties with minimal secant degree and linear systems of maximal dimension on surfaces

In this paper we prove, using a refinement of Terracini's Lemma, a sharp lower bound for the degree of (higher) secant varieties to a given projective variety, which extends the well known lower bound for the degree of a variety in terms of its dimension and codimension in projective space. Moreover we study varieties for which the bound is attained proving some general properties related to tangential projections, e.g. these varieties are rational. In particular we completely classify surfaces (and curves) for which the bound is attained. It turns out that these surfaces enjoy some maximality properties for their embedding dimension in terms of their degree or sectional genus. This is related to classical beautiful results of Castelnuovo and Enriques that we revise here in terms of adjunction theory.     

Generation of k-jets on toric varieties

The notion of a k-convex -support function for a toric variety is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the -support function . Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k. Received October 22, 1997; in final form January 12, 1998     

Global residues for sparse polynomial systems

We consider families of sparse Laurent polynomials f₁,…,f_n with a finite set of common zeros Z_f in the torus Tⁿ=(C−{0})ⁿ. The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over Z_f. We present a new symbolic algorithm for computing the global residue as a rational function of the coefficients of the f_i when the Newton polytopes of the f_i are full-dimensional. Our results have consequences in sparse polynomial interpolation and lattice point enumeration in Minkowski sums of polytopes.     

On the geometry of complete intersection toric varieties

In this paper we give a geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to 2n − 2, is exactly 2n − 2 if and only if the cone is a bipyramidal cone, where n > 1 is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones. Received: 4 July 2005     

Abelian surfaces in projective toric 4-folds

We show fundamental properties on embedding of abelian surfaces into projective toric 4-folds, and study the case of the toric Del Pezzo 4-fold from the viewpoint of the moduli space of abelian surfaces with polarization of type (1, 5). Received: 31 January 2005; revised: 15 April 2005     

Classifying smooth lattice polytopes via toric fibrations

We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n?2d+1. This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill.     

On varieties of almost minimal degree III: Tangent spaces and embedding scrolls

Let X⊂P^r be a variety of almost minimal degree which is the projected image of a rational normal scroll from a point p outside of . In this paper we study the tangent spaces at singular points of X and the geometry of the embedding scrolls of X, i.e. the rational normal scrolls Y⊂P^r which contain X as a codimension one subvariety.     

Minimal generators of toric ideals of graphs

Let IG be the toric ideal of a graph G. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal IG.     

On Euler–Jaczewski sequence and Remmert–Van de Ven problem for toric varieties

Let X be a complete toric variety and Y a smooth projective variety with . We prove that, if is a surjective morphism then . Received: 15 May 2001; in final form: 22 October 2001/ Published online: 4 April 2002     

Moduli stacks of stable toric quasimaps

We construct new “virtually smooth” modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental's compactifications, when the complex structure of the curve is allowed to vary and markings are included, and are the toric counterpart of the moduli spaces of stable quotients introduced by Marian, Oprea, and Pandharipande to compactify spaces of maps to Grassmannians. A brief discussion of the resulting invariants and their (conjectural) relation with Gromov-Witten theory is also included.     

Differential operators on toric varieties and Fourier transform

We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point of the affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of .     

On toric varieties which are almost set-theoretic complete intersections

We describe a class of affine toric varieties V that are set-theoretically minimally defined by binomial equations over fields of any characteristic.     

Cohomology of toric bundles

Let p: E B be a principal bundle with fibre and structure group the torus T ( ^*)ⁿ over a topological space B. Let X be a nonsingular projective T-toric variety. One has the X-bundle : E(X) B where E(X) = E × _T X, ([e,x]) = p(e). This is a Zariski locally trivial fibre bundle in case p: E B is algebraic. The purpose of this note is to describe (i) the singular cohomology ring of E(X) as an H ^* (B;)-algebra, (ii) the topological K-ring of K ^* (E(X)) as a K ^* (B)-algebra when B is compact. When p : E B is algebraic over an irreducible, nonsingular, noetherian scheme over , we describe (iii) the Chow ring of A ^* (E(X)) as an A ^* (B)-algebra, and (iv) the Grothendieck ring $\mathcal K$⁰ (E (X)) of algebraic vector bundles on E (X) as a $\mathcal K$⁰(B)-algebra.     

Remarks on abelian surfaces in nonsingular toric Fano 4-folds

In this paper, we investigate whether the 124 nonsingular toric Fano 4-folds admit totally nondegenerate embeddings from abelian surfaces or not. In consequence, we determine the possibilities of these embeddings, except for the remaining 18 nonsingular toric Fano 4-folds. Received: 12 July 2002     

On the Lüroth semigroup of curves lying on a smooth quadric

For a smooth irreducible complete algebraic curveC the “gaps” are the integersn such that every linear series of degreen has at least a base point. The Lüroth semigroup S_C of a curveC is the subsemigroup ofN whose elements are not gaps. In this paper we deal with irreducible smooth curves of type (a, b) on a smooth quadricQ. The main result is an algorithm by which we can say if some integer λ∈N is a gap or is in S_C. In the general case there are integers λ which are undecidable. For curves such as complete intersection, arithmetically Cohen-Macaulay or Buchsbaum, we are able to describe explicitly “intervals” of gaps and “intervals” of integers which belong to S_C. For particular cases we can completely determine S_C, by giving just the type of the curve (in particular the degree and the genus). Work done with financial support of M.U.R.S.T. while the authors were members of G.N.S.A.G.A. of C.N.R.