The separability of the Gauss map versus the reflexivity

In projective algebraic geometry, various pathological phenomena in positive characteristic have been observed by several authors. Many of those phenomena concerning the behavior of embedded tangent spaces seem to be controlled by the separability of (the extension of function fields defined by) the Gauss map, or by the reflexivity with respect to the projective dual for a projective variety. The purpose of this paper is to survey the studies on the relationship between the separability of the Gauss map and the reflexivity for a projective variety: Is the separability of the Gauss map equivalent to the reflexivity for a projective variety?     

Higher Gauss maps of Veronese varieties—A generalization of Boole’s formula and degree bounds for higher Gauss map images

The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.     

Developable varieties with all singularities at infinity

A projective variety is called developable if the image of its Gauss map has a smaller dimension than the variety itself. Developable varieties are always singular, and requiring that all singularities lie in a hyperplane puts a severe restriction on them. Here we refine a theorem of Wu and Zheng stating that such varieties are the union of cones if the dimension of the Gauss image is less than or equal to four. Afterwards we study their singular locus. Finally, we describe the geometry of such varieties whose Gauss image has dimension two. Received: Received: 8 November 2000 / Revised version: 15 May 2001     

The separability of the Gauss map and the reflexivity for a projective surface

It is known that if a projective variety X in P ^N is reflexive with respect to the projective dual, then the Gauss map of X defined by embedded tangent spaces is separable, and moreover that the converse is not true in general. We prove that the converse holds under the assumption that X is of dimension two. Explaining the subtleness of the problem, we present an example of smooth projective surfaces in arbitrary positive characteristic, which gives a negative answer to a question raised by S. Kleiman and R. Piene on the inseparability of the Gauss map.      

Any algebraic variety in positive characteristic admits a projective model with an inseparable Gauss map

We determine the values attained by the rank of the Gauss map of a projective model for a fixed algebraic variety in positive characteristic p. In particular, it is shown that any variety in p>0 has a projective model such that the differential of the Gauss map is identically zero. On the other hand, we prove that there exists a product of two or more projective spaces admitting an embedding into a projective space such that the differential of the Gauss map is identically zero if and only if p=2.     

Projective varieties admitting an embedding with Gauss map of rank zero

We introduce an intrinsic property for a projective variety as follows: there exists an embedding into some projective space such that the Gauss map is of rank zero, which we call (GMRZ) for short. It turns out that (GMRZ) imposes strong restrictions on rational curves on projective varieties: In fact, using (GMRZ), we show that, contrary to the characteristic zero case, the existence of free rational curves does not imply that of minimal free rational curves in positive characteristic case. We also focus attention on Segre varieties, Grassmann varieties, and hypersurfaces of low degree. In particular, we give a characterisation of Fermat cubic hypersurfaces in terms of (GMRZ), and show that a general hypersurface of low degree does not satisfy (GMRZ).     

Subvarieties of generic hypersurfaces in a nonsingular projective toric variety

We investigate the subvarieties contained in generic hypersurfaces of projective toric varieties and prove two main theorems. The first generalizes Clemens’ famous theorem on the genus of curves in hypersurfaces of projective spaces to curves in hypersurfaces of toric varieties and the second improves the bound in the special case of toric varieties in a theorem of Ein on the positivity of subvarieties contained in sufficiently ample generic hypersurfaces of projective varieties. Both depend on a hypothesis which deals with the surjectivity of multiplication maps of sections of line bundles on the toric variety. We also obtain an infinitesimal Torelli theorem for hypersurfaces of toric varieties.     

On the Ideals of Secant Varieties of Segre Varieties

We establish basic techniques for determining the ideals of secant varieties of Segre varieties.We solve a conjecture of Garcia, Stillman, and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.     

On the dimensions of secant varieties of Segre-Veronese varieties

This paper explores the dimensions of higher secant varieties to Segre-Veronese varieties. The main goal of this paper is to introduce two different inductive techniques. These techniques enable one to reduce the computation of the dimension of the secant variety in a high-dimensional case to the computation of the dimensions of secant varieties in low-dimensional cases. As an application of these inductive approaches, we will prove non-defectivity of secant varieties of certain two-factor Segre-Veronese varieties. We also use these methods to give a complete classification of defective sth Segre–Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre–Veronese varieties.     

On the Geometry of Point-Plane Constraints on Rigid-Body Displacements

In this paper the rigid-body displacements that transform a point in such a way that it remains on a particular plane are studied. These sets of rigid displacements are referred to as point-plane constraints and are given by the intersection of the Study quadric of all rigid displacements with another quadric in 7-dimensional projective space. The set of all possible point-plane constraints comprise a Segre variety. Two different classes of problems are investigated. First instantaneous kinematics, for a given rigid motion there are points in space which, at some instant, have no torsion or have no curvature to some order. The dimension and degrees of these varieties are found by very simple computations. The corresponding problems for point-sphere constraints are also found. The second class of problems concern the intersections of several given constraints. Again the characteristics of these varieties for different numbers of constraints are found using very simple techniques.     

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes, we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective \mathbb Q{\mathbb Q}-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev’s structure theorem for the cone of nef curves of projective terminal threefolds.     

On a relation between spectral theory of lens spaces and Ehrhart theory

In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens space is introduced. It is proved that if two lens spaces are isospectral then the dimension of global sections of powers of a natural line bundle on these two toric varieties are equal and they have the same general intersection number. Also, harmonic polynomial representation of the group SO($n$) is used to provide a more elementary proof for a theorem of Lauret, Miatello and Rossetti on isospectrality of lens spaces.     

On d-Dimensional Dual Hyperovals

d-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.     

Algebraic cocycles on normal, quasi-projective varieties

This paper extends to quasi-projective varieties earlier work by the author and H. Blaine Lawson concerning spaces of algebraic cocycles on projective varieties. The topological monoid C_r(Y) (U) of effective cocycles on a normal, quasi-projective variety U with values in a projective variety Y consists of algebraic cycles on U×Y equi-dimensional of relative dimension r over U. A careful choice of topology enables the establishment of various good properties: the definition is essentially algebraic, the group completion Z_r (Y) (U) has 'sensible' homotopy groups, the construction is contravariant with respect to U, convariant with respect to Y, and there is a natural 'quality map" to the topological group of cycles on U×Y. The fundamental theorem presented here is the extension of Friedlander-Lawson duality to this context: the duality map Z_r (Y) (U) to Z_{_r+m} (U × Y) is a homotopy equivalence provided that both U and Y are smooth (where m=dim U). Various application are given, especially the determination of the homotopy types of certain topological groups of algeb raic morphisms.     

Segre embeddings and the canonical image of a curve

We prove that there is no g for which the canonical embedding of a general curve of genus g lies on the Segre embedding of any product of three or more projective spaces.     

Differential Forms on Quasihomogeneous Noncomplete Intersections

In this article, we discuss a few simple methods for computing the Poincaré series of modules of differential forms given on quasihomogeneous noncomplete intersections of various types. Among them are curves associated with a semigroup, bouquets of such curves, affine cones over rational or elliptic curves, and normal determinantal and toric varieties, including some types of quotient singularities, as well as cones over the Veronese embedding of projective spaces or over the Segre embedding of products of projective spaces, rigid singularities, fans, etc. In many cases, correct formulas can be derived without resorting to analysis of complicated resolvents or using computer systems of algebraic calculations. The obtained results allow us to compute the basic invariants of singularities in an explicit form by means of elementary operations on rational functions.     

Dually Degenerate Varieties and the Generalization of a Theorem of Griffiths–Harris

The dual variety X^* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X^* is a hypersurface in the dual space (P^N)^*. If dimX^*<N–1, then the variety X is called dually degenerate. The authors refine these definitions for a variety XP^N with a degenerate Gauss map of rankr. For such a variety, in the general case, the dimension of its dual variety X^* is N–l–1, where l=n–r, and X is dually degenerate if dimX^*<N–l–1. In 1979 Griffiths and Harris proved that a smooth variety XP^N is dually degenerate if and only if all its second fundamental forms are singular. The authors generalize this theorem for a variety XP^N with a degenerate Gauss map of rankr. Mathematics Subject Classification (2000) 53A20.     

Existence of a non‐reflexive embedding with birational Gauss map for a projective variety

We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙ^N to re‐embed into some projective space ℙ^M so as to be non‐reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙ^N is identically zero; hence the projective varietyX re‐embedded in ℙ^M yields a negative answer to Kleiman–Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙ^N with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙ^N satisfying the assumption. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the p-adic meromorphy of the function field height zeta function

In this brief note, we will investigate the number of points of bounded height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the p-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is p-adic meromorphic.