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).     

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     

Complexity of Chow varieties and number of morphisms on surfaces of general type

We provide a general estimate for the number of irreducible components of a Chow variety, the variety that parametrizes algebraic cycles of given dimension and degree contained in a projective variety. The result is then applied to obtain an upper bound for the finite number of surfaces of general type that are images of a fixed surface. Received: 29 January 1998 / Revised version: 24 June 1998     

A tropical approach to secant dimensions

Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. The approach is especially successful for toric varieties such as Segre-Veronese embeddings. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; and indeed, no Segre-Veronese embeddings are known where the tropical lower bound does not give the correct dimension. Short self-contained introductions to secant varieties and the required tropical geometry are included.     

Zero cycles on homogeneous varieties

In this paper we study the group A₀(X) of zero-dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety X. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of étale subalgebras of central simple algebras which we construct. This allows us to show A₀(X)=0 for certain classes of homogeneous varieties for groups of each of the classical types, extending previous results of Swan/Karpenko, of Merkurjev, and of Panin.     

Higher order bad loci

Zero-schemes on smooth complex projective varieties, forcing all elements of ample and free linear systems to be reducible, are studied. Relationships among the minimal length of such zero-schemes, the positivity of the line bundle associated with the linear system, and the dimension of the variety are established. Bad linear spaces are also investigated.     

Generalized Greatest Common Divisors,Divisibility Sequences,and Vojta’s Conjecture for Blowups

We apply Vojta’s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta’s conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.     

Projective Reed–Muller type codes on rational normal scrolls

In this paper we study an instance of projective Reed–Muller type codes, i.e., codes obtained by the evaluation of homogeneous polynomials of a fixed degree in the points of a projective variety. In our case the variety is an important example of a determinantal variety, namely the projective surface known as rational normal scroll, defined over a finite field, which is the basic underlining algebraic structure of this work. We determine the dimension and a lower bound for the minimum distance of the codes, and in many cases we also find the exact value of the minimum distance. To obtain the results we use some methods from Gröbner bases theory.     

The minimal degree of plane models of double covers of smooth curves

If X is a smooth curve such that the minimal degree of its plane models is not too small compared with its genus, then X has been known to be a double cover of another smooth curve Y under some mild condition on the genera. However there are no results yet for the minimal degrees of plane models of double covers except some special cases. In this paper, we give upper and lower bounds for the minimal degree of plane models of the double cover X in terms of the gonality of the base curve Y and the genera of X and Y. In particular, the upper bound equals to the lower bound in case Y is hyperelliptic. We give an example of a double cover which has plane models of degree equal to the lower bound.     

Some remarks on varieties with projectively isomorphic hyperplane sections

We study here the projective varieties with the property that there exists a projective isomorphism between two of their generic hyperplane sections. The case of surfaces had already been studied by Fubini and Fano in the 1920s. The latter gave the list of all (possibly signular) surfaces with projectively isomorphic hyperplane sections. The proof, however, was essentially wrong. By means of a different approach, we are able to supply a proof of Fano's claims. Moreover, we show some general properties of varieties with projectively isomorphic hyperplane sections: they have uniruled hyperplane sections and are related to varieties with small dual varieties. In particular we are able to conclude that threefolds with projectively isomorphic hyperplane sections either have rational sections or are P²-bundles over a curve.A member of GNSAGA of CNR.     

Singular Poisson-Kähler geometry of Scorza varieties and their secant varieties

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.     

Pluricanonical maps of varieties of maximal Albanese dimension

Let X be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of in terms of the set . An immediate consequence of this is that the Kodaira dimension is invariant under smooth deformations. We then study the pluricanonical maps . We prove that if X is of general type, is generically finite for and birational for . More generally, we show that for the image of is of dimension equal to and for , is the stable canonical map. Received July 7, 2000 / Published online April 12, 2001     

复射影直纹面（II）

本文利用几何直纹面及曲线上秩－２局部自由层的一些性质，讨论射影空间中的直纹面一些特性，给出了非正则直纹面次数的下界并讨论了个维射影空间中次数接近下界的非正则直纹面的结构，如奇点集的结构，底曲线的结构及纤维束次数等，完全确定了这类曲面的几何结构。     

Microlocal study of topological Radon transforms and real projective duality

Various topological properties of projective duality between real projective varieties and their duals are obtained by making use of the microlocal theory of (subanalytically) constructible sheaves developed by Kashiwara [M. Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985) 193-209] and Kashiwara-Schapira [M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss., vol. 292, Springer, Berlin-Heidelberg-New York, 1990]. In particular, we prove in the real setting some results similar to the ones proved by Ernström in the complex case [L. Ernström, Topological Radon transforms and the local Euler obstruction, Duke Math. J. 76 (1994) 1-21]. For this purpose, we describe the characteristic cycles of topological Radon transforms of constructible functions in terms of curvatures of strata in real projective spaces.     

On the finite generation of the effective monoid of rational surfaces

We give a numerical criterion for ensuring the finite generation of the effective monoid of the surfaces obtained by a blowing-up of the projective plane at the supports of zero dimensional subschemes assuming that these are contained in a degenerate cubic. Furthermore, this criterion also ensures the regularity of any numerically effective divisor on these surfaces. Thus the dimension of any complete linear system is computed. On the other hand, in particular and among these surfaces, we obtain ringed rational surfaces with very large Picard numbers and with only finitely many integral curves of strictly negative self-intersection. These negative integral curves except two (−1)-curves are all contained in the support of an anticanonical divisor. Thus almost all the geometry of such surfaces is concentrated in the anticanonical class.     

Some remarks on the study of good contractions

LetX be a complex projective variety with log terminal singularities admitting an extremal contraction in terms of Minimal Model Theory, i.e. a projective morphism φ:X→Z onto a normal varietyZ with connected fibers which is given by a (high multiple of a) divisor of the typeK _x+rL, wherer is a positive rational number andL is an ample Cartier divisor. We first prove that the dimension of anu fiberF of φ is bigger or equal to (r-1) and, if φ is birational, thatdimF≥r, with the equalities if and only ifF is the projective space andL the hyperplane bundle (this is a sort of “relative” version of a theorem of Kobayashi-Ochiai). Then we describe the structure of the morphism φ itself in the case in which all fibers have minimal dimension with the respect tor. If φ is a birational divisorial contraction andX has terminal singularities we prove that φ is actually a “blow-up”.     

Laplace equations,Lefschetz properties and line arrangements

In this note we extend the main result in [6] on artinian ideals failing Lefschetz properties, varieties satisfying Laplace equations and existence of suitable singular hypersurfaces. Moreover we characterize the minimal generation of ideals generated by powers of linear forms by the configuration of their dual points in the projective plane and we use this result to improve some propositions on line arrangements and Strong Lefschetz Property at range 2 in [6]. The starting point was an example in [3]. Finally we show the equivalence among failing SLP, Laplace equations and some unexpected curves introduced in [3].     

A note on non-reduced Picard schemes

The Picard scheme of a smooth curve and a smooth complex variety is reduced. In this note we discuss which classes of surfaces in terms of the Enriques-Kodaira classification can have non-reduced Picard schemes and whether there are restrictions on the characteristic of the ground field. It turns out that non-reduced Picard schemes are uncommon in Kodaira dimension κ≤0, that this phenomenon can be bounded for κ=2 (general type) and that it is as bad as can be for κ=1.     

Seshadri constants and the generation of jets

In this paper we explore the connection between Seshadri constants and the generation of jets. It is well known that one way to view Seshadri constants is to consider them as measuring the rate of growth of the number of jets that multiples of a line bundle generate. Here we ask, conversely, what we can say about the number of jets once the Seshadri constant is known. As an application of our results, we prove a characterization of projective space among all Fano varieties in terms of Seshadri constants.     

Isolated rational curves on K3-fibered Calabi–Yau threefolds

In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙ ^{n 1}}×ℙ¹. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space. Received: 14 October 1997 / Revised version: 18 January 1998