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).     

Imposing Singular Points and Many Nodes to Curves on Surfaces

 Let S be a smooth projective surface. Here we study the conditions imposed to curves of a fixed very ample linear system by a general union of types of singularities τ when most of connected components of τ are ordinary double points. This problem is related to the existence of “good” families of curves on S with prescribed singularities, most of them being nodes, and to the regularity of their Hilbert scheme. Received 6 July 2000; in revised form 16 June 2001     

Mixed multiplicities,Hilbert polynomials and homaloidal surfaces

We investigate the relationship among several numerical invariants associated to a (free) projective hypersurface V : the sequence of mixed multiplicities of its Jacobian ideal, the Hilbert polynomial of its Milnor algebra, and the sequence of exponents when V is free. As a byproduct, we obtain explicit equations for some of the homaloidal surfaces in the projective 3‐dimensional space constructed by C. Ciliberto, F. Russo and A. Simis.     

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.     

Automorphisms of Fermat-like varieties

 We study the automorphisms of some nice hypersurfaces and complete intersections in projective space by reducing the problem to the determination of the linear automorphisms of the ambient space that leave the algebraic set invariant. Received: 18 December 2000 / Revised version: 22 October 2001     

On complete families of curves with a given fundamental group in positive characteristic

In this paper we prove that complete families of smooth and projective curves of genus g≥2 in characteristic p>0 with a constant geometric fundamental group are isotrivial.     

LG/CY correspondence: The state space isomorphism

We prove the mirror duality conjecture for the mirror pairs constructed by Berglund, Hübsch, and Krawitz. Our main tool is a cohomological LG/CY correspondence which provides a degree-preserving isomorphism between the cohomology of finite quotients of Calabi–Yau hypersurfaces inside a weighted projective space and the Fan–Jarvis–Ruan–Witten state space of the associated Landau–Ginzburg singularity theory.     

Graded Cohen–Macaulay rings of wild Cohen–Macaulay type

We give sufficient conditions for a standard graded Cohen–Macaulay?ring, or equivalently, an arithmetically Cohen–Macaulay?projective variety, to be Cohen–Macaulay?wild in the sense of representation theory. In particular, these conditions are applied to hypersurfaces and complete intersections.     

Motives of some Fano varieties

We study the Fano varieties of projective k-planes lying in hypersurfaces and investigate the associated motives. The first author is partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The second author is partially supported by TüBİTAK-BDP funds and Bilkent University research development funds.     

Classes caractéristiques des intersections complètes

We prove various inequalities for characteristic classes of smooth projective varieties which are complete intersections. We relate such classes to symmetric functions of the degrees of the hypersurfaces that intersect onX, by means of symmetric functions that are invariant by translation, and use positivity properties and inequalities between Newton classes. This extends previous results of H. S. Tai.

     

Irregularity of certain algebraic fiber spaces

Let X, Y be smooth complex projective varieties, and be a fiber space whose general fiber is a curve of genus g. Denote by q _f the relative irregularity of f. It is proved that , if f is not generically trivial; moreover, if either a) f is non-constant and the general fiber is either hyperelliptic or bielliptic or b) q(Y)= 0, then , and the bound is best possible. A classification of fiber surfaces of genus 3 with q _f = 2 is also given in this note. Received: 19 March 1997 / Revised version: 29 October 1997     

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     

Linear and Steiner Bundles on Projective Varieties

We generalize the theory of Horrocks monads to ACM varieties, and use the generalization to establish a cohomological characterization of linear and Steiner bundles on projective space and on quadric hypersurfaces. We also characterize Steiner bundles on the Grassmannian G(1, 4) of lines in ?⁴. Finally, we study linear resolutions of bundles on ACM varieties, and characterize linear homological dimension on quadric hypersurfaces.     

Asymptotic purity for very general hypersurfaces of ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> × ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> of bidegree (<Emphasis Type="Italic">k,k</Emphasis>)

For a complex irreducible projective variety, the volume function and the higher asymptotic cohomological functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this paper, we study the vanishing properties of these functions on hypersurfaces of ℙ ⁿ × ℙ ⁿ . In particular, we show that very general hypersurfaces of bidegree (k, k) obey a very strong vanishing property, which we define as asymptotic purity: at most one asymptotic cohomological function is nonzero for each divisor. This provides evidence for the truth of a conjecture of Bogomolov and also suggests some general conditions for asymptotic purity.     

A bound on the Castelnuovo-Mumford regularity for curves

Recall that a projective curve in with ideal sheaf is said to be n-regular if for every integer and that in this case, it is cut out scheme-theoretically by equations of degree at most n. The purpose here is to show that an irreducible, reduced, projective curve of degree d and large arithmetic genus satisfies a smaller regularity bound than the optimal one . For example, if then a curve is -regular unless it is embedded by a complete linear system of degree . Received: 29 May 2000 / Published online: 24 September 2001     

Real hypersurfaces in quaternionic projective space

Summary This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We obtain three types of such real hypersurfaces. Two of them are known. Third type is new and in its study the first example of proper quaternion CR-submanifold appears. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. Finally we study the Ricci tensor of a real hypersurface of quaternionic projective space and classify pseudo-Einstein, almost-Einstein and Einstein real hypersurfaces.     

Cubic hypersurfaces admitting an embedding with Gauss map of rank 0

We give a characterization of Fermat cubic hypersurfaces of dimension greater than 2 in characteristic 2 in terms of the property, called (GMRZ), that a projective variety admits an embedding whose Gauss map is of rank 0. In contrast to the higher dimensional case, for cubic surfaces the above characterization is no longer true. Moreover, we prove that the process of blowing up at points preserves the property (GMRZ), and that every smooth rational surface in fact satisfies (GMRZ) in the characteristic 2 case.