Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds

We prove that the moduli spaces of polarized Abelian threefolds with polarizations of types D=(1,1,2),(1,2,2),(1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space which parametrizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A^-1. Dedicated to the memory of Fabio BardelliMathematics Subject Classification (2000) Primary: 14K10; Secondary: 14H10, 14H30, 14D07     

Halphen pencils on weighted Fano threefold hypersurfaces

On a general quasismooth well-formed weighted hypersurface of degree Σ _i=1⁴ a _i in ℙ(1, a ₁, a ₂, a ₃, a ₄), we classify all pencils whose general members are surfaces of Kodaira dimension zero.      

Linear Systems on Fano Threefolds. I

Abstract

In this note, we classify all the polarized Fano threefold (X, H) with Bs|H|¬ = ?. As corollaries we obtained that (1) the very ample part of the conjecture of Fujita holds for smooth Fano threefolds and (2) global Seshadri constants of ample divisors on Fano threefolds are bounded from below by 1 except three types of polarized Fano threefolds.     

The de Rham Bundle on a Compactification of Moduli Space of Abelian Varieties

In this paper, we show that the Chern classes c _k of the de Rham bundle defined on any good toroidal compactification of the moduli space of Abelian varieties of dimension g are zero in the rational Chow ring of , for g=4, 5 and k>0.     

Classification of arithmetically Gorenstein divisors on some Fano varieties

We find all the arithmetically Gorenstein divisors on Fano varieties of dimension n and index r with 3 ≤ n ≤ 2r − 1 and ρ(Y) ≥ 2, where Y is smooth, connected, subcanonical and linearly normal.      

Existence of Indecomposable Rank Two Vector Bundles on Higher Dimensional Toric Varieties

In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ? ⁿ is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ? ⁿ .     

A Peculiar Property of Exceptional Weyl Groups

Let W be a Weyl group and P W, a parabolic subgroup. In this paper, we give the decomposition of the permutation representation Ind _P ^W 1 into irreducibles for each exceptional W and maximal parabolic P. We find that there is an 'extra' common irreducible component which appears for exceptional groups and not for classical groups. This work is motivated by the study of Prym varieties and integrable systems.     

Complete toric varieties with reductive automorphism group

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.     

On the Vanishing Orders of Vector Fields on Fano Varieties of Picard Number 1

We show that the vanishing order of a non-zero vector field at a generic point of a smooth Fano variety of Picard number 1 cannot exceed the dimension of the Fano variety. Furthermore, if there exist only finitely many rational curves of minimal degree through a generic point of the Fano variety, we show that a non-zero vector field cannot vanish at a generic point of the Fano variety.     

Hilbert Modular Threefolds of Arithmetic Genus One

In 1981, Weisser proved that there are exactly four Galois cubic number fields with Hilbert modular threefolds of arithmetic genus one. In this paper, we extend Weisser's work to cover all cubic number fields. Our main result is that there are exactly 33 fields with Hilbert modular threefolds of arithmetic genus one. These fields are enumerated explicitly.     

Primitive Contractions of Calabi–Yau Threefolds I

We construct examples of primitive contractions of Calabi–Yau threefolds with exceptional locus being ?¹ × ?¹, ?², and smooth del Pezzo surfaces of degrees ≤ 5. We describe the images of these primitive contractions and find their smoothing families. In particular, we give a method to compute the Hodge numbers of a generic fiber of the smoothing familly of each Calabi–Yau threefold with one isolated singularity obtained after a primitive contraction of type II. As an application, we get examples of natural conifold transitions between some families of Calabi–Yau threefolds.     

The Classification of (1,<Emphasis Type="Italic">k</Emphasis>)-Defective Surfaces

In this paper we give the full classification of surfaces X such that the family of lines contained in a (k+1)-secant to X has dimension smaller than the expected.     

On Positive Sasakian Geometry

A Sasakian structure =(\xi,\eta,\Phi,g) on a manifold Mis called positiveif its basic first Chern class c₁( ) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This provides us with a new technique for proving the existence of positive Ricci curvature metrics on certain odd dimensional manifolds. As an example we give a completely independent proof of a result of Sha and Yang that for every nonnegative integer kthe 5-manifolds k#(S ²×S ³) admits metrics of positive Ricci curvature.     

Identifying quadric bundle structures on complex projective varieties

In this paper we characterize smooth complex projective varieties that admit a quadric bundle structure on some dense open subset in terms of the geometry of certain families of rational curves.