Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety Y \subsetneq X{Y \subsetneq X} . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

The Grothendieck ring of varieties and piecewise isomorphisms

Let K ₀(Var _k ) be the Grothendieck ring of algebraic varieties over a field k. Let X, Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X ₁, ..., X _n , Y ₁, ..., Y _n into locally closed subvarieties such that X _i is isomorphic to Y _i for all i ≤ n), then [X] = [Y] in K ₀(Var _k ). Larsen and Lunts ask whether the converse is true. For characteristic zero and algebraically closed field k, we answer positively this question when dim X ≤ 1 or X is a smooth connected projective surface or if X contains only finitely many rational curves.     

Osculating spaces to secant varieties

We generalize the classical Terracini’s Lemma to higher order osculating spaces to secant varieties. As an application, we address with the so-called Horace method the case of thed-Veronese embedding of the projective 3-space. This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy).     

Vector bundles over normal varieties trivialized by finite morphisms

Let Y be a normal and projective variety over an algebraically closed field k and V a vector bundle over Y. We prove that if there exist a k-scheme X and a finite surjective morphism g : X → Y that trivializes V then V is essentially finite.     

On the projectively almost-factorial varieties

Summary A projectively normal subvariety (X,O _X) ofP ^N(k), k an algebraically closed field of characteristic zero, will be said to be projectively almost-factorial if each Weil divisor has a multiple which is a complete intersection in X. The main result is the following: (X,O _X) is projectively almost-factorial if and only if for all x ∈ X the local ringO _x is almost-factorial and the quotient ofPic(X) modulo the subgroup generated by the class ofO _X (1) is torsion. We also prove the invariance of the projective almost-factoriality up to isomorphisms and state some relations between the projective almost-factoriality (resp. projective factoriality) of X and the almost-factoriality (resp. factoriality) of the affine open subvarieties. Finally we discuss some consequences of the main result in the case k=ℂ: in particular we prove that the Picard group of a projectively almost-factorial variety is isomorphic to the Néron-Severi group, hence finitely generated. Entrata in Redazione il 23 aprile 1976. AMS(MOS) subject classification (1970): Primary 14C20, 13F15.     

Local rigidity of quasi-regular varieties

For a G-variety X with an open orbit, we define its boundary ∂ X as the complement of the open orbit. The action sheaf S _X is the subsheaf of the tangent sheaf made of vector fields tangent to ∂ X. We prove, for a large family of smooth spherical varieties, the vanishing of the cohomology groups H ⁱ (X, S _X ) for i > 0, extending results of Bien and Brion (Compos. Math. 104:1–26, 1996). We apply these results to study the local rigidity of the smooth projective varieties with Picard number one classified in Pasquier (Math. Ann., in press).     

Multiplicative properties of projectively dual varieties

We present a general formula for the dimension of the projectively dual of the product of two projective varietiesX ₁ andX ₂, in terms of dimensions ofX ₁,X ₂ and their projective duals (Theorem 0.1). The proof is based on the formula due to N. Katz expressing the dimension of the dual variety in terms of the rank of certain Hessian matrix. Some consequences and related results are given, including the “Cayley trick” from [3] and its dual version. Partially supported by the NSF (DMS-9102432) Partially supported by the NSF (DMS-9104867) This article was processed by the author using the Springer-Verlag T_EE mamath macro package 1990.     

Complemented copies of ℓ<Subscript>p</Subscript> spaces in tensor products

We give sufficient conditions on Banach spaces X and Y so that their projective tensor product X ⊗_π Y, their injective tensor product X ⊗_ɛ Y, or the dual (X ⊗_π Y)^* contain complemented copies of ℓ_p.     

Kodaira dimension of irregular varieties

Let f:X→Y be an algebraic fiber space with general fiber F. If Y is of maximal Albanese dimension, we show that κ(X)≥κ(Y)+κ(F).     

The singular locus of the secant varieties of a smooth projective curve

Let X be a smooth irreducible non-degenerated projective curve in some projective space P^N. Let r be a positive integer such that 2r + 1 < N and let S_r(X) be the r-th secant variety of X. It is a variety of dimension 2r + 1. In this paper we prove that the singular locus is the (r - 1)-th secant variety S_{r- 1}(X) if X does not have any (2r + 2)-secant 2r-space divisor. Received: 26 November 2002     

Additive group actions on Danielewski varieties and the cancellation problem

Given complex algebraic varieties X and Y of the same dimension, the Cancellation Problem asks if an isomorphism between X  ×  and Y  ×  induces an isomorphism between X and Y. Iitaka and Fujita (J. Fac. Sci. Univ. 24:123–127, 1977) established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski constructed a counterexample using smooth rational affine surfaces. His construction was further generalized by Fieseler (Comment. Math. Helvetici 69:5–27, 1994) and Wilkens (C.R. Acad. Sci. Paris Sér. I Math. 326(9):1111–1116, 1998) to describe a larger class of affine surfaces. Here we introduce higher-dimensional analogues of these surfaces. By studying algebraic actions of the additive group on certain of these varieties, we obtain new counterexamples to the Cancellation Problem in every dimension d ≥ 2.     

Secant varieties of toric varieties

Let X_P be a smooth projective toric variety of dimension n embedded in P^r using all of the lattice points of the polytope P. We compute the dimension and degree of the secant variety . We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties X_A embedded using a set of lattice points A⊂P∩Zⁿ containing the vertices of P and their nearest neighbors.     

Severi varieties

R. Hartshorne conjectured and F. Zak proved (cf [6,p.9]) that any smooth non-degenerate complex algebraic variety with satisfies denotes the secant variety of X; when X is smooth it is simply the union of all the secant and tangent lines to X). In this article, I deal with the limiting case of this theorem, namely the Severi varieties, defined by the conditions and . I want to give a different proof of a theorem of F. Zak classifying all Severi varieties. F. Zak proves that there exists only four Severi varieties and then realises a posteriori that all of them are homogeneous; here I will work in another direction: I prove a priori that any Severi variety is homogeneous and then deduce more quickly their classification, satisfying R. Lazarsfeld et A. Van de Ven's wish [6, p.18]. By the way, I give a very brief proof of the fact that the derivatives of the equation of Sec(X), which is a cubic hypersurface, determine a birational morphism of . I wish to thank Laurent Manivel for helping me in writing this article. Received in final form: 29 March 2001 / Published online: 1 February 2002     

Nash problem for stable toric varieties

The main result of this paper is the proof of the Nash conjecture for stable toric varieties. We also introduce the Nash problem for pairs and prove it in the case of pairs (X, Y) of a toric variety X and a proper closed T ‐invariant subset Y containing Sing(X) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On multisecants and the Green-Lazarsfeld index of projective varieties

It is well known that a nondegenerate projective subvariety X ì \mathbb P^r{X \subset \mathbb {P}^r} of degree d and codimension c > 1 has minimal degree (i.e., d = c + 1) if and only if index(X) ≥ c if and only if X has no multisecant c-space. In this paper we extend this result by classifying varieties with index(X) ≥ c − s or with no multisecant (c − s)-space for s = 1 and 2.     

Irregularity of certain algebraic fiber spaces

LetX, Y be smooth complex projective varieties, andf: X →Y be a fiber space whose general fiber is a curve of genusg. Denote byq _f the relative irregularity off. It is proved thatq _f ≤5g+1 / 6, iff 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, thenq _f ≤g+1 / 2, and the bound is best possible. A classification of fiber surfaces of genus 3 withq _f =2 is also given in this note. Project supported by China Postdoctoral Science Foundation     

The Jacobian of a nonorientable Klein surface

Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surfaceY is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms onY quotiented by the torsion-free part of the first integral homology ofY. Denote byX the double cover ofY given by orientation. The Jacobian ofY is identified with the space of all degree zero holomorphic line bundlesL overX with the property thatL is isomorphic to σ*/-L, where σ is the involution ofX.