Complex surfaces polarized by an ample and spanned line bundle of genus three

Let X be a complex connected projective nonsingular algebraic surface endowed with an ample line bundle L, which is spanned by its global sections. Pairs (X, L) as above, with sectional genus g(X, L)=1+(L·(K _X L))/2=3 are classified by means of the main techniques of adjunction theory.     

Bipolarized threefolds with hyperelliptic curve sections

A classification of smooth complex projective threefoldsX polarized by two very ample line bundlesL andM is given, under the assumption that two general elements of |L| and |M| intersect transversally along a smooth hyperelliptic curve.

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Sunto Si fornisce una classificazione delle terne (X,L,M), doveX è una varietà algebrica proiettiva complessa liscia di dimensione 3 edL,M sono due fibrati lineari molto ampi suX, tali che l’intersezione di due elementi generici di |L| ed |M| sia una curva iperellittica.

     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

On nef reductions of projective irreducible symplectic manifolds

Let X be a projective irreducible symplectic manifold and L be a non trivial nef divisor on X. Assume that the nef dimension of L is strictly less than the dimension of X. We prove that L is semiample. Partially supported by Grant-in-Aid no. 15740002 (Japan Society for Promotion of Sciences)     

The <Emphasis Type="Italic">L</Emphasis>-series of a cubic fourfold

We study the L-series of cubic fourfolds. Our main result is that, if X/C is a special cubic fourfold associated to some polarized K3 surface S, defined over a number field K and satisfying , then X has a model over K such that the L-series of the primitive cohomology of X/K can be expressed in terms of the L-series of S/K. This allows us to compute the L-series for a discrete dense subset of cubic fourfolds in the moduli spaces of certain special cubic fourfolds. We also discuss a concrete example.     

A degree bound for globally generated vector bundles

Let E be a globally generated vector bundle of rank e ≥ 2 over a reduced irreducible projective variety X of dimension n defined over an algebraically closed field of characteristic zero. Let L := det(E). If deg(E) := deg(L) = L ⁿ  > 0 and E is not isomorphic to , we obtain a sharp bound

Generation of k-jets on toric varieties

The notion of a k-convex -support function for a toric variety is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the -support function . Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k. Received October 22, 1997; in final form January 12, 1998     

Extensions of regular mappings and the ?ojasiewicz exponent at infinity

Let F:V→Cm be a regular mapping, where V⊂Cn is an algebraic set of positive dimension and m?n?2, and let L_∞(F) be the ?ojasiewicz exponent at infinity of F. We prove that F has a polynomial extension G:Cn→Cm such L_∞(G)=L_∞(F). Moreover, we give an estimate of the degree of the extension G. Additionally, we prove that if then for any β∈Q, β?L_∞(F), the mapping F has a polynomial extension G with L_∞(G)=β. We also give an estimate of the degree of this extension.     

Vector bundles on ruled Fano 3-folds

Let be a ruled Fano 3-fold. The goal of this paper is to compute the dimension, prove the irreducibility and smoothness and describe the structure of the moduli space M _L (2;c ₁,c ₂) of L-stable, rank 2 vector bundles E on X with certain Chern classes and for a suitable polarization L closely related to c ₂. More precisely, we will cover the study of some moduli spaces M _L (2;c ₁,c ₂) such that the generic point is given as a non-trivial extension of line bundles. This work nicely reflects the general philosophy that moduli spaces inherits a lot of geometrical properties of the underlying variety. Received: 16 February 1999 / Revised version: 2 July 1999     

Numerically positive divisors on algebraic surfaces

Numerically positive line bundles on a complex projective smooth algebraic surfaceS are studied. In particular for any such line bundleL Pic(S) we prove the following facts: (i)g(L) 0 and (ii)L is ample ifg(L) 1,g standing for the arithmetic genus. Some applications are discussed. We also investigate numerically positive non-ample line bundlesL withg(L)=2.     

Simultaneous generation of jets on K3 surfaces

Let L be an ample line bundle on a K3 surface. We give a sharp bound on n for which nL is k-jet ample.Received: 27 December 2002     

On complex manifolds polarized by an ample line bundle of sectional genus q ( X ) + 2

Let (X,L) be a polarized manifold with dim X = n. In this paper, we classify (X,L) with n = 3, , and g(L)=q(X) + 2. Moreover we also classify (X,L) with , g(L)=q(x) + 2, and . Received February 12, 1999     

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”.     

A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over and let L be a line bundle on X. In this paper we shall construct a moduli space parametrizing -cohomology L-twisted Higgs pairs, i.e., pairs where E is a vector bundle on X and . If we take , the canonical line bundle on X, the variety is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section , one can define, in a natural way, a Poisson structure on . We also analyze the relations between this Poisson structure on and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bo1] in the case of curves. Received November 4, 1997; in final form May 28, 1998     

Heights for line bundles on arithmetic varieties

Let X be an arithmetic variety and L be an element of the Néron-Severi group of its generic fiber X _K . Then there are only finitely many line bundles on X, generically belonging to L, such that the degrees of on the irreducible components of the special fibers of X and the height of are bounded. The concept of a height used here is recalled. Several elementary properties of this height are proven. Received: 9 March 1996     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

Fonction L unité d'un groupe de Barsotti-Tate

The purpose of this article is to give a cohomological formula for the unit-root part of the L-function associated to a Barsotti-Tate group G on a scheme S over a field of characteristic p when G extends to some compactification of S. This is an analogue of a part of a conjecture of Katz according to wich the L-function of an F-crystal should be expressed in terms of the p-adic etale sheaf corresponding to the unit-root part of the crystal. In order to carry out this project, we use the technics of [E-LS II] wich require in our case an extension of the Dieudonné crystalline theory ([B-B-M]) to “crystal of level mG” in the sense of Berthelot. We show that the unit-root L-function of the Dieudonné crystal associated to G can be expressed in terms of the syntomic cohomology of the Ext group of G by the constant sheaf.

Received: 24 March 1997 / Revised version: 6 January 1998     

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S. Received: 28 March 2000 / Revised version: 20 October 2000     

Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this paper, we provide a lower bound for h⁰(m(K_X+L)) under the assumption that κ(K_X+L)≥0. In particular, we get the following: (1) if 0≤κ(K_X+L)≤2, then h⁰(K_X+L)>0 holds. (2) If κ(K_X+L)=3, then h⁰(2(K_X+L))≥3 holds. Moreover we get a classification of (X,L) with κ(K_X+L)=3 and h⁰(2(K_X+L))=3 or 4.