Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

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     

On the 2 very ampleness of the adjoint bundle

Let S be a smooth projective surface over C polarized by a 2-very ample line bundle L=O _S(L), i.e. for any 0-dimensional subscheme (Z,O _Z ) of length 3 the restriction map Γ(L)→Γ(L⊗O _Z) is a surjection. This generalization of very ampleness was recently introduced by M. Beltrametti and A.J. Sommese. The authors prove that, if L·L≥13, the adjoint line bundleK _S⊗L is 2-very ample apart from a list of well understood exceptions and up to contracting down the smooth rational curves E such that E·E=−1, L·E=2. The appendix contains an inductive argument in order to extend the result in higher dimension.     

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.     

Complex curves of genus three, Kummer surfaces and Quillen metrics

Let C be a smooth, complex, projective curve of genus 3. By choosing an unramified double covering of C, the Abel-Prym map yields an embedding of C into a Kummer surface K when C is non-hyperelliptic. We compute the Quillen metric on the determinant of the cohomologies of with respect to the metric on C induced from the flat Kähler metric on K. For the computation of the Quillen metric, we show the exact self-duality of the Heisenberg-invariant Kummer's quartic surfaces.     

Curves and 0-cycles on projective surfaces

LetC be a smooth curve with ag _n ¹ , i.e. a linear system of dimension 1 and degreen, lying on a smooth projective surfaceS. Let :S P ^N be the map associated to the line bundleK _S +[C] and letD be a general divisor of the given linear systemg _n ¹ . LetV be the linear space spanned by the image ofD through . We study the case in whichn:=dimV=1 and in general we discuss the case in whichn is small. The starting point is an analysis of the adjunction map using Bogomolov-Reider-Serrano techniques; several results from curve theory are also needed.     

On the complexity of the projective classification of surfaces

LetX be a complex, connected, projective surface. LetL be a very ample line bundle onX, i.e. there is an embedding :X P _c with . In this article we study projective classification for surfaces when the independent variable is large.     

Segre classes of a vector bundle spanned by global sections

We show that the Segre polynomial determines the minimal number of sections spanning a vector bundle spanned by global sections.The author was an NSERC Postdoctoral Fellow while preparing this paper.     

Classification of polarized manifolds admitting homogeneous varieties as ample divisors

We give a complete classification of smooth polarized varieties (X, L) such that the linear system |L| has a homogeneous member A.     

Notes on very ample vector bundles on 3-folds

Let be a very ample vector bundle of rank two on a smooth complex projective threefold X. An inequality about the third Segre class of is provided when is nef but not big, and when a suitable positive multiple of defines a morphism X → B with connected fibers onto a smooth projective curve B, where K_X is the canonical bundle of X. As an application, the case where the genus of B is positive and has a global section whose zero locus is a smooth hyperelliptic curve of genus ≧ 2 is investigated, and our previous result is improved for threefolds. Received: 27 January 2005; revised: 26 March 2005     

On complex n-folds polarized by an ample line bundle L with .

Let (X L) be a polarized manifold with dim X = n≥3 and dim Bs |L|≤0. In this paper, we classify (X,L) with _g(L) = q(X) +m and h^o(L) ≥ n + m.     

On non-vanishing of cohomologies of generalized Raynaud polarized surfaces

We consider a family of slightly extended version of Raynaud’s surfaces X over the field of positive characteristic with Mumford-Szpiro type polarizations Z, which have Kodaira non-vanishing H¹(X,Z⁻ⁿ)≠0 for all 1≤n≤N with some N≥1. The surfaces are at least normal but smooth under a special condition. We also give a fairly large family of non-Mumford-Szpiro type polarizations Z_a,b with Kodaira non-vanishing.     

Pluricanonical maps of stable log surfaces

Stable surfaces and their log analogues are the type of varieties naturally occurring as boundary points in moduli spaces. We extend classical results of Kodaira and Bombieri to this more general setting: if (X,Δ)$(X,\mathrm{\Delta })$ is a stable log surface with reduced boundary (possibly empty) and I   is its global index, then 4I(_KX+Δ)$4I(K_X+\mathrm{\Delta })$ is base-point-free and 8I(_KX+Δ)$8I(K_X+\mathrm{\Delta })$ is very ample.     

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 partially ample adjoint divisors

We study the MMP (minimal model program) invariant properties of partially ample adjoint divisors. Using these invariant properties, we prove that the three notions of (numerically, cohomologically, asymptotically) partial ampleness coincide for big divisors on Q$\mathbb{Q}$-factorial Fano type varieties and more generally Mori dream spaces. This gives a partial answer to a question raised by Totaro.     

The d-very ampleness of adjoint line bundles on quasi-elliptic surfaces

In this paper, we give a numerical criterion of Reider-type for the d-very ampleness of the adjoint line bundles on quasi-elliptic surfaces, and meanwhile we give a new proof of the vanishing theorem on quasi-elliptic surfaces emailed from Langer and show that it is the optimal version.