On the Lüroth semigroup of curves lying on a smooth quadric

For a smooth irreducible complete algebraic curveC the “gaps” are the integersn such that every linear series of degreen has at least a base point. The Lüroth semigroup S_C of a curveC is the subsemigroup ofN whose elements are not gaps. In this paper we deal with irreducible smooth curves of type (a, b) on a smooth quadricQ. The main result is an algorithm by which we can say if some integer λ∈N is a gap or is in S_C. In the general case there are integers λ which are undecidable. For curves such as complete intersection, arithmetically Cohen-Macaulay or Buchsbaum, we are able to describe explicitly “intervals” of gaps and “intervals” of integers which belong to S_C. For particular cases we can completely determine S_C, by giving just the type of the curve (in particular the degree and the genus). Work done with financial support of M.U.R.S.T. while the authors were members of G.N.S.A.G.A. of C.N.R.     

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.     

Specialization of the torsion subgroup of the Chow group

An example is given in which specialization is not injective. Partial support by the NSF (DMS-0200012) gratefully acknowledged.     

On the motive of certain subvarieties of fixed flags

We compute the Chow motive of certain subvarieties of the Flags manifold and show that it is an Artin motive.     

On Characteristic Classes of Determinantal Cremona Transformations

We compute the multidegrees and the Segre numbers of general determinantal Cremona transformations, with generically reduced base scheme, by specializing to the standard Cremona transformation and computing its Segre class via mixed volumes of rational polytopes. Dedicated to the memory of Shiing-Shen Chern     

A relation between the parabolic Chern characters of the de Rham bundles

In this paper, we consider the weight i de Rham–Gauss–Manin bundles on a smooth variety arising from a smooth projective morphism for . We associate to each weight i de Rham bundle, a certain parabolic bundle on S and consider their parabolic Chern characters in the rational Chow groups, for a good compactification S of U. We show the triviality of the alternating sum of these parabolic bundles in the (positive degree) rational Chow groups. This removes the hypothesis of semistable reduction in the original result of this kind due to Esnault and Viehweg.     

Adequate equivalence relations and Pontryagin products

Let A be an abelian variety over a field k. We consider CH₀(A) as a ring under Pontryagin product and relate powers of the ideal I⊆CH₀(A) of degree zero elements to powers of the algebraic equivalence relation. We also consider a filtration F⁰⊇F¹⊇… on the Chow groups of varieties of the form T×_kA (defined using Pontryagin products on A×_kA considered as an A-scheme via projection on the first factor) and prove that F^r coincides with the r-fold product (F¹)^*r as adequate equivalence relations on the category of all such varieties.     

Torsion cohomology classes and algebraic cycles on complex projective manifolds

Atiyah and Hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold, which are not Poincaré dual to an algebraic cycle. We notice that the order of these classes must be small compared to the dimension of the manifold. However, building upon a construction of Kollár, one can provide such examples with arbitrary high prime order, the dimension being fixed. This method also provides examples of torsion algebraic cycles, which are non trivial in the Griffiths group, and lie in a arbitrary high level of the H. Saito filtration on Chow groups.     

On syzygies of projective bundles over abelian varieties

Syzygies or $N_p$-property of an ample line bundles on abelian varieties are well known. In this paper, we study defining equations and syzygies among them of projective bundles over abelian varieties. We prove an analogue of Pareschi's theorem (or Lazarsfeld's conjecture) on abelian varieties, extended to projective bundles over an abelian variety.     

On a notion of toric special linear systems

In this note, we study linear systems on complete toric varieties X with an invariant point whose orbit under the action of $\mathrm{Aut}(X)$ contains the dense torus T of X. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations.     

Zero-cycles on self-product of modular curves

We use the elements in K-cohomology groups which are constructed by Flach and Mildenhall to obtain a finiteness result for the torsion part of the Chow group of a self-product of a modular curve.     

Non-trivial elements in the Abel–Jacobi kernels of higher-dimensional varieties

The purpose of this paper is to construct non-trivial elements in the Abel–Jacobi kernels in any codimension by specializing correspondences with non-trivial Hodge-theoretical invariants at points with different transcendence degrees over a subfield in C$\mathbb{C}$.     

Okounkov bodies and Seshadri constants

Okounkov bodies, which are closed convex sets defined for big line bundles, have rich information on the line bundles. On the other hand, Seshadri constants are invariants which measure the positivity of line bundles. In this paper, we prove that Okounkov bodies give lower bounds of Seshadri constants.     

Algebraic cycles and realification of complex projective varieties

We investigate some properties of underlying real algebraic structure on complex projective varieties.     

On a conjecture of Cox and Katz

This note shows that a certain toric quotient of the quintic Calabi-Yau threefold in provides a counterexample to a recent conjecture of Cox and Katz concerning nef cones of toric hypersurfaces. Received: 8 February 2001; in final form: 17 September 2001 / Published online: 1 February 2002     

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.     

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.