Enrique Arrondo

Parametric generalized offsets to hypersurfaces

In this paper we extend the classical notion of offset to the concept of generalized offset to hypersurfaces. In addition, we present a complete theoretical analysis of the rationality and unirationality of generalized offsets. Characterizations for deciding whether the generalized offset to a hypersurface is parametric or it has two parametric components are given. As an application, an algorithm to analyse the rationality of the components of the generalized offset to a plane curve or to a surface, and to compute rational parametrizations of its rational components, is outlined.

Genus formula for generalized offset curves

In this paper, we present a formula for computing the genus of irreducible generalized offset curves to projective irreducible plane curves with only affine ordinary singularities over an algebraically closed field. The formula expresses the genus of the offset by means of the degree and the genus of the original curve.

Vector bundles on fano 3-folds without intermediate cohomology

A well known result of G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964; Zbl 0126.16801)] says that a vector bundle on a projective space has no intermediate cohomology if and only if it decomposes as a direct sum of line bundles. It is also known that only on projective spaces and quadrics there is, up to a twist by a line bundle, a finite number of indecomposable vector bundles with no intermediate cohomology [see R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165-182 (1987; Zbl 0617.14034) and also H. Kn¨orrer, Invent. Math. 88, 153-164 (1987; Zbl 0617.14033)]. In the paper under review the authors deal with vector bundles without intermediate cohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-folds they consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5 and smooth 3-dimensional linear sections of G(1, 4) P9. A complete classification of rank two vector bundles without intermediate cohomology on such 3-folds is given. In fact the authors prove that, up to a twist, there are only three indecomposable vector bundles without intermediate cohomology. Vector bundles of rank greater than two are also considered. Under an additional technical condition, the authors characterize the possible Chern classes of such vector bundles without intermediate cohomology.

Schwarzenberger bundles of arbitrary rank on the projective space

We define Schwarzenberger bundles on any smooth projective variety X . We introduce the notions of jumping pairs of a Steiner bundle E on X and determine a bound for the dimension of its jumping locus. We completely classify Steiner bundles whose set of jumping pairs have maximal dimension, proving that they are all Schwarzenberger bundles. Introduction Inspired by a particular family of rank n vector bundles on P which had been introduced by Schwarzenberger in [Sch61], Steiner bundles on the projective space P were defined by Dolgachev and Kapranov in [DK93]. In this paper, as well as in [Val00] and [AO01], the authors assign a Steiner bundle on P to a certain configuration of hyperplanes. Moreover, given a family of special hyperplanes, which Vallès, Ancona and Ottaviani designate by unstable hyperplanes, they are able to reconstruct a Steiner bundle E and determine whether E is a Schwarzenberger bundle. Schwarzenberger bundles on the projective space of arbitrary rank were recently introduced in [Arr10]. In this work, a generalization of unstable hyperplanes of Steiner bundles, now called jumping hyperplanes, is given. After determining a range for the dimension of the jumping locus, Steiner bundles whose locus has maximal dimension are classified and proved to be all Schwarzenberger. In light of the definition of a Steiner bundle on smooth projective varieties X given in [MRS09], the results in [Arr10] were extended in [AM12] for the Grassmannian variety. Our goal in the present paper is to generalize the results in [Arr10] and [AM12] to any smooth projective variety. In Section 1 we recall the definition of Steiner bundles on smooth projective varieties and give an equivalent definition using linear algebra. In Section 2 we propose a definition of Schwarzenberger bundles on smooth projective varieties (see Definition 2.1), which generalizes the one given in [Arr10] and [AM12]. In Section 3 we define jumping pair for a Steiner bundle on X (see Definition 3.1), endow the set of all jumping pairs with the structure of an projective variety and give a lower bound for its dimension. In Section 4 we obtain an upper bound for the dimension of the jumping variety by studying its tangent space at a fixed jumping pair (see Theorem 4.1). In Section 5 we prove a complete classification of Steiner bundles whose jumping locus has maximal dimension and show that they all are Schwarzenberger bundles (see Theorem 5.1). Acknowledgements. The three authors were partially supported by Fundação para a Ciência e Tecnologia, project “Geometria Algébrica em Portugal”, PTDC/MAT/099275/2008; and by

Another Elementary Proof of the Nullstellensatz

1. P. Flajolet and A. Odlyzko, Singularity analysis of generating functions, SIAM J. Discrete Math. 3 (1990) 216-240. 2. R. E. Greenwood, The number of cycles associated with the elements of a permutation group, this Monthly 60 (1953) 407-409. 3. J. Konvalina, A unified interpretation of the binomial coefficients, the Stirling numbers, and the Gaussian coefficients, this Monthly 107 (2000) 901-910. 4. L. Milne-Thomson, The Calculus of Finite Differences, Chelsea, New York, 1981.

Appendix to“congruences of small degree in G(1,4)”: pfaffian linkage in codimension three and applications to congruences

We introduce the notion of Pfaffian linkage in codimension three and give sufficient conditions for the linked variety to be smooth. As a result, we are able to construct smooth congruences of lines in P4 whose existence was an open problem.

CLASSIFICATION OF n-DIMENSIONAL SUBVARIETIES OF G(1, 2n) THAT CAN BE PROJECTED TO G(1 ,n +1 )

A structure theorem is given for n-dimensional smooth subvarieties of the Grassmannian G(1, N); with N >= n + 3, that can be isomorphically projected to G(1, n + 1). A complete classification in the cases N = 2n + 1 and N = 2n follows, as a corollary.