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Enrique Arrondo

Publications and source records attributed to Enrique Arrondo.

16 recordsLinked to original sources
arXiv2024

Representation theory of finite groups through (basic) algebraic geometry

We introduce a new approach to representation theory of finite groups that uses some basic algebraic geometry and allows to do all the theory without using characters. With this approach, to any finite group $G$ we associate a finite number of points and show that any field containing the coordinates of those points works fine as the ground field for the representations of $G$. We apply this point of view to the symmetric group $S_d$, finding easy equations for the different symmetries of functions in $d$ variables. As a byproduct, we give an easy proof of a recent result by Tocino that states that the hyperdeterminant of a $d$-dimensional matrix is zero for all but two types of symmetry.

math.RT↗
arXiv2024

Cohomological characterization of Universal Bundles of G(1,n)

We characterize directs sums of twists of symmetric powers of the universal quotient bundle over the Grassmannian of lines. We use a method that could be used for analogue results on any arbitrary variety, and that should give stronger results than using the standard technique of Beilinson's spectral sequence.

math.AG↗
arXiv2024

On the vanishing of the hyperdeterminant under certain symmetry conditions

Given a vector space $V$ over a field $\K$ whose characteristic is coprime with $d!$, let us decompose the vector space of multilinear forms $V^*\otimes\overset{\text(d)}{\ldots}\otimes V^*=\bigoplus _λW_λ(X,\K)$ according to the different partitions $λ$ of $d$, i.e. the different representations of $S_d$. In this paper we first give a decomposition $W_{(d-1,1)}(V,\K)=\bigoplus_{i=1}^{d-1}W_{(d-1,1)}^i(V,\K)$. We finally prove the vanishing of the hyperdeterminant of any $F\in(\bigoplus_{λ\ne(d),(d-1,1)})\oplus W_{(d-1,1)}^i(V,\K)$. This improves the result in [10] and [1], where the same result was proved without this new last summand.

math.AG↗
arXiv2019

Skew-Symmetric Tensor Decomposition

We introduce the ``skew apolarity lemma'' and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in {$\bigwedge^dV_{\mathbb{C}}$ with $d\leq 3$ and $\dim V_{\mathbb{C}} \leq 8$}. New algorithms to compute the rank and a minimal decomposition of a tritensor are also presented.

math.AG↗
arXiv2013

Schwarzenberger bundles on smooth projective varieties

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.

math.AG↗
arXiv2012

Jumping pairs of Steiner bundles

In this work we introduce the definition of Schwarzenberger bundle on a Grassmannian. Recalling the notion of Steiner bundle, we generalize the concept of jumping pair for a Steiner bundle on a Grassmannian. After studying the jumping locus variety and bounding its dimension, we give a complete classification of Steiner bundle with jumping locus of maximal dimension, which all are Schwarzenberger bundles.

math.AG↗
arXiv2009

Cohomological Characterization of Vector Bundles on Grassmannians of Lines

We introduce a notion of regularity for coherent sheaves on Grassmannians of lines. We use this notion to prove some extension of Evans-Griffith criterion to characterize direct sums of line bundles. We also give a cohomological characterization of exterior and symmetric powers of the universal bundles of the Grassmannian.

math.AG↗
arXiv2008

Schwarzenberger bundles of arbitary rank on the projective space

We introduce a generalized notion of Schwarzenberger bundle on the projective space. Associated to this more general definition, we give an ad-hoc notion of jumping subspaces of a Steiner bundle on ${\Bbb P^n}$ (which in rank $n$ coincides with the notion of unstable hyperplane introduced by Vallès, Ancona and Ottaviani). For the set of jumping hyperplanes, we find a sharp bound for its dimension. We also classify those Steiner bundles whose set of jumping hyperplanes have maximal dimension and prove that they are generalized Schwarzenberger bundles.

math.AG↗
arXiv2006

A home-made Hartshorne-Serre correspondence

We provide an elementary proof of the Hartshorne-Serre correspondence for constructing vector bundles from local complete intersection subschemes of codimension two. This will be done, as in the correspondence of hypersurfaces and line bundles, by patching together local determinantal equations in order to produce sections of a vector bundle.

math.AG↗
arXiv2006

On the Picard group of low-codimension subvarieties

We introduce a method to determine if n-dimensional smooth subvarieties of an ambient space of dimension at most 2n − 2 inherit the Picard group from the ambient space (as it happens when the ambient space is a projective space, according to results of Barth and Larsen). As an application, we give an affirmative answer (up to some mild natural numerical conditions) when the ambient space is a Grassmannian of lines (thus improving results of Barth, Van de Ven and Sommese) or a product of two projective spaces of the same dimension.

math.AG↗
arXiv1998

Vector bundles on Fano 3-folds without intermediate cohomology

We study the vector bundles without intermediate cohomology on Fano threefolds of index two, degree d=3,4,5 and Betti number one. We obtain a complete characterization in the case of rank-two vector bundles. For arbitrary rank, we give all possible Chern classes of such vector bundles, under some general condition.

math.AG↗
arXiv1998

Vector bundles on G(1,4) without intermediate cohomology

We characterize the vector bundles on G(1,4) that have no intermediate cohomology. We obtain them from extensions of the universal bundles and others related with them. In particular, we get a characterization of the universal vector bundles from their cohomology.

math.AG↗
arXiv1993

Classification of smooth congruences with a fundamental curve

We give a classification and a construction of all smooth $(n-1)$-dimensional varieties of lines in ${\bf P}\sp n$ verifying that all their lines meet a curve. This also gives a complete classification of $(n-1)$-scrolls over a curve contained in $G(1,n)$.

alg-geom↗