arXiv subjects
Publications and source records attributed to Laurent Gruson.
In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many further unirationality results for special moduli spaces of rational curves on quadric hypersurfaces and del Pezzo surfaces are obtained via explicit birational methods.
We construct a Z_3-slice to the natural action of PGL(V) on the variety of smooth arcs in the projective space P(V).
We classify rank two vector bundles on P3 with Buchsbaum index equal to three and also give some results on the H1-module of "negative instanton"bundles.
We give a detailed analysis of the semisimple elements, in the sense of Vinberg, of the third exterior power of a 9-dimensional vector space over an algebraically closed field of characteristic different from 2 and 3. To a general such element, one can naturally associate an Abelian surface X, which is embedded in 8-dimensional projective space. We study the combinatorial structure of this embedding and explicitly recover the genus 2 curve whose Jacobian variety is X. We also classify the types of degenerations of X that can occur. Taking the union over all Abelian surfaces in Heisenberg normal form, we get a 5-dimensional variety which is a birational model for a genus 2 analogue of Shioda's modular surfaces. We find determinantal set-theoretic equations for this variety and present some additional equations which conjecturally generate the radical ideal.
Let E be a stable rank two normalized vector bundle on P3. If H is a general plane we show that h^0(E_H(1)) \leq 2+c_1. It follows that h^0(E(1)) \leq 2+c_1.
We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.
Let X be a smooth, connected, dimension n, quasi-projective variety imbedded in \PP_N. Consider integers {k_1,...,k_r}, with k_i>0, and the Hilbert Scheme H_{k_1,...,k_r}(X) of aligned, finite, degree \sum k_i, subschemes of X, with multiplicities k_i at points x_i (possibly coinciding). The expected dimension of H_{k_1,...,k_r}(X) is 2N-2+r-(\sum k_i)(N-n). We study the locus of points where H_{k_1,...,k_r}(X) is not smooth of expected dimension and we prove that the lines carrying this locus do not fill up \PP_N
We work over an algebraically closed field of arbitrary characteristic. Ellingsrud-Peskine proved that smooth surfaces in P^4 are subject to strong limitations. Their whole argument is derived from the fact that the sectional genus of surfaces of degree d lying on a hypersurface of degree s varies in an interval of length \frac{d(s-1)^2}{2s}. The aim of the present paper is to show that for smooth codimension two subvarieties of P^n, n\geq 5, one can get a similar result with an interval whose length depends only on s. The main point is Lemma 1.1 whose proof is a direct application of the positivity of N_X(-1) (where N_X is the normal bundle of X in P^n). We get a series of (n-3) inequalities; the first one of which being in the paper of Ellingsrud-Peskine, the second was obtained in a preliminary version (arXiv:math.AG/0406497) by an essentially equivalent but more geometric argument. Then we first derive two consequences: 1) roughly speaking, (Thm. 2.1, Remark 2.2) the family of "biliaison classes" of smooth subvarieties of P^5 lying on a hypersurface of degree s is limited; 2) the family of smooth codimension two subvarieties of P^6 lying on a hypersurface of degree s is limited (Thm. 1.4). The result quoted in 1) is not effective, but 2) is. In the last section we try to obtain precise inequalities connecting the usual numerical invariants of a smooth subcanonical subvariety X of P^n, n\geq 5 (the degree d, the speciality index e, the least degree, s, of an hypersurface containing X). In particular we prove (Thm. 3.12): s\geq n+1.
In this article we study genus 5 curves with a fixed point free involution. We give two geometrical caracterisations of these curves amoung all genus 5 curves. One of these was conjectured by Arbarello, Cornalba, Griffiths and Harris in their book on curves. The second one is given by some particular embeddings corresponding to the points of the Prym variety associated to the situation.