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Giorgio Ottaviani

Publications and source records attributed to Giorgio Ottaviani.

65 records · Page 4Linked to original sources
arXiv2006

Induction for secant varieties of Segre varieties

This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective $t$-secant varieties to Segre varieties for t < 7. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of P^n. We determine the set of p for which unbalanced Segre varieties have defective p-secant varieties. In addition, we show that the Segre varieties P^1 x P^1 x P^n x P^n and P^2 x P ^3 x P^3 are deficient and completely describe the dimensions of their secant varieties. In the final section we propose a series of conjectures about defective Segre varieties.

math.AG↗
arXiv2005

Resolutions of homogeneous bundles on P^2

In this paper we characterize minimal free resolutions of homogeneous bundles on P^2. Besides we study stability and simplicity of homogeneous bundles on P^2 by means of their minimal free resolutions; in particular we give a criterion to see when a homogeneous bundle is simple by means of its minimal free resolution in the case the first bundle of the resolution is irreducible.

math.AG↗
arXiv2005

Quivers and the cohomology of homogeneous vector bundles

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE type as the cohomology of a complex explicitly described. The main tool is the equivalence between the category of homogeneous bundles and the category of representations of a certain quiver ${\cal Q}_X$ with relations, whose vertices are the dominant weights of the reductive part of $P$. This equivalence was found in some cases by Bondal, Kapranov and Hille and we find the appropriate relations for any Hermitian symmetric variety, computing them explicitly for Grassmannians.

math.AG↗
arXiv2001

The Binet-Cauchy Theorem for the Hyperdeterminant of boundary format multidimensional Matrices

Let $A$, $B$ be multidimensional matrices of boundary format respectively $\prod_{i=0}^p(k_i+1)$, $\prod_{j=0}^q(l_j+1)$. Assume that $k_p=l_0$ so that the convolution $A\ast B$ is defined. We prove that $Det (A\ast B)=Det(A)^α\cdot Det(B)^β$ where $α= \frac {l_0!}{l_1!... l_q!}$, $β=\frac {(k_0+1)!}{k_1! ... k_{p-1}!(k_p+1)!}$ and $Det$ is the hyperdeterminant. When $A$, $B$ are square matrices this formula is the usual Binet-Cauchy Theorem computing the determinant of the product $A\cdot B$. It follows that $A \ast B$ is nondegenerate if and only if $A$ and $B$ are both nondegenerate. We show by a counterexample that the assumption of boundary format cannot be dropped.

math.AG↗
arXiv2001

Group action on instanton bundles over $\PP^3$

Denote by MI(k) the moduli space of k-instanton bundles E of rank 2 on $\PP^3=\PP(V)$ and by $Z_k(E)$ the scheme of k-jumping lines. We prove that $[E]\in MI(k)$ is not stable for the action of SL(V) if $Z_k(E)\neq\emptyset$. Moreover $\dim Sym(E)\ge 1$ if $length Z_k(E)\ge 2$. We prove also that E is special if and only if $Z_k(E)$ is a smooth conic. The action of SL(V) on the moduli of special instanton bundles is studied in detail.

math.AG↗
arXiv2001

Nondegenerate multidimensional matrices and instanton bundles

In this note we prove that the moduli space of rank $2n$ symplectic instanton bundles on ${\PP^{2n+1}}$, defined from the well known monad condition, is affine. This result was not known even in the case $n=1$, where the real instanton bundles correspond to self dual Yang Mills $Sp(1)$-connections over the 4-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles.

math.AG↗
arXiv2000

Syzygies of Veronese embeddings

We prove that the Veronese embedding of P^n of degree d with n\ge 2, d\ge 3 does not satisfy property N_p (according to Green and Lazarsfeld) if p\ge 3d-2. We conjecture that the converse holds. This is true for n=2 and for n=d=3.

math.AG↗
arXiv1997

Boundedness for codimension two submanifolds of quadrics

Arrondo, Sols and De Cataldo proved that there are only finitely many families of codimension two subvarieties not of general type in the smooth quadric of dimension $n+2$ for $n\ge 2 $, $n\neq 4$. In this paper we drop the assumption $n\neq 4$ from the previous result (obviously the assumption $n\ge 2$ cannot be removed).

alg-geom↗
arXiv1995

On singularities of ${\cal M}_{I\!\! P^3}(c_1,c_2)$

Let ${\cal M}_{I\!\! P^3}(c_1,c_2)$ be the moduli space of stable rank-$2$ vector bundles on $I\!\! P^3$ with Chern classes $c_1$, $ c_2$. We prove the following results. 1) Let $0 \le β< γ$ be two integers, ($γ\ge 2)$, such that $2γ-3β>0$; then ${\cal M}_{I\!\! P^3}(0,2γ^2-3β^2)$ is singular (the case $β=0$ was previously proved by M. Maggesi). 2) Let $0 \le β< γ$ be two odd integers ($γ\ge 5)$, such that $2γ-3β+1>0$; then ${\cal M}_{I\!\! P^3}(-1,2(γ/2)^2-3(β/2)^2+1/4)$ is singular. In particular ${\cal M}_{I\!\! P^3}(0,5)$, ${\cal M}_{I\!\! P^3}(-1,6)$ are singular.

alg-geom↗
arXiv1994

The Tangent Space at a Special Symplectic Instanton Bundle on $P_{2n+1}$

Mathematical instanton bundles on $ P_3$ have their analogues in rank--$2n$ instanton bundles on odd dimensional projective spaces $ P_{2n+1}$. The families of special instanton bundles on these spaces generalize the special 'tHooft bundles on $ P_3$. We prove that for a special symplectic instanton bundle $ E$ on $ P_{2n+1}$ with $c_2=k$ $h^1End( E) = 4(3n-1) k + (2n-5)(2n-1)$. Therefore the dimension of the moduli space of instanton bundles grows linearly in $k$. The main difference with the well known case of $ P_3$ is that $h^2End( E)$ is nonzero, in fact we prove that it grows quadratically in $k$. Special symplectic instanton bundles turn out to be singular points of the moduli space. Such bundles $ E$ are $SL(2)$--invariant and the result is obtained regarding the cohomology groups of $ E$ as $SL(2)$--representations.

alg-geom↗