ON SYMMETRIC AND SKEW-SYMMETRIC DETERMINANTAL VARIETIES (original) (raw)

THE CONNECTEDNESS OF SYMMETRIC AND SKEW-SYMMETRIC DEGENERACY LOCI: EVEN RANKS

A degeneracy locus is the set of points where a vector-bundle map has rank at most a given integer. Such a set is symmetric or skew-symmetric according as whether the vector-bundle map is symmetric or skew-symmetric. We prove a connectedness result, first conjectured by Fulton and Lazarsfeld, for skew-symmetric degeneracy loci and for symmetric degeneracy loci of even ranks.

On the stratication of secant varieties of Veronese varieties via symmetric rank

When considering σ r (X), the variety of r-secant P r−1 to a projective variety X, one question which arises is what are the possible values of the X-rank of points on σ r (X), apart from the generic value r? This geometric problem is of particular relevance (also for Applied Math) when X is a variety parameterizing some kind of tensors. We study here the case when X is a Veronese variety (i.e. the case of symmetric tensors). We find the complete description of the rank strata in some cases, and we give algorithms which compute the symmetric rank.

On the locus of points of high rank

arXiv (Cornell University), 2017

Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any GL n orbit of a homogeneous form.

Symmetric powers of the cotangent bundle and classification of algebraic varieties

Lecture Notes in Mathematics, 1979

We introduce and study coordinate-wise powers of subvarieties of P n , i.e. varieties arising from raising all points in a given subvariety of P n to the r-th power, coordinate by coordinate. This corresponds to studying the image of a subvariety of P n under the quotient of P n by the action of the finite group Z n+1 r. We determine the degree of coordinate-wise powers and study their defining equations, in particular for hypersurfaces and linear spaces. Applying these results, we compute the degree of the variety of orthostochastic matrices and determine iterated dual and reciprocal varieties of power sum hypersurfaces. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with a degenerate eigenspectrum.

On the Geometry of Conic Bundles Arising in Adjunction Theory

Mathematische Nachrichten, 2009

Besana, Conic Bundles 225 1) Yis a smooth, 2) There exists an ample line bundle .A on Ysuch that H = K , @ & , 3) Let X 3 E 1L'"-3) 1 be a generic smooth threefold section of (X, L). If K , , @ Llx3 is spanned by global sections then K,0L@"'"-2) is spanned. I n particular this is true The author would like to express his deepest gratitude to his adviser Professor ANDREW J. SOMMESE, under whose direction this work was produced as part of the author's Ph. D. thesis. i;r 2 5. 2. Notation and background material The notation used in this work is mostly standard from Algebraic Geometry. Good references are [16] and [14]. 0.1. The ground field is always the field C of complex numbers. All varieties and manifolds and their dimensions are over C. Unless otherwise stated all varieties are supposed to be projective. P" denotes the n-dimensional complex projective space and C* the multiplicative group of non zero complex numbers. 0.2. Given a projective n-dimensional variety X, 0, denotes its structure sheaf and Pic(X) denotes the group of line bundles over X. Line bundles, vector bundles and CARTIER divisors are denoted by capital letters, sometimes script, as L., M, g ' , & ... Invertible sheaves, line bundles and divisors are used interchangeably as customary. If D is a divisor on X , the associated invertible sheaf is denoted O,(D) or sometimes O(D) when there is no possibility of confusion regarding the ambient variety. Let LEPic(X), then the following notation is used: L. C the intersection number of L with a curve C ; L " the degree of L; L@' the line bundle L O. .. O L s-times; ILI the complete linear system of effective divisors associated with L; LI,. the restriction of L to a subvariety Y c X. If L. C 2 0 for every irreducible curve C, the line bundle L is said to be nef. A nef line bundle L is big if L"' > 0. 0.3. Given a vector bundle 6' on an n-dimensional projective variety X , its dual is denoted &*. If rank 6 = 1 sometimes the dual is denoted-8. The maximum exterior power A"& is denoted det 8. P(8) denotes the projectivized bundle over X according &*-0 to [16] convention, i.e. P(&)=-. Recall that for any line bundle 9 it is C* 0.4. Given the rank-two vector bundle O @ O (n) over PI, the geometrically ruled surface P(O@O(n)) is denoted by F,. A fiber of the natural projection to PI is denoted by f or v. The image of the fundamental section is denoted by E or 1. For fundamental results on these surfaces see [16]. 0.5. If X is a smooth n-dimensional variety, K , or K denotes the canonical bundle, i.e. K , = AnT: where T, is the holomorphic tangent space. If X is normal the dualizing 1261 J. WAVRIK, Deformations of BANACH (branched) coverings of complex manifolds. American Journal of Mathematics, 90 (1968) 926-960 Unioersi/y of Norre Dame Norre Dame, I N U.S.A.