On the Hilbert scheme of Palatini threefolds

Fania, Maria Lucia; Mezzetti, Emilia

doi:10.1515/advg.2002.017

Cited by 14 publications

(14 citation statements)

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“…In the present paper we generalise the result of [19] by proving that it holds for all m ≥ 4 and all k ≥ m − 1. Moreover, we prove that, for m = 3, the map ρ is generically injective for all k ≥ 4.…”

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confidence: 56%

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Skew-symmetric matrices and Palatini scrolls

Faenzi

Fania

2009

Math. Ann.

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cited By (since 1996)1International audienceWe prove that, for m greater than 3 and k greater than m - 2, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension 2k is birational to the Hilbert scheme of Palatini scrolls in P2k-1. For m = 3 and k > 3, this Grassmannian is proved to be birational to the set of pairs (ε, Y), where Y is a smooth plane curve of degree k and ε is a stable rank-2 bundle on Y whose determinant is oscriptY(k-1). © Springer-Verlag 2009

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confidence: 56%

“…A similar statement holds for the Palatini scroll in P 5 : the main result of [19] states that ρ is a birational map if m = 4, n = 5. On the other hand, it was proved in [4], and in fact classically known to Fano (see [20]), that the map ρ is generically 4 : 1 in case m = 3, n = 5.…”

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confidence: 60%

Skew-symmetric matrices and Palatini scrolls

Faenzi

Fania

2009

Math. Ann.

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“…On the other hand, a Palatini scroll X is obtained from only one map φ [FM02]. The geometrical interpretation of the congruence Γ in this case is that the lines of Γ are the 4-secant lines of X not contained in X, so unicity follows.…”

Section: Grassmannians Of Lines: Linear Sections and Focal Propertiesmentioning

confidence: 99%

Geometry of Lines and Degeneracy Loci of Morphisms of Vector Bundles

Mezzetti

2016

From Classical to Modern Algebraic Geometry

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ABSTRACT. Corrado Segre played a leading role in the foundation of line geometry. We survey some recent results on degeneracy loci of morphisms of vector bundles where he still is of profound inspiration. GRASSMANNIANS OF LINES: LINEAR SECTIONS AND FOCAL PROPERTIES1.1. Classical point of view: work of Corrado Segre on Grassmannians. The geometry of families of lines in the projective space, classically called " line geometry", has been one of the first objects of investigation of Corrado Segre, and a fil rouge of his research throughout his career. His graduation thesis was published in 1883, in two articles, in " Memorie dell'Accademia delle Scienze di Torino". In the first article [Seg83a], corresponding to the first two chapters of the thesis, Segre systematically studies the (hyper)quadrics, in the second one [Seg83b] he develops the geometry of the Klein quadric in P 5 , i.e. the Grassmannian G(1, 3) of lines in P 3 . He considers linear and quadratic sections of the Grassmannian, called linear and quadratic complexes when of dimension three, and linear and quadratic congruences when of dimension two, and moreover ruled surfaces. In particular he studies the notion of focal locus of a congruence, roughly speaking the set of points where two "infinitely near" lines of the family meet.We want to mention then a short article published by Corrado Segre in 1888 [Seg88], where he considers the line geometry in a projective space of any dimension n. Here Segre states a few basic facts on the focal points of a congruence of lines in P n , meaning now a family of dimension n − 1.

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“…It results that the lines of the congruence B are the 4-secant lines of X and X is the fundamental locus of B ( [FM02]). …”

Section: Introductionmentioning

confidence: 99%

On linear spaces of skew-symmetric matrices of constant rank

Manivel

Mezzetti

2005

manuscripta math.

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We consider the problem of constructing matrices of linear forms of constant rank by focusing on the associated vector bundles on projective spaces. Important examples are given by the classical Steiner bundles, as well as some special (duals of) syzygy bundles that we call Drézet bundles. Using the classification of globally generated vector bundles with small first Chern class on projective spaces, we are able to describe completely the indecomposable matrices of constant rank up to six; some of them come from rigid homogeneous vector bundles, some other from Drézet bundles related either to plane quartics or to instanton bundles on P 3 .

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On the Hilbert scheme of Palatini threefolds

Abstract: We study the Hilbert scheme of Palatini threefolds X in the projective space of dimension five. We prove that such a scheme has an irreducible component containing X which is birational to the Grassmannian G(3,14) and we determine the exceptional locus of the birational map

Cited by 14 publications

References 10 publications

Skew-symmetric matrices and Palatini scrolls

Skew-symmetric matrices and Palatini scrolls

Geometry of Lines and Degeneracy Loci of Morphisms of Vector Bundles

On linear spaces of skew-symmetric matrices of constant rank

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