Linear spaces of matrices of constant rank and instanton bundles

Boralevi, Ada; Faenzi, Daniele; Mezzetti, Emilia

doi:10.1016/j.aim.2013.08.018

Cited by 14 publications

(19 citation statements)

References 23 publications

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“…The algorithm returns an N × N block matrix (6,4), (10,5), (11,6), (12,7), (13,8), (14,9), and (15, 10), (c) −x 4 on (2, 1), (7,3), (8,4), (16,8), (17,9), and (18, 10), (d) −x 5 on (3, 1), (9,4), and (19,10). In Part 2 the algorithm builds a 20 × 10 matrix L with nonzero elements that satisfies det(M ) = −p.…”

Section: Discussionmentioning

confidence: 99%

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Uniform Determinantal Representations

Boralevi

Doornmalen²,

Draisma

et al. 2017

SIAM J. Appl. Algebra Geometry

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Abstract. The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak and Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.

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Section: Discussionmentioning

confidence: 99%

“…There is a considerable body of work devoted to giving lower and upper bounds for such dimensions, both in the case of bounded and constant rank, but these bounds are rarely sharp; see, among many other references, [15,20,38,42] and the more recent works on skew-symmetric matrices of constant rank [4,25].…”

Section: Spaces Of Singular Matrices Letmentioning

confidence: 99%

Uniform Determinantal Representations

Boralevi

Doornmalen²,

Draisma

et al. 2017

SIAM J. Appl. Algebra Geometry

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“…These problems have been considered in [BFM13]. The main result is a method to contruct a skew-symmetric matrix A of linear forms on P 3 , having constant rank r = n − 1, i.e.…”

Section: Maximal Dimensionmentioning

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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“…Almost all (that is, all except one) known examples of dimension 4 were produced by the two authors together with D. Faenzi in [3], and they have c = 4 and 6. For these values it is easy to see that 4 is an effective bound.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…Dans le cas c 2 1 − 4c 2 0, on prouve l'existence de fibrés globalement engendrés qui ne peuvent pas correspondre à des matrices du type ci-dessus, certains même définissant un plongement de P 2 dans une Grassmannienne. Notre résultat étend des travaux antérieurs sur le cas c 1 3. happened for example in [14,16,15,3]. On the other side, linear systems have been proved useful in approaching some classical problems in geometry: examples of this phenomenon are, among many others, [4,10,5].…”

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Planes of matrices of constant rank and globally generated vector bundles

Boralevi¹,

Mezzetti²

2015

Ann. inst. Fourier

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We consider the problem of determining all pairs (c1, c2) of Chern classes of rank 2 bundles that are cokernel of a skew-symmetric matrix of linear forms in 3 variables, having constant rank 2c1 and size 2c1 + 2. We completely solve the problem in the "stable" range, i.e. for pairs with c 2 1 − 4c2 < 0, proving that the additional condition c2 ≤ c 1 +1 2 is necessary and sufficient. For c 2 1 − 4c2 ≥ 0, we prove that there exist globally generated bundles, some even defining an embedding of P 2 in a Grassmannian, that cannot correspond to a matrix of the above type. This extends previous work on c1 ≤ 3.2010 Mathematics Subject Classification. 14J60, 15A30. Key words and phrases. Skew-symmetric matrices, constant rank, globally generated vector bundles. Research partially supported by MIUR funds, PRIN 2010-2011 project "Geometria delle varietà algebriche", and by Università degli Studi di Trieste -FRA 2013 project "Geometria e topologia delle varietà".

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Linear spaces of matrices of constant rank and instanton bundles

Cited by 14 publications

References 23 publications

Uniform Determinantal Representations

Uniform Determinantal Representations

Geometry of Lines and Degeneracy Loci of Morphisms of Vector Bundles

Planes of matrices of constant rank and globally generated vector bundles

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