On the Hilbert scheme of degeneracy loci of twisted differential forms

We prove that, for 3 < m < n-1, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension n is birational to the Hilbert scheme of the degeneracy loci of m global sections of Omega(2), the twisted cotangent bundle on P^{n-1}. For 3=m<n-1 and n odd, this Grassmannian is proved to be birational to the set of Veronese surfaces parametrized by the Pfaffians of linear skew-symmetric matrices of order n.Comment: 19 pages. Minor corrections, exposition improved. To appear in Trans. Amer. Math. So

Degeneracy loci of twisted differential forms and linear line complexes

open1noWe prove that the Hilbert scheme of degeneracy loci of pairs of global sections of , the twisted cotangent bundle on , is unirational and dominated by the Grassmannian of lines in the projective space of skew-symmetric forms over a vector space of dimension n. We provide a constructive method to find the fibers of the dominant map. In classical terminology, this amounts to giving a method to realize all the pencils of linear line complexes having a prescribed set of centers. In particular, we show that the previous map is birational when n = 4.openFabio TanturriTanturri, Fabi

The unirationality of the Hurwitz schemes H_10,8 and H_13,7

open2siWe show that the Hurwitz scheme Hg; d parametrizing d-sheeted simply branched covers of the projective line by smooth curves of genus g, up to isomorphism, is unirational for ðg; dÞ ¼ ð10; 8Þ and ð13; 7Þ. The unirationality is settled by using liaison constructions in P1 P2 and P6 respectively, and through the explicit computation of single examples over a finite field.openHanieh Keneshlou; Fabio TanturriKeneshlou, Hanieh; Tanturri, Fabi

Orbital degeneracy loci and applications

Degeneracy loci of morphisms between vector bundles have been used in a wide variety of situations. We introduce a vast generalization of this notion, based on orbit closures of algebraic groups in their linear representations. A preferred class of our orbital degeneracy loci is characterized by a certain crepancy condition on the orbit closure, that allows to get some control on the canonical sheaf. This condition is fulfilled for Richardson nilpotent orbits, and also for partially decomposable skew-symmetric three-forms in six variables. In order to illustrate the efficiency and flexibility of our methods, we construct in both situations many Calabi--Yau manifolds of dimension three and four, as well as a few Fano varieties, including some new Fano fourfolds.Comment: To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5

Matrix factorizations and curves in P^4

open2siLet C be a curve in 4 and X be a hypersurface containing it. We show how it is possible to construct a matrix factorization on X from the pair (C,X) and, conversely, how a matrix factorization on X leads to curves lying on X. We use this correspondence to prove the unirationality of the Hurwitz space H12,8 and the uniruledness of the Brill-Noether space W13,91. Several unirational families of curves of genus 16 ≤ g ≤ 20 in 4 are also exhibited.openFrank-Olaf Schreyer; Fabio TanturriSchreyer, Frank-Olaf; Tanturri, Fabi

The geometry of the Coble cubic and orbital degeneracy loci

The Coble cubics were discovered more than a century ago in connection with genus two Riemann surfaces and theta functions. They have attracted renewed interest ever since. Recently, they were reinterpreted in terms of alternating trivectors in nine variables. Exploring this relation further, we show how the Hilbert scheme of pairs of points on an abelian surface, and also its Kummer fourfold, a very remarkable hyper-Kähler manifold, can very naturally be constructed in this context. Moreover, we explain how this perspective allows us to describe the group law of an abelian surface, in a strikingly similar way to how the group structure of a plane cubic can be defined in terms of its intersection with lines

Polyvector fields for Fano 3-folds

peer reviewedWe compute the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of Fano 3-folds. This is the first step in understanding the non-trivial Gerstenhaber algebra structure of this invariant, and yields some initial insights in the classification of Poisson structures on Fano 3-folds of higher Picard rank

Fano 3-folds from homogeneous vector bundles over Grassmannians

We rework the Mori–Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians