A scheme related to the Brauer loop model

We introduce the_Brauer loop scheme_ E := {M in M_N(C) : M\cp M = 0}, where \cp is a certain degeneration of the ordinary matrix product. Its components of top dimension, floor(N^2/2), correspond to involutions \pi in S_N having one or no fixed points. In the case N even, this scheme contains the upper-upper scheme from [Knutson '04] as a union of (N/2)! of its components. One of those is a degeneration of the_commuting variety_ of pairs of commuting matrices. The_Brauer loop model_ is a quantum integrable stochastic process introduced in [de Gier--Nienhuis '04], and some of the entries of its Perron-Frobenius eigenvector were observed (conjecturally) to match the degrees of the components of the upper-upper scheme. We extend this, with proof, to_all_ the entries: they are the degrees of the components of the Brauer loop scheme. Our proof of this follows the program outlined in [Di Francesco--Zinn-Justin '04]. In that paper, the entries of the Perron-Frobenius eigenvector were generalized from numbers to polynomials, which allowed them to be calculated inductively using divided difference operators. We relate these polynomials to the multidegrees of the components of the Brauer loop scheme, defined using an evident torus action on E. In particular, we obtain a formula for the degree of the commuting variety, previously calculated up to 4x4 matrices.Comment: 31 pages, 4 figures; v2 has tiny edits and an extra circle actio

Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type

Let X be a compact Kahler holomorphic-symplectic manifold, which is deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface. Let L be a nef line-bundle on X, such that the 2n-th power of c_1(L) vanishes and c_1(L) is primitive. Assume that the two dimensional subspace H^{2,0}(X) + H^{0,2}(X), of the second cohomology of X with complex coefficients, intersects trivially the integral cohomology. We prove that the linear system of L is base point free and it induces a Lagrangian fibration on X. In particular, the line-bundle L is effective. A determination of the semi-group of effective divisor classes on X follows, when X is projective. For a generic such pair (X,L), not necessarily projective, we show that X is bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion sheaves, each with pure one dimensional support, on a projective K3 surface.Comment: 34 pages. v3: Reference [Mat5] and Remark 1.8 added. Incorporated improvement to the exposition and corrected typos according to the referees suggestions. To appear in the proceedings of the conference Algebraic and Complex Geometry, Hannover 201

Automorphism group of the moduli space of parabolic vector bundles over a curve

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 26-10-2018The main objective of this thesis is the computation of the automorphism group of the moduli space of parabolic vector bundles over a smooth complex projective curve. We will start by de ning the notion of a parabolic -module { a module over a sheaf of rings of di erential operators with a parabolic structure at certain marked points { and building their moduli space. This will provide us a common theoretical framework that allows us to work with several kinds of moduli spaces of bundles with parabolic structure such parabolic vector bundles, parabolic (L-twisted) Higgs bundles, parabolic connections or parabolic -connections. As an application, we build the parabolic Hodge moduli space and the parabolic Deligne{Hitchin moduli space. Then, we will address the computation of the automorphism group of the moduli space of parabolic bundles. Let X and X0 be irreducible smooth complex projective curves with sets of marked points D X and D0 X0 and genus g 6 and g0 6 respectively. LetM(X; r; ; ) be the moduli space of rank r stable parabolic vector bundles on (X;D) with parabolic weights and determinant . We classify the possible isomorphisms : M(X; r; ; ) ���! M(X0; r0; 0; 0). First, a Torelli type theorem is proved, implying that for to exist it is necessary that (X;D) = (X0;D0) and r = r0. Then we prove that the possible isomorphisms are generated by automorphisms of the pointed curve (X;D), tensorization with suitable line bundles, dualization of parabolic vector bundles and Hecke transformations at the parabolic points. These results are extended to birational equivalences : M(X; r; ; ) 99K M(X0; r0; 0; 0) which are de ned over \big" open subsets. The particular case of \concentrated" weights (corresponding to \small" stability parameters) is studied further. In this case Hecke transformations give rise to birational morphisms that do not extend to automorphisms of the moduli space. Moreover, an analysis of the stability chambers for the weights allows us to determine an explicit computable presentation of the group of automorphisms of the moduli space for arbitrary generic weights. Finally, the automorphism group of the moduli space of framed bundles over a smooth complex projective curve X of genus g > 2 with a framing over a point x 2 X is also described. It is shown that this group is generated by pullbacks using automorphisms of the curve X that x the marked point x, tensorization with certain line bundles over X and the action of PGLr(C) by composition with the framing