On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Abstract

Let M (n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M (n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms M or P (G, M (n, ξ)) and the moduli space of stable bundles over X × G, where G is the Grassmannian G(n − r, C n ). Moreover, we give sufficient conditions for M or 2ns (P 1 , M (n, ξ)) to be non-empty, when s ≥ 1.

On the Hilbert scheme of the moduli space of torsion free sheaves on surfaces

“…In [20] Narasimhan and Ramanan used elementary transformations of vector bundles on curves to introduce certain subvarieties in the moduli space of vector bundles which they called Hecke cycles. Brambila-Paz and the first author also used m-elementary transformations to describe a non-singular open set of the Hilbert scheme of the moduli space of vector bundles on a curve [2]. Coskun and Huizenga have been used elementary transformations to study priority sheaves since that they are well-behaved under elementary modifications [4,5,6].…”

“…The aim of this section is to define an embedding from G(E x , m) into the moduli space M X,H (n; c 1 , c 2 + m) of torsion free sheaves. Generalizing some techniques of [2] and [20] we establish a closed embedding φ z :…”

“…Hence, Narasimhan and Ramanan computed a bound for the dimension of the Hecke component and proved that is non-singular in those points defined by Hecke cycles. Moreover, when X is a curve and m ≥ 2, Brambila-Paz and Mata-Gutiérrez in [2] generalized the construction of Hecke cycles using Grassmannians and defined Hecke Grassmannians. They proved that the corresponding Hecke component is non-singular and a bound for its dimension was given.…”

“…The proof of this Theorem follows some ideas and techniques of [2] and [20]. For a fixed vector bundle E and a point x ∈ X we determine a closed embedding φ z : Proposition 3.4).…”

Geometry of Moduli Stacks of $(k,l)$-stable vector bundles over an algebraic curve

“…With the results described above we can now also define a general Hecke correspondence for the moduli stacks Bun k,l X (n, d) of (k, l)-stable vector bundles. Hecke correspondence have been defined and used in many contexts (see, [BM,Go2,Hl,LOZ]). In particular, Hoffmann in [Ho] described a Hecke correspondence for the moduli stack of all vector bundles over an algebraic curve using the evaluation map transformation and he constructed a vector bundle over any given open substack.…”