Déformations des extensions larges de faisceaux

Abstract

Let X be a projective smooth irreducible polarized variety over the field of complex numbers. Typical examples of wide extensions are vector bundles E that have a subsheaf F whose slope is much bigger than the slope of E/F, and such that F and E/F are stable. We study the deformations of such bundles. The case of unstable rank 2 bundles has been considered by S. A. Strømme on ‫ސ‬ 2 , and by C. Bȃnicȃ on ‫ސ‬ 3 . We build moduli spaces of wide extensions, and if the dimension of X is greater than 2, it may even happen that we obtain fine moduli spaces.

Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder-Narasimhan type

“…Remark 8.2. Moduli spaces of unstable bundles of rank 2 on the projective plane have been constructed by Strømme in [22] and this has been generalised to sheaves with Harder-Narasimhan filtrations of length two over smooth projective varieties by Drézet in [4].…”

Moduli of sheaves supported on curves of genus two in a quadric surface

“…It is then easy to see that Φ(τ ) = σ (see [6,Proposition 4.3.1]). It follows that for an extension (11) associated to σ, the sheaf F has zero-dimensional torsion if and only if σ ∈ Im(Φ).…”

Moduli of unstable bundles of HN-length two with fixed algebra of endomorphisms

“…Some moduli spaces of unstable bundles have been constructed by adding some extra information: the moduli spaces of unstable vector bundles of rank 2 and 3 were constructed in [14] and in [38], respectively, when dim X = 1 and the algebra of endomorphisms is fixed; in [40] when X is the projective plane and in [3] when X = P 3 (C), in both cases they consider the degree of instability. J. M. Drezet studied in [19] the case of very unstable bundles when dim X > 2. In [26] the authors construct the moduli spaces of pure sheaves with fixed Harder-Narasimhan type which have some additional data called an m-rigidification.…”