Infinitesimal deformations of some Quot schemes

Abstract

Let E be a vector bundle on a smooth complex projective curve C of genus at least two. Let Q(E, d) be the Quot scheme parameterizing the torsion quotients of E of degree d. We compute the cohomologies of the tangent bundle T Q(E,d) . In particular, the space of infinitesimal deformations of Q(E, d) is computed. Kempf and Fantechi computed the space of infinitesimal deformations of]). We also explicitly describe the infinitesimal deformations of Q(E, d).

Extension groups of tautological bundles on symmetric products of curves

“…In Section 2, we recall some basic definitions and facts on symmetric powers and punctual Quot schemes of curves, and on tautological bundles on these varieties. In the short Section 3, we deduce Theorem 1.2 from a result of [BGS22].…”

“…Biswas, Gangopadhyay, and Sebastian [BGS22] computed the cohomology of the tangent bundle of the punctual Quot scheme. Theorem 1.2 is a rather easy corollary of one of their auxiliary results, describing the push-forward of the universal quotient sheaf along the Quot-Chow morphism.…”

“…In [BGS22], the cohomology of the tangent bundle of Quot d was computed. The tangent bundle is not a tautological bundle.…”

“…This is Theorem 3.2 together with Proposition 2.2. d) , we write Quot D for the fibre of µ : Quot d → C (d) over the point D. One key to obtain Proposition 3.1 and related results in [BGS22] is to study the restriction of the universal sheaves on C × Quot d to the fibres of the Quot-Chow morphism by means of a certain birational morphism g x : S x → Quot D (note that in [BGS22], and also in [GS20] where it was studied first, this morphism is denoted by d) , an order x = (x 1 , . .…”