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Let X, S be two smooth projective varieties and X → S be a smooth family of projective curves of genus ≥ 2 over an algebraically closed field k and let E be a vector bundle of rank r ≥ 3 over X and P(E) be its projectivization. Fix d ≥ 1. Let Q(E, d) be the relative quot scheme of torsion quotients of E of degree d. Then we show that the identity component of the group of automorphisms of Q(E, d) over S is isomorphic to the identity component of the group of automorphisms of P(E) over S. As a corollary, the identity component of the automorphism group of flag scheme of filtrations of torsion quotients of O r C , where r ≥ 3 and C a smooth projective curve of genus ≥ 2 is computed.

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).

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