Automorphisms of the Quot Schemes Associated to Compact Riemann Surfaces

Abstract

Let X be a compact connected Riemann surface of genus at least two. Fix positive integers r and d. Let Q denote the Quot scheme that parametrizes the torsion quotients of O ⊕r X of degree d. This Q is also the moduli space of vortices for the standard action of U(r) on C r . The group PGL(r, C) acts on Q via the action of GL(r, C) on O ⊕r X . We prove that this subgroup PGL(r, C) is the connected component, containing the identity element, of the holomorphic automorphism group Aut(Q). As an application of it, we prove that the isomorphism class of the complex manifold Q uniquely determines the isomorphism class of the Riemann surface X.

On the Kähler structures over Quot schemes, II

“…Since this action of PGL(r, C) on Q is effective, we have PGL(r, C) ⊂ Aut 0 (Q) . It is known that PGL(r, C) = Aut 0 (Q) [BDH,Theorem 1.1]. Since the standard action on CP r−1 of any element A ∈ PGL(r, C) has a fixed point, the action of A on Q has a fixed point.…”

Nef cones of some Quot schemes on a Smooth Projective Curve

“…As a corollary we get the following result. When g = 1 note that µ 12), such that O Γ defines a map D → C (2) and F = F /T 0 (F ) is a line bundle on D which is globally generated. Hence…”

“…In [15], the author computed the Nef(C (d ) ) in the case when C is a very general curve of even genus and d = gon(C ) − 1. In [11] Nef(C (2) ) is computed in the case when C is very general and g is a perfect square. In [5] Nef(C (2) ) was computed assuming the Nagata conjecture.…”

“…In [11] Nef(C (2) ) is computed in the case when C is very general and g is a perfect square. In [5] Nef(C (2) ) was computed assuming the Nagata conjecture. We refer the reader to [12,Section 1.5] for more such examples and details.…”

“…In [1], the Betti cohomologies of Q(O n C , d ) are computed, Q(O n C , d ) has been interpreted as the space of higher rank divisors of rank n, and an analogue of the Abel-Jacobi map was constructed. In [2] the automorphism group scheme of Q(Q n C , d ) was computed in the case when the genus of C satisfies g (C ) > 1 and a Torelli theorem for these Quot schemes was proved. In [3] the Brauer group of Q(O n C , d ) is computed.…”