Automorphisms of relative Quot schemes

Abstract

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.

Nef cones of some Quot schemes on a Smooth Projective Curve

“…Recall the definition of η from equation (7), this is a section of Φ. For a line bundle L on C we have a line bundle G d ,L over C (d ) (see [15, page 8] for notation).…”

“…The first being a line in the fiber of Φ : Q → C (d ) , see Definition 12, which was denoted [l ]. Recall the section η of Φ from equation (7), taking L to be the trivial bundle. The second curve is η * ([l ]), where [l ] is from Definition 2.…”

“…By Lemma 22 and Lemma 23 we get that B O (d −1), Q is nef. To show B O (d −1), Q is not ample, consider a section η : C (d ) → Q constructed as in (7) with L the trivial bundle. Let p i denote the two projections from P 1 × PW * .…”

“…In [3] the Brauer group of Q(O n C , d ) is computed. In [7], the automorphism group scheme of Q(E , d ) was computed in the case when either rk E ≥ 3 or E is semistable and genus of C satisfies g (C ) > 1. In [8], the S-fundamental group scheme of Q(E , d ) was computed.…”

Infinitesimal deformations of some Quot schemes

“…From this it follows that the maximal connected subgroup of Aut(Q(O ⊕r , d)) is PGL(r, C) = Aut(O ⊕r )/C * . More generally, if either E is semistable or r 3, then H 0 (Q(E, d), T Q(E,d) ) = H 0 (X, End(E))/C [Gan19], and hence the maximal connected subgroup of Aut(Q(E, d)) is Aut(E)/C * .…”