On Automorphisms of Moduli Spaces of Parabolic Vector Bundles

Abstract

Fix n ≥ 5 general points p1, . . . , pn ∈ P 1 , and a weight vector A = (a1, . . . , an) of real numbers 0 ≤ ai ≤ 1. Consider the moduli space MA parametrizing rank two parabolic vector bundles with trivial determinant on P 1 , p1, . . . , pn which are semistable with respect to A. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space MA. It is isomorphic to Z

On the birational geometry of spaces of complete forms I: collineations and quadrics

“…Since such a fiber is isomorphic to (P n2 ) r2 × • • • × (P n h ) r h we conclude by induction on h. while PsAut(X (1)) ∼ = Aut(X (1)) ∼ = P GL (4). Furthermore, the pseudo-automorphism group of Q(n) is given by PsAut(Q(n)) ∼ = P GL(n + 1) S 2 if n 2 while PsAut(Q(1)) ∼ = Aut(Q(1)) ∼ = P GL (3).…”

“…First of all, note that by Theorem 4.11 the sections ofD 1 , D 2 , D 3 , E 1 , E 2 , E 3 are homogeneous generators ofCox(X (3)) with respect to the usual grading on Pic(X (3)) as displayed in(4.19). Furthermore, by(4.20) in the proof of Proposition 4.18 we have that Mov(X(3)…”

“…The Kontsevich moduli space M 0,0 (P 2 , 2), parametrizing degree two stable maps from a nodal rational curve to P 2 , is isomorphic to the space of complete conics Q (2) [19, Section 0.4]. Then Theorem 7.5 says in particular that Aut(M 0,0 (P 2 , 2)) ∼ = P GL(3) S 2 , where P GL (3) acts by motions of the target P 2 , and the involution of S 2 associates to a conic its dual.…”

“…For instance, Nef(X (3)) has three extremal rays that we will denote by D 1 , D 2 , D 2 . The divisors D i are big and so they induce birational models X (3)(D i ) of X (3). A general point of X (3)(D 1 ) corresponds to a twisted cubic, a general point of X (3)(D 2 ) corresponds to a rational normal quartic in the Grassmannian of lines of P 3 parametrizing tangent lines to a twisted cubic, and a general point of X (3)(D 3 ) corresponds to a twisted cubic parametrizing osculating planes to another twisted cubic.…”

“…Note that X (n) 1 is Fano for n ∈ {1, 2} while X (3) 1 is weak Fano, that is −K X (3)1 is nef and big, but not Fano. Now, applying the techniques developed in [7,17,18,34,35,37,38] to deal with automorphisms of moduli spaces of curves and stable maps and in [2,3] for moduli of parabolic vector bundles, we will finally compute the pseudo-automorphism group of the spaces of complete forms. We will need the following preliminary result.…”