Upper bounds for the essential dimension of the moduli stack of SL n -bundles over a curve

Abstract

We find upper bounds for the essential dimension of various moduli stacks of SLn-bundles over a curve. When n is a prime power, our calculation computes the essential dimension of the stack of stable bundles exactly and the essential dimension is not equal to the dimension in this case.

The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

“…Our aim is to compute an upper bound on the essential dimension of Bun r,d X,D . When the divisor is empty, the moduli stack coincides with Bun r,d X , the moduli stack of vector bundles of rank r and degree d over X. Dhillon and Lemire found bounds on the essential dimension of Bun r,d X in [DL09]. Our results improve the upper bound obtained there; this is explained in Remark 13.3.…”

“…Remark 13.3. The main result of [DL09] shows that ed(Bun r,d X ) ≤ ⌊h g (r)⌋ + g . The function h g (r) is defined recursively by h g (1) = 1 and h g (r) − h g (r − 1) = r 3 − r 2 + r 4 4 (g − 1) + r 2 + r 2 g 2 4 + 1 4 .…”

“…Unlike the Jordan-Hölder filtration, the socle filtration is Galois invariant, so it exists over the base field. This sidesteps the major difficulty in [DL09].…”

“…To pass from the essential dimension of Bun r,d,s X,D to that of Bun r,d,ss X,D , the moduli stack of semistable vector bundles of rank r and degree d with parabolic structure along D, we use the socle filtration, a finite filtration of a semistable parabolic bundle with polystable parabolic bundles as quotients (see § 8 for details). This was the key to improving the bounds in [DL09] as there we used the the Jordan-Hölder filtration, a finite filtration of a semistable parabolic bundle with stable parabolic bundles as quotients. Unlike the Jordan-Hölder filtration, the socle filtration is Galois invariant, so it exists over the base field.…”

On the essential dimension of coherent sheaves

“…We then apply our general results to the special case of vector bundles of fixed rank r and degree d over a smooth projective curve C. Theorem 7.3 gives the essential dimension of the moduli stack Bun C,r,d in this case, modulo the now famous conjecture of Colliot-Thélène, Karpenko and Merkurjev [8, §1]. Our result improves the upper bounds on this essential dimension given in [9] and in [3].…”