Abstract:
This thesis is divided into two distinct projects.
(I) Stratified Bundles on Hilbert Scheme of Points on a Surface:
Let k be an algebraically closed field of characteristic p>3, and let S be a smooth projective surface over k with a k-rational point x. For n≥2, let S
[n]
denote the Hilbert scheme of n points on S. We compute the fundamental group scheme π
alg
(S
[n]
,
n
~
x
), defined by the Tannakian category of stratified bundles on S
[n]
.
(II) Criteria for Rationality of Moduli of Chains:
Let X be a compact Riemann surface of genus 2. We study the birational geometry of the moduli of holomorphic chains of type t on X, which are stable with respect to a fixed parameter θ. For suitable t and θ, we establish criteria for the rationality of these moduli spaces.
