Problems on Hilbert schemes and quiver bundles [HBNI Th280]

Saurav Holme Choudhury

dc.contributor.author	Saurav Holme Choudhury
dc.date.accessioned	2026-06-24T09:50:37Z
dc.date.available	2026-06-24T09:50:37Z
dc.date.submitted	2025-11
dc.identifier.uri	https://dspace.imsc.res.in/xmlui/handle/123456789/921
dc.description.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.	en_US
dc.description.tableofcontents	Notation ........................................................................ 21 1. Introduction .................................................................. 23 1.1 Arrangement of the Thesis ............................................. 26 2. Stratified Bundles on the Hilbert Scheme of n Points on a Surface .................................................................................. 29 2.1 Tannakian Formalism .................................................. 30 2.1.1 Neutral Tannakian Categories: Recovering an Affine Group Scheme from its Representations ............................................... 33 2.2 Stratified Bundles ...................................................... 36 2.2.1 The Group Scheme π alg (X,x) .......................... 39 2.3 The Hilbert Scheme of n Points on a Surface S ................. 41 2.4 The Functor Between Tannakian Categories ......................... 43 2.4.1 The Homomorphism ................................................ 50 2.5 Isomorphism of Fundamental Group Schemes ....................... 51 2.5.1 Faithfully Flat .................................................... 52 2.5.2 Closed Immersion .................................................. 55 2.6 Relation with the Étale Fundamental Group of S [n] .......... 58 3. Criterion for Rationality of Moduli of Chains ......................... 61 3.1 Introduction ............................................................ 61 3.2 Preliminaries ............................................................ 63 3.2.1 Preliminaries on Projective Bundles and Rationality Results .... 63 3.2.2 Preliminaries on Chains ............................................ 63 3.3 Moduli Spaces of Chains with Stable Components ................. 67 3.4 Non-emptiness of N θ s (r,d) .................................. 71 3.4.1 The Stack of Hecke Correspondences .............................. 71 3.4.2 The Stack of Injective Chains ...................................... 73	en_US
dc.publisher.publisher	The Institute of Mathematical Sciences
dc.subject	Hilbert schemes	en_US
dc.subject	Quiver bundles	en_US
dc.title	Problems on Hilbert schemes and quiver bundles [HBNI Th280]	en_US
dc.type.degree	Ph.D	en_US
dc.type.institution	HBNI	en_US
dc.description.advisor	Jaya N. Iyer
dc.description.pages	80p.	en_US
dc.type.mainsub	Mathematics	en_US
dc.type.hbnibos	Mathematical Sciences	en_US