Construction of Hilbert and Quot Schemes

TL;DR

This paper provides an expository overview of Grothendieck's construction of Hilbert and Quot Schemes, highlighting their importance in modern algebraic geometry and their development through key mathematicians.

Contribution

It offers a detailed exposition of the foundational construction of Hilbert and Quot Schemes, including historical context and subsequent advancements.

Findings

Clarifies the construction techniques of Hilbert and Quot Schemes

Highlights their role in deformation theory and moduli problems

Summarizes developments by Mumford, Altman, and Kleiman

Abstract

This is an expository account of Grothendieck's construction of Hilbert and Quot Schemes, following his talk `Techniques de construction et theoremes d'existence en geometrie algebriques IV : les schemas de Hilbert', Seminaire Bourbaki 221 (1960/61), together with further developments by Mumford and by Altman and Kleiman. Hilbert and Quot schemes are fundamental to modern Algebraic Geometry, in particular, for deformation theory and moduli constructions. These notes are based on a series of six lectures in the summer school `Advanced Basic Algebraic Geometry', held at the Abdus Salam International Centre for Theoretical Physics, Trieste, in July 2003.