The aim of these talks is to understand how to construct Hilbert and Quot schemes, which play a vital role on moduli problems. Roughly speaking, Hilbert schemes classify all closed subschemes of a given scheme and Quot schemes classify all coherent subsheaves of a given coherent sheaf on a given scheme. Since Quot schemes are generalization of Hilbert schemes and the construction is the same, we mainly focus on Quot schemes.

 

Talk 1. Preliminary.

A brief introduction to the notion we will be using and sketch the construction of Quot schemes.

 

Talk 2 Construction of Quot schemes.

In this talk, we assume some facts on Castelnuovo-Mumford regularity and flattening stratifications. Then we construct Grassmannians first and see how Quot schemes can be embedded into a suitable Grassmannian as a closed subscheme.

 

Talk 3 Castelnuovo-Mumford regularity.

 

Talk 4 Flattening stratifications.