Pre-talk Title: The Picard Scheme
Abstract: The set of all line bundles on a variety form a group known as the Picard group. Like with many objects in algebraic geometry, the Picard group can be given a scheme structure in which it parametrizes families of line bundles on X, this is called the Picard scheme. In this pre-talk, we will discuss the basic properties of the Picard scheme like its dimension, smoothness, connected components, etc.
Title: Moduli of Very Ample Line Bundles
Abstract: A line bundle is very ample if the map into projective space that it defines is an embedding. Given that there is a moduli space for all line bundles, the Picard scheme, it is natural to ask if there is a moduli space of very ample line bundles. We will discuss some complications that arise from very ampleness not being a locally closed property and show how we can modify the moduli problem to fix this.