Virtual Techniques
Lecture 1. Basics in intersection theory.
Moduli spaces of stable maps are usually singular with different components of different dimensions. In order to do intersection theory on such spaces one needs to construct a "virtual fundamental class". I will give examples of such unexpectedly large moduli spaces and I will explain the general idea behind the construction of virtual fundamental classes. For this I will review the construction of Fulton--Macpherson intersection products and briefly explain how to generalise it to construct virtual fundamental classes.
Lecture 2. Obstructions: examples.
I will give examples of obstruction theories and try to give as much geometric motivation as possible.
Lecture 3. Working with virtual classes: examples.
I will explain that virtual fundamental classes behave well with respect to pull-backs and certain push-forwards. This allows us to compare virtual classes of different types of moduli spaces such as stable maps and stable quotients or to compare invariants of different targets such as Gromov-Witten invariants of blow-ups.