# Juan Martin Perez Bernal: An overview of the construction of moduli spaces, from coherent sheaves to principal bundles

**Abstract:** In the construction of moduli spaces two very important tools come up frequently, namely a strategy sometimes called Langton’s algorithm to prove the properness of the moduli functor and GIT adjacent methods to show the existence of the moduli spaces.

We will shortly review the important theorems underlying both of these tools in the case of the moduli space of vector bundles over a curve X.

More precisely, we will show that Langton's algorithm in this case is just the valuative criterion of properness of the moduli functor that associates to each parameter space S the set of isomorphism classes of families of vector bundles over SxX. Additionally, we will see how the existence of Quot schemes provides the perfect environment to develop notions of stability and a corresponding GIT quotient.

Lastly we will tackle which technical problems arise when trying to generalize the previous question first to higher dimensional projective varieties X, then when trying to generalize to principal G-bundles. Here we will talk about how Langton's algorithm is applied to the higher dimensional case and how it can be modified to the G-principal case. In doing this it will become apparent how torsion free sheaves and their generalization known as singular principal G-bundles are necessary to have projective moduli spaces.

The talk will also be broadcasted via Webex. The Webex link is:

https://fu-berlin.webex.com/join/juanmartip95

Further information on the research seminar: https://userpage.fu-berlin.de/~aschmitt/FSSoSe22.html

### Time & Location

May 03, 2022 | 04:15 PM

SR 120, Arnimallee 3