##### Department of Mathematics,

University of California San Diego

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### Math 208 - Seminar in Algebraic Geometry

## Kristin DeVleming

#### UCSD

## Moduli of surfaces in $\mathbb{P}^3$

##### Abstract:

For fixed degree $d$, one could ask for a meaningful compactification of the moduli space of smooth degree $d$ surfaces in $\mathbb{P}^3$. In other words, one could ask for a parameter space whose interior points correspond to [isomorphism classes of] smooth surfaces and whose boundary points correspond to degenerations of these surfaces. Motivated by Hacking's work for plane curves, I will discuss a KSBA compactification of this space by considering a surface $S$ in $\mathbb{P}^3$ as a pair $(\mathbb{P}^3, S)$ satisfying certain properties. We will study an enlarged class of these pairs, including singular degenerations of both $S$ and the ambient space. The moduli space of the enlarged class of pairs will be the desired compactification and, as long as the degree $d$ is odd, we can give a rough classification of the objects on the boundary of the moduli space.

### October 5, 2018

### 2:00 PM

### AP&M 5829

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