*Fall 2022*

**Geometric Invariant Theory**

This seminar is constructed to promote an environment for graduate students working on algebraic geometry and neighboring fields to give expository talks on geometric invariant theory. The following is a tentative plan for topics: Affine quotients, stability, classical invariants of smooth hypersurfaces in projective space, projective quotients, linearizations, variation of quotients, flips & flops, review on smooth projective curves, stable vector bundles on curves, moduli functors, vector bundles on curves, and moduli space of stable sheaves on curves. For security reasons, meeting location will be shared via the group's Zulip server. If you want the meeting information for a particular talk, contact Pat Lank.

Abstracts

**Jonathan Smith - Invariant Theory Case Studies - **Before invoking invariant theory to construct quotient varieties in future meetings, we will introduce it via some natural and beautiful examples. We will find a parameter space for affine plane conics, discuss the Hilbert series, explore a few classical binary invariants, and overview some invariants for general plane curves (affine and projective). The talk will focus primarily on results and general ideas and neglect technicalities. I hope to offer a response to the question, “Why study invariant theory?”

**Pat Lank - Affine quotients and stability - **We start by discussing examples of the failure of the orbit space of an algebraic group $G$ acting on a variety $X$ being Zariski closed in affine space, i.e. being identified as a variety. By posing a mild condition on $G$, namely being linearly reductive, a robust theory starts to become established and interesting structural statements are proved. This introduces the notion of an affine quotient morphism $\Phi:X \to X//G$ which is a correction to this mentioned failure, and it is shown that it is surjective and yields a one-to-one correspondence between points in $X//G$ and $G$-closed orbits in $X$. Next, $G$-stability is discussed and it is shown that set of $G$-stable points $X^s \subset X$ is an open subset. Furthermore, the restriction of the affine quotient morphism to the $G$-stable points yields a ‘geometric quotient’ in the sense that its fibers are the $G$-orbits in $X^s$.

**Bailey Heath - Classical Invariants and Stability in Projective Space - **In this talk, we discuss classical invariants of smooth hypersurfaces in $\mathbb{P}^n$, such as the discriminant of degree-$d$ forms on $\mathbb{P}^n$. We then discuss stability of smooth hypersurfaces in $\mathbb{P}^n$, which is highlighted by the key result that any smooth hypersurface defined by an element of $k\left[x_0,x_1,...,x_n\right]$ (for $k$ an algebraically closed field of characteristic 0) of degree at least 3 is invariant under only a finite subset of $\operatorname{GL}\left(n+1, k\right)$. To conclude, we will explore the examples of binary quartics and plane cubics.

**Shreya Sharma - Moduli space of semistable hypersurfaces in $\mathbb{P}^n$ and the projective quotient - **The moduli space of smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$, $U_{n,d}/GL(n+1)$ is constructed by the action of $GL(n+1)$ on the affine variety $U_{n,d}:= \mathbb{V}_{n,d}\setminus\{\text{disc}= 0\}$. Gluing together all such affine quotients obtained by the action of $GL(n+1)$ on $\mathbb{V}_{n,d}\setminus \{F=0\}$ as $F$ runs through all classical invariants gives us a projective variety. It turns out that this projective variety itself is a quotient(called projective quotient) and is a moduli space of semistable hypersurfaces of degree $d$ in $\mathbb{P}^n$. We show this construction and motivate the 'projective quotient'.
This is a continuation of last week's talk and will serve as a motivation for next week's talk on projective quotients.