*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.

**Matthew Booth - Extending the idea of a quotient: from values to ratios - **Let $G$ be an algebraic group acting on an affine variety $X$. In Chapter 5, we constructed a quotient variety of $X$ by $G$ in the following way: for ``suitable'' $G$-invariant functions $f_1,\dots,f_n\in k[X]^G$, consider the map $\psi:X\rightarrow\mathbb{A}^{n}$ via $\psi(x)=\big(f_1(x),\dots,f_n(x)\big)$, and take the image of this map for the quotient. Our goal will be to improve this construction by choosing $f_0,f_1,\dots,f_n$ with the requirement that $\frac{f_i}{f_j}$ be a $G$-invariant rational function for all $1\leq i,j\leq n$. (We do not require the functions themselves to be $G$-invariant.) This gives rise to a rational map from $X$ into $\mathbb{P}^n$ via $x\mapsto\big(f_0(x):f_1(x):\cdots:f_n(x)\big)$, and we shall take the image of this map to arrive at a projective quotient.
More generally, for a linearly reductive group $G$ acting on an affine variety $X=\text{Spm}\ R$ and a choice of character $\chi\in\text{Hom}(G,\mathbb{G}_{\mathfrak{m}})$, we construct a map $\Phi_{\chi}:X\rightarrow X/\!/_{\chi}G:=\text{Proj}\bigoplus_{m\in\mathbb{Z}}R_{\chi^m}^G$, the $\textit{Proj quotient map}$ in direction $\chi$. (The notation $R_{\chi}^G$ refers to the set of semi-invarinants of weight $\chi$ inside $R$.) In particular, the choice of $\chi=1$ (the trivial character) will make the Proj quotient coincide with the affine quotient. Along the way, we will see some answers to the so-called $\textit{Italian problem}$: if an algebraic variety $X$ has a quotient under the action of $G$, does this quotient have function field equal to the invariant function field of $X$ under the group action?

**Uttaran Dutta - Moduli problems and vector bundles - **I will be going over the basics starting with Serre duality, Riemann-Roch on curves, I will briefly discuss the correspondence between vector bundles and locally free sheaves, and then after defining rank and degree of a vector bundles, I will define semistability introduced by Mumford in order to construct the moduli space of a algebraic vector bundles over a smooth projective curve. I will then talk about moduli problems, fine and coarse moduli spaces, and will try to explain exactly why we need semistability (spoiler: so that our moduli problem is bounded). I will finish the talk giving you an overview of how we are going to construct our moduli space.

**Pat Lank - Stability for vector bundles - **Within this talk, which is a continuation of the previous, we will discuss basic definitions and properties of stability for vector bundles on curves. This includes an introduction to the moduli space of vector bundles on a curve and equivalent characterizations for (semi)stability. If time permits, then there will be an elementary discussion about generalizing these ideas to higher dimensions.