Algebra, Geometry, and Number Theory - Student Seminar

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.