Algebraic geometry & Commutative Algebra Student Seminar at University of South Carolina
Schedule
 1/23/23

 Speaker: Pat Lank “Singularity theory via pairs”

 Abstract: Within this talk, we will have an introduction to understanding singularities via pairs for an algebraic variety. Attempting to not become lost in technicality, I will start by defining rational singularities, and describe an interesting characterization on how to detect them in the derived category. There are many natural examples of algebraic varieties which have rational singularities, and we will see a few. From this, one can continue to sharpen this notion to other singularity types, and I will sketch how this is done by discussing log discrepancies. This motivates the definition of (Kawamata) log canonical singularities, and if time permits, I will describe examples and characterizations of such.
 2/2/23

 Speaker: Jonathan Smith “An Invariant in Equivariant Birational Geometry”

 Abstract: The Cremona group of dimension \(n\) over a field \(k\), denoted \(Cr_n(k)\), is the group of birational automorphisms of \(\mathbb{P}^n_k\). Suppose \(\text{char}(k) = 0\) and \(k = \bar{k}\). Then given a subgroup \(G\) in \(Cr_n(k)\), we can resolve indeterminacy and singularities so that \(G\) acts regularly on a smooth rational projective variety \(X\). Understanding conjugacy of subgroups in \(Cr_n(k)\) amounts to studying \(G\)birational equivalence of these varieties with group actions. To differentiate between two \(G\)varieties up to \(G\)birational equivalence, we can use equivariant birational invariants. We will discuss an equivariant birational invariant determined by M. Kontsevich, V. Pestun, and Yu. Tschinkel in 2019 and calculate the values of the invariant for some lowdimensional varieties. This talk is expository and will primarily follow a paper of B. Hassett, A. Kresch, and Yu. Tschinkel from 2020.
 2/16/23

 Speaker: Bailey Heath “Minkowski’s Bound On Orders of Finite Rational Matrix Groups”

 Abstract: A classical result in the study of finite subgroups of the general linear group \(\operatorname{GL}_n\left(\mathbb{Q}\right)\) is Minkowski’s 1887 bound on their orders, and the goal of this talk will be to prove this result. Our proof, which will follow notes by Serre, will also include a proof of a special case of the JordanZassenhaus Theorem, which states that every finite subgroup of \(\operatorname{GL}_n\left(\mathbb{Q}\right)\) is conjugate to a finite subgroup of \(\operatorname{GL}_n\left(\mathbb{Z}\right)\)
 3/16/23


 Abstract: I will start by defining what a Mori dream space is. I will explain the consequences of each of the conditions in the definition. As a result we will see that Mori dream space allows us to tackle the geometric arguments about a space from a conmbinatorial view. For example, I will show that the cone of Nef divisors of a MDS, \(\operatorname{Nef}(X)\) is a rational polyhedron, and there is a bijection between the rational contractions of \(X\) and the faces of \(\operatorname{Nef}(X).\) As a consequence of the final condition we will see that the cone of effective curves has a decomposition into ‘Mori chambers’ and if I get time I will show the existence of flips as well. All of this will ultimately give us a huge resemblance between VGIT and the geometry of MDS. This is very powerful as then studying all the contractions of a MDS will essentially be equal to studying the GIT quotients of that space. In the end I will describe a couple of results (without proof) about Mori dream surfaces: when a Del Pezzo surface and K3 surface is MDS, and we will see that studying the Cox rings of these spaces is enough to answer these questions.
 3/22/23


 Abstract: I will mainly talk about Mori Dream Surfaces. The talk will not be complete as I do not have all the pieces, but with some assumption I will go ahead. One of the famous results of Hu and Keel says that for a normal projective \(\mathbb{Q}\)factorial variety \(X\), being a MDS is equivalent to the fact that the Cox ring of \(X\) is finitely generated. Then we have the result: If \(X\) is a normal projective \(\mathbb{Q}\)factorial surface and \(\operatorname{Pic}(X)\) is finitely generated, then Cox ring of \(X\) is finitely generated iff \(\operatorname{Eff}(X)\) is a polyhedral and \(\operatorname{Nef}(X)=\operatorname{SAmple}(X).\) We will prove a couple of results studying how the Nef, Semiample and Effective cones look like on a surface. We will try to find more equivalent criterions on surfaces for which we have \(\operatorname{Nef}(X)=\operatorname{SAmple}(X).\) At the end we will see that in most of the cases finding whether the Effective cone is a polyhedral is enough. Using this and some facts about K3 surfaces, we will be able to see that a K3 surface is a MDS iff its automorphism group is finite. One can also show in similar lines, that a Weak Del Pezzo surface is MDS. There are many detailed articles studying the generators of Cox(Del Pezzo suface), which characterizes Mori Dream Del Pezzo surfaces completely.