Algebra, Geometry, and Number Theory Seminar

Abstracts

Mohammed Alabbood - Classification in PG(3,q) of classes of smooth cubic surfaces up to Eckardt points

In this seminar, we will classify classes of smooth cubic surfaces in PG(3,q) up to Eckardt points where q is prime and q>7. By considering configurations of 6 points in general position in the projective plane PG(2,q), we can describe subsets of projective space PG(3,q) that correspond to non-singular cubic surfaces with m Eckardt points. Recall that a non-singular cubic surface, say X, can be viewed as the blow up of PG(2,q) at 6 points in general position. Furthermore, there are 45 tritangent planes on X. Classification of cubic surfaces with m Eckardt points have been studied by Segre. However, we give another way to classify these cubic surfaces by defining an operation on the set of all triples of lines on the cubic surface that correspond to 45 tritangent planes on X.

Mohammed Alabbood - PhD Defense

In our thesis, we use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3,q) from 6 points in general position in PG(2,q) for q=17,19,23,29,31. We classify the cubic surfaces with twenty-seven lines in three dimensional space (up to e-invariants) by introducing computational and geometrical procedures for the classification. All elliptic and hyperbolic lines on a non-singular cubic surface in PG(3,q) for q=17,19, 23, 29, 31 are calculated. We define an operation on triples of lines on a non-singular cubic surface with 27 lines which help us to determine the exact value of the number of Eckardt point on a cubic surface. Moreover, we discuss the irreducibility of classes of smooth cubic surfaces in PG(19,C), and we give the corresponding codimension of each class as a subvariety of PG(19,C).

Owen Biesel - A fine moduli space of enriched structures

The enriched structures on a stable curve over a field parameterize the different ways of smoothing that curve, but it is less clear how to define the set of enriched structures for a family of stable curves over an arbitrary base scheme. We give a new, more general definition of enriched structure that reduces to the original for stable curves over a separably closed field, and show that the resulting fine moduli space has a universal property in terms of Néron models for the universal curve's Jacobian. This is joint work with David Holmes.

Philip Engel - Tilings of Riemann Surfaces

A k-differential on a Riemann surface is a nonzero section of the kth tensor power of the canonical bundle. When k = 3, 4, or 6 the moduli space of k-differentials contains a natural discrete subset: Surfaces tiled by hexagons, squares, or triangles respectively. It is possible to compute the volume of the moduli space by enumerating these tiled surfaces, using techniques from representation theory pioneered by Eskin and Okounkov. After summarizing this work, I will describe recent work joint with P. Smillie on the k = 5 case and the enumeration of Penrose tilings.

Luigi Ferraro - Hopf algebra actions on some AS-regular algebras of small GK-dimension

The classical Chevalley-Shephard-Todd Theorem gives a characterization of when a group acting linearly on the commutative polynomial ring has a ring of invariants that is isomorphic to a polynomial ring. Understanding when group actions (or more generally, Hopf actions) on AS-regular algebras give AS-regular invariant rings has proven to be a difficult problem. We provide some new examples of Hopf actions on some AS-regular algebras such that the ring of invariants is also AS-regular.

Jesse Kass - Counting nodal curves over an arithmetically interesting field

Given two general degree d complex polynomials f(x,y) and g(x,y), the equation f(x,y) + t g(x,y) defines a singular curve for exactly 3 (d-1)^2 complex values of t. This is a classical result that has been generalized in many ways to results counting complex curves, but the problem of generalizing it by replacing the complex numbers with a more arithmetically interesting field, such as the rational numbers or a finite field or..., has only been taken up recently. In my talk, I will explain results due to the speaker, Kirsten Wickelgren, and Marc Levine in the latter direction.

This talk is practice for talks at Geometry & Arithmetic of Surfaces Workshop.

Jesse Kass - An introduction to counting curves arithmetically

A long-standing program in algebraic geometry focuses on counting the number of curves in special configuration such as the lines on a cubic surface (27) or the number of conic curves tangent to 5 given conics (3264). While many important counting results have been proven purely in the language of algebraic geometry, a major modern discovery is that curve counts can often be interpreted in terms of algebraic topology and this topological perspective reveals unexpected properties.

One problem in modern curve counting is that classical algebraic topology is only available when working over the real or complex numbers. A successful solution to this problem should produce curve counts over fields like the rational numbers in such a way as to record interesting arithmetic information. My talk will explain how to derive such counts using ideas from A1-homotopy theory. The talk will focus on joint work with Kirsten Wickelgren on the cubic surface and results on more general hypersurfaces by Marc Levine.

There will be no pretalk, and the talk will only be 30 minutes.

Tracy Huggins - Multiple Perspectives on Essential Dimension

Given a finite group G and a field k, can G be realized as the Galois group of a field extension of k? This is known as the inverse Galois problem and is known to be difficult to answer in general.

The essential dimension of a finite group G over k measures the complexity of G by probing a relaxed form of the inverse Galois problem. Essential dimension can be approached from multiple perspectives, which together place essential dimension at the intersection of study of étale algebras, versal polynomials, and classifying functors. I will discuss a few of these different perspectives during my talk

Alicia Lamarche - Exceptional collections of toric varieties associated to root systems

Given a root system R, one can construct a toric variety X(R) by taking the maximal cones of X(R) to be the Weyl chambers of R. The automorphisms of R act on X(R); and a natural question arises: can one decompose the derived category of coherent sheaves on X(R) in a manner that is respected by Aut(R)? Recently, Castravet and Tevelev constructed full exceptional collections for D^b(X(R)) when R is of type A_n. In this talk, we'll discuss progress towards answering this question in the case where R is of type D_n, with emphasis on the 'base' case of D_4.

Yoav Len - Lifting Tropical Intersections

My talk is concerned with combinatorial aspects of intersection theory. When tropicalizing algebraic varieties, each of their intersection points maps to a tropical intersection point. Characterizing this locus is a fundamental problem in tropical geometry. In my talk, I will appeal to non-Archimedean and polyhedral geometry to characterize the locus in various cases. The solution leads to a combinatorial tool for counting multitangent hyperplanes of algebraic varieties, detecting dual defects, and for computing Newton polygons of dual varieties.

Keller Vandebogert - More on Compressed k-algebras

This will be the technical version of my CMS talk. The pretalk will introduce nonstandard terminology and give some examples (ie, Tor-algebras, Buchsbaum-Eisenbud resolution, compressed, inverse systems, etc). The main talk will go over some recent results regarding the structure theory of Artinian compressed k-algebras and outline the techniques involved.

Robert Vandermolen - An explicit kernel construction for Grassmannian flops

We will discuss the current work in producing an explicit kernel that induces the derived equivalence which arises from the Grassmannian flop. Specifically we will see that the essential image of the functor associated to this kernel aligns with a(n) (exceptional) collection first studied by Kappronav. Further we will explore some interesting geometric properties which the kernel of this functor enjoys and their curious ties to Geometric Invariant Theory. Future directions for a more general technique in producing these interesting kernels, will also be discussed. The pre-talk will discuss the previous work of Ballard et. al. in a similar kernel and it's geometric connections to the homology of projective spaces.

Last semester's seminar.