Date 
Speaker 
Title/Abstract 
Feb 2 
Wenbo Niu (UArk) 
Global generation of adjoint line bundles
In this talk, I will give a survey on the problem of global generation of adjoint line bundles. It is also known as Fujita’s conjecture. We will discuss Reider’s theorem for surface case. Then we will see how one can use vanishing theorem to study higher dimensional case. The conjecture is still open for dimension larger than 5.

Feb 9 
Lance Miller (UArk) 
Diferential forms for singular schemes
This talk will survey ways of approaching the construction of algebraic differential forms for singular schemes. Notably, many distinct approaches and reviewing their basic properties, we will discuss a systematic framework in Voevodsky's htopology. This was pioneered by Huber and Joerder in characteristic 0 and by Huber, Kelly, and Kebekus in positive characteristic.

Feb 16 
Wenbo Niu (UArk) 
Global generation of adjoint line bundles (part 2)
Continuation.

Feb 23 
Bernd Ulrich (Purdue) 
Residual Intersections: Socles and Duality
A wellknown formula expresses the socle of an Artinian complete intersection of characteristic zero in terms of a Jacobian determinant.
We generalize this formula to rings that are neither Artinian nor
complete intersections. We deduce our formula from an explicit
description of the Dedekind complementary module of residual
intersections. This is a report on joint work with David Eisenbud.

Mar 2 
Botong Wang (Wisc) 
Hodge theory in the enumeration of points, lines, planes, etc.
Given n points on the plane, by connecting each pair of them, one obtains either one line or at least n lines. This is an old theorem in enumerative combinatorial geometry due to de Bruijn and Erdos. In the first part of the talk, we will present a higher dimensional generalization of this theorem, which confirms a "topheavy" conjecture of Dowling and Wilson in 1975. The key idea is to relate some combinatorial quantities to the cohomology ring of algebraic varieties and to use the hard Lefschetz theorem of intersection cohomology groups. In the second part, I will discuss some work in progress further generalizing the result to nonrealizable matroids. This is joint work with Tom Braden, June Huh, Jacob Matherne and Nick Proudfoot.

Mar 9 
Youngsu Kim
(UArk) 
Defining ideals of Rees algebras
In this talk, we present a method of finding defining ideals of certain Rees algebras. Our primary focus is on Rees algebras which arise as the coordinate rings of (closed) graphs of rational maps between projective spaces. Our main result is that under suitable conditions, we can find all the generating degrees and the minimal number of generators of defining ideals. This is joint work with Vivek Mukundan.

Mar 23 
NA 
Spring Break
No Seminar.

Mar 30 
Jaiung Jun
(Binghamton) 
Geometry of hyperfields
In this talk, we introduce rather exotic algebraic structures called hyperrings and hyperfields. We first review the basic definitions and examples of hyperrings, and illustrate how hyperfields can be employed in algebraic geometry to show that certain topological spaces (underlying topological spaces of schemes, Berkovich analytification of schemes, and real schemes) are homeomorphic to sets of rational points of schemes over hyperfields.

Apr 2 
Paolo Mantero
(U Ark) 
Singularities of Reeslike algebras
Recently, J. McCullough and I. Peeva provided the first counterexamples to the Regularity Conjecture. These examples are obtained by a construction dubbed Reeslike algebra. Since there is still hope that the Regularity Conjecture may hold under additional assumptions (e.g. in the smooth case), it is natural to ask: How "bad" can the singularities of Reeslike algebras be? How are singularities affected by the Reeslike algebra construction?
In this talk, based on joint work with J. McCullough and L.E. Miller, we will provide quantitative and qualitative answers to these questions.

Apr 13 
Matthew Mastronei
(UIUC) 
Koszul almost complete intersections
Let R=S/I be a quotient of a standard graded polynomial ring S by an ideal I generated by quadrics. If R is Koszul, a question of Avramov, Conca, and Iyengar asks whether the Betti numbers of R over S can be bounded above by binomial coefficients on the minimal number of generators of I. Motivated by previous results for Koszul algebras defined by three quadrics, we give a complete classification of the structure of Koszul almost complete intersections and, in the process, give an affirmative answer to the above question for all such rings.

Apr 27: 10:45am 
Roi DocampoÁlvarez
(OU) 
Differentials on the arc space
We study the sheaf of Kähler differentials on the arc space of an algebraic variety. We obtain explicit formulas that can be used effectively to understand the local structure of the arc space. The approach leads to new results as well as simpler and more direct proofs of some of the fundamental theorems in the literature. The main applications include: an interpretation of Mather discrepancies as embedding dimensions of certain points in the arc space, a new proof of a version of the birational transformation rule in motivic integration, a new proof of the curve selection lemma for arc spaces, and a description of Nash blowups of jet schemes. This is joint work with Tommaso de Fernex.
