The Schedule

Bernd Ulrich, Purdue University

Title:
Ideals in the linkage class of a complete intersection

Abstract:
Linkage (or liaison) has been used since the nineteenth century to study and classify curves in projective three-space and, more generally, varieties in projective space or homogeneous ideals in polynomial rings. Of particular importance have been licci ideals, ideals that can be linked to a complete intersection in a finite number of steps. It is known that the Castelnuovo-Mumford regularity of a licci ideal forces a very strict upper bound for the initial degree of the ideal. Now, in joint work with Craig Huneke and Claudia Polini, we conjecture that it also bounds the number of generators of the ideal, and we prove this conjecture in many cases. In addition, we provide new sufficient conditions for an ideal to be licci, for classes of ideals of height three and for ideals containing a maximal regular sequence of quadrics. The talk will also explain connections with recent work by Guerrieri, Ni, Weyman and by Jelisiejew, Ramkumar, Sammartano.

Anna Brosowsky, University of Nebraska

Title: 

Some two variable limit F-signatures

Abstract: 

The F-signature is a positive characteristic ring invariant measuring how nice the ring is, on a scale of "not even strongly F-regular" to "regular". One can also consider the F-signature function of a ring element, which carries information about the F-signature, Hilbert-Kunz multiplicity, and F-pure threshold. As a further generalization, given a polynomial in characteristic 0, the limit F-signature function, comes from considering the F-signature mod p, and taking an appropriately scaled limit as the primes p go to infinity. One downside of these invariants is they are quite difficult to compute in practice. In this talk, we will present a formula for the limit F-signature function for a polynomial of the form f=x^ay^b(x+y)^c and we will overview two different ways this can be computed.

Janet Page, North Dakota State University

Title: 

Smooth surfaces with maximally many lines

Abstract: 

How many lines can lie on a smooth surface of degree d?  In 1849, Cayley and Salmon proved their famous result that every smooth projective surface of degree 3 contains 27 lines.  In higher degrees, we know that not all surfaces of degree d have the same number of lines (many have no lines at all), but Segre proved an upper bound: a smooth surface of degree d > 2 over the complex numbers has at most (d-2)(11d-6) lines.  However, over fields of positive characteristic, there are smooth projective surfaces which break Segre’s upper bound.  In this talk, I’ll give a new upper bound which holds over any field.  In addition, we’ll fully classify those surfaces of degree d which contain the maximal number of lines possible.  This talk is based on joint work with Tim Ryan and Karen Smith.

Stephen Landsittel, University of Missouri

Title: 

Epsilon Multiplicity is a Limit of Amao Multiplicities

Abstract: 

In a 2014 paper, Cutkosky proved a volume equals multiplicity formula for the multiplicity of an m-primary ideal. We will discuss a generalization of this result to the epsilon multiplicity.

Rankeya Datta, University of Missouri

Title:
Finite generation of extensions of valuation rings

Abstract:
In this talk we will characterize when a generically finite local extension of valuation rings is essentially of finite type,  based on the ramification-theoretic invariant called defect.  The presence of defect in prime and mixed characteristics has emerged to be one of the main obstacles to proving the outstanding local uniformization theorem in these settings.  The relation between essential finite generation and defect was conjectured by Knaf.

Reception at Cutkosky-Srinivasan residence

Day 2

September 29

Rui-jie Yang, University of Kansas

Title:
Localization along a hypersurface and hodge theory

Abstract:

In commutative algebra, a useful way to understand a hypersurface is to consider the localization of the structure sheaf of the ambient space along this hypersurface. However, it is not a coherent O-module, but rather a coherent module over the sheaf of differential operators. This is where the theory of D-modules enters into the picture. In recent years, pioneered by the work of Mustață-Popa and Schnell, it is realized that there are extra structures on the localization module coming from Hodge theory, or more precisely Saito’s theory of mixed Hodge modules, a theory of D-modules with “mixed Hodge structures”. In this talk, I will introduce this circle of ideas. 

Furthermore, I will discuss a joint work with Dougal Davis on a formula of the Hodge filtration on the Kashiwara-Malgrange V-filtration, using ideas from unitary representation theory of real Lie groups. This in turn gives a complete understanding of the Hodge filtration on the localization module. As an application, in a joint work with Andras Lorincz, for multiplicity-free spaces, we completely describe the Hodge and weight filtrations in terms of the representation-theoretic information. This generalizes the work of Perlman-Raicu in the case of generic determinantal 

Aleksandra Sobieska, Marshall University 

Title: 

Finite and Infinite Resolutions for Numerical Semigroup Rings via Apery Specializations

Abstract: 

Each numerical semigroup S with smallest positive element m corresponds to an integer point in a polyhedral cone K_m, known as the Kunz cone. The faces of K_m form a stratification of numerical semigroups that has been shown to respect a number of algebraic properties of S. In this talk, I will describe how the structure of the Kunz cone informs the minimal free resolution of the numerical semigroup ring k[S] over the polynomial ring, and the minimal free resolution of the ground field k over k[S]. The Kunz structure allows for extremely explicit resolutions to be written down in both cases. This talk is based on joint work with Benjamin Braun, Tara Gomes, Ezra Miller, Christopher O’Neill, and Eduardo Torres Davila.

Shahriyar Roshan-Zamir, University of Nebraska 

Title:

Interpolation in weighted projective spaces.

Abstract:

Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree d singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992--1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this talk, we use commutative algebra to prove analogous statements in the weighted projective space, a natural generalization of the projective space. I will also introduce an inductive procedure for weighted projective planes, similar to that originally due to A. Terracini from 1915, to demonstrate the only example of a weighted projective plane, with mild assumptions, where the analogue of the Alexander-Hirschowitz theorem holds without any exceptions. Furthermore, I will give interpolation bounds for an infinite family of weighted projective planes. 

Krishna Hanumanthu, Chennai Mathematical Institute

Title: 

Some questions on Seshadri constants.

Abstract: 

Let X be a projective variety, and let L be an ample line bundle on X. For a point x in X, the Seshadri constant of L at x is defined as the infimum, taken over all curves C passing through x, of the ratios  \frac{L.C}{m}, where L.C denotes the intersection product of L and C, and m is the multiplicity of C at x. This concept was introduced by J.-P. Demailly in 1990, inspired by Seshadri’s ampleness criterion. Seshadri constants provide insights into both the local behavior of L at x and certain global properties of X.

The notion of Seshadri constants has been extended in various directions. Two important generalizations include replacing L with a vector bundle of arbitrary rank and replacing the point x with a finite collection of points. We will give an overview of the current research in this area and discuss some recent results on Seshadri constants of vector bundles as well as multi-point Seshadri constants on surfaces.