Algebra Seminar, Spring 2017
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Date 
Speaker 
Title/Abstract 
1/23 
N/A 
Organizational meeting

1/30 
William Taylor (UArk) 
Interpolating Between HilbertSamuel and HilbertKunz Multiplicity HilbertSamuel multiplicity and HilbertKunz multiplicity each give a numerical measure of the complexity of an ideal in a local ring of positive characteristic. While their definitions and properties are similar, and there are certain wellknown inequalities relating them, they have historically been treated mostly independently. In this talk we define a single function that interpolates between the HilbertSamuel multiplicity of one ideal and the HilbertKunz multiplicity of another ideal as a real parameter varies. We will see that certain wellknown theorems regarding the two original multiplicities are special cases of more general theorems. We will also see a relationship between the new multiplicity function and the Fthreshold. Finally, we will demonstrate a method of computing the new multiplicity for normal toric rings as a volume in real space.

2/6 
William Taylor (UArk) 
Interpolating Between HilbertSamuel and HilbertKunz Multiplicity HilbertSamuel multiplicity and HilbertKunz multiplicity each give a numerical measure of the complexity of an ideal in a local ring of positive characteristic. While their definitions and properties are similar, and there are certain wellknown inequalities relating them, they have historically been treated mostly independently. In this talk we define a single function that interpolates between the HilbertSamuel multiplicity of one ideal and the HilbertKunz multiplicity of another ideal as a real parameter varies. We will see that certain wellknown theorems regarding the two original multiplicities are special cases of more general theorems. We will also see a relationship between the new multiplicity function and the Fthreshold. Finally, we will demonstrate a method of computing the new multiplicity for normal toric rings as a volume in real space.

Wed. 2/15 
Daniel Juda (UArk) 
The ainvariant and HilbertKunz multiplicity for rings of invariants of cyclic pgroups In this talk we review basic facts about rings of invariants for cyclic pgroups and use them to bound the ainvariant of these rings of invariants in terms of the underlying representation theory. We also present a result analogous to Noether's bound on the top degree of a homogeneous generating set for such rings of invariants. Using this, we give a bound on the HilbertKunz multiplicity.

3/6 
Luigi Lombardi (Stony Brook) 
A decomposition theorem for direct images of pluricanonical bundles to abelian varieties. TBA

Fri. 3/17 
Eric Canton (UNL) 
Log discrepancies at semivaluations: semicontinuity and detecting strong Fregularity
We have long known, or implicitly used, that to a potential splitting phi of the Frobenius homomorphism on a ring R of positive characteristic one can assign a divisor on Spec(R), and these divisors describe the Frobenius splitting behavior of phi at height one primes. We generalize this process and assign log discrepancies to valuations on Spec(R/P) for primes P of R; such valuations are called semivaluations of R. The set of semivaluations on R admits a natural topology, and we prove that our log discrepancy is lowersemicontinuous in this topology, inducing a lowersemicontinuous function on Spec(R) considered with the constructible topology. We also show that (minimal) log discrepancies on semivaluations can be used to characterize the Frobenius splitting behavior of phi at any point of Spec(R).

4/3 
Mike DiPasquale (Ok. St.) 
Free multibraid arrangements and resolutions Freeness of the module of derivations is a central topic in arrangement theory and is closely linked to freeness of the module of multiderivations. In this talk we will describe some recent progress in understanding freeness of multibraid arrangements. It turns out that the most interesting class of free multiplicities arise from signedeliminable graphs (this is due to AbeNuidaNumata), generalizing a classical result of Stanley. We give a numerical freeness criterion which yields that, on a large central cone, the AbeNuidaNumata multiplicities are `most likely' the only free ones. We also show that free multibraid arrangements give rise to an interesting class of free complexes which resolve an underlying ideal generated by powers of linear forms. This work is partially joint with Chris Francisco, Jeff Mermin, and Jay Schweig. No knowledge of arrangements will be assumed.

4/24 
Alessio Sammartano (Purdue) 
Extremal free resolutions for a given Hilbert polynomial
Let (R,m,k) be a polynomial ring over a field or a complete intersection defined by powers of the variables. Let P(t) be an admissible Hilbert polynomial for factor rings of R. We investigate the existence of saturated ideals I of R such that R/I achieves the largest Betti numbers over R and PoincarĂ© series of k among all saturated ideals whose Hilbert polynomial of the factor ring is P(t). This is joint work with Giulio Caviglia.
