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Polynomial growth of Betti sequences over local rings
Luchezar Avramov, University of Nebraska
Abstract:
The asymptotic patterns of the Betti sequences of finitely generated modules over a local ring R reflect and affect the nature
of its singularity. For instance, these sequences are eventually zero if and only if R is regular, and they are eventually constant
if and only if R is a hypersurface section of a regular ring. The talk is about rings over which every Betti sequences is eventually
given by some polynomial of degree at most c. We conjecture that these are precisely the hypersurface sections of complete
intersection rings of codimension c and multiplicity 2c. It will be shown that this condition is sufficient, and that it is also necessary if c ≤ 4 or if R is homogeneous. The talk is based on joint work with Alexandra Seceleanu and Zheng Yang.
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A-hypergeometric solution sheaves
Christine Berkesch Zamaere, University of Minnesota
Abstract:
A-hypergeometric systems are the D-module counterparts of toric ideals, and their behavior is linked closely to the combinatorics of
toric varieties. I will discuss recent work that aims to explain the behavior of the solutions of these systems as their parameters vary.
Our goal, which can be achieved in special cases, is to stratify the parameter space so that solutions are locally analytic within each
(connected component of a) stratum, and this turns out to be closely related to certain local cohomology modules.
This is joint work with Jens Forsgard and Laura Matusevich.
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A generalization of the Abhyankar Jung theorem to associated graded rings of valuations
Dale Cutkosky, University of Missouri
Abstract:
Suppose that R → S is an extension of local domains and ν* is a valuation dominating S.
We consider the natural extension of associated graded rings along the valuation grν*(R) → grν*(S). We give examples showing that in general, this extension does not share good properties of the extension R → S, but after enough blow ups above the valuations, good properties of the extension R → S are reflected in the extension of associated graded rings. Stable properties of this extension (after blowing up) are much better in characteristic zero than in positive characteristic. Our main result is a generalization of the Abhyankar-Jung theorem which holds for extensions of associated graded rings along the valuation, after enough blowing up.
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On some recent consequences of Serre's conditions (Si)
Hailong Dao, University of Kansas
Abstract:
Serre's condition (S2) is quite familiar to commutative algebraists as part of the condition for normality. On the surface it appears to be a rather weak condition. In this talk we will discuss some surprising consequences of (Si) for i=2 and higher. These consequences involve cohomological dimension, depth, h-vector, and in the square-free monomial case, the Castelnuovo-Mumford regularity of relevant ideals. We will also discuss some intriguing open questions on the tightness of these statements. This talk is based on recent joint work and discussions with
D. Eisenbud, K. Han, S. Takagi, and M. Varbaro.
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The Order Ideal Conjecture and edge Homomorphism
Sankar Dutta, University of Illinois at Urbana-Champaign
Abstract:
The purpose of this talk is two-fold : a) To present different aspects/special cases of the order ideal conjecture over regular/non-regular local rings
and b) to present equivalence of the following:
1) The non-vanishing of a specific edge homomorphism of a spectral sequence originating from the associativity of Hom and Tensor product, 2) the validity of the monomial conjecture and 3) the validity of the order ideal conjecture on regular local rings.
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Asymptotic Syzygies
Lawrence Ein, University of Illinois at Chicago
Abstract:
We'll discuss joint work with Rob Lazarsfeld. Let X be a smooth projective variety and L be a sufficiently very ample line bundle
on X. We are interested in the shape of the minimal resolution of the coordinate ring of R(X, L).
In particular, we are interested in finding out which Koszul groups Torp(R, k)p+q are nonzero.
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Kakeya problems over finite fields
Daniel Erman, University of Wisconsin
Abstract:
The Kakeya Needle Problem has its origins in harmonic analysis, but it has led to a number of interesting related questions about algebra and geometry over finite fields. I'll first give background on this famous problem. Then I'll talk about recent work of myself and Jordan Ellenberg which uses degeneration techniques to make progress on some of the related algebraic questions.
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Finiteness questions about local cohomology: an introduction
Mel Hochster, University of Michigan
Abstract:
There are many open questions
concerning the associated primes of a Noetherian
ring R with support in an arbitrary ideal, and the
minimal primes among these. It is not known whether
the set of minimal primes is always finite. The set
of associated primes can be infinite, but there are
many known cases in which is finite, especially if
R is regular. We survey what is known about these
questions, and present some recent work with
Luis Nunez Betancourt in characteristic p in which
finiteness results are obtained (some of these results
have been proved independently by M. Katzman and
W. Zhang).
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The Associativity Formula and the Projection Formula
Steve Kleiman, MIT
Abstract:
In the 1950s, Samuel developed the Intersection Theory of Cycles in
Algebraic Geometry by reducing to algebraic counterparts about the
Samuel multiplicity, the normalized leading coefficient of the
Hilbert-Samuel polynomial. Reversing history, we'll give new proofs of
the Associativity Formula and the Projection Formula for the Samuel
multiplicity by reducing to geometric counterparts, which have been
proved directly primarily by Fulton. Happily, similar proofs yield
generalizations of the two formulas to the Buchsbaum-Rim multiplicity.
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DeRham cohomology of a complete local ring of equicharacteristic zero
Gennady Lyubeznik, University of Minnesota
Abstract:
In his paper on de Rham cohomology of algebraic varieties R. Hartshorne defined de Rham cohomology for complete local rings of
equicharacteristic zero as well. I am going to report on the very recent work of my student Nicholas Switala who substantially refined
R. Hartshorne's results in the complete local case. In particular, Nick proved that what is well-defined in the complete local case is
not only de Rham cohomology itself but an entire Hodge-to-de Rham spectral sequence. Nick's proof uses the theory of D-modules.
As part of his proof he develops a theory of Matlis duality for D-modules that is of independent interest.
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Lech's conjecture in dimension three
Linquan Ma, Purdue University
Abstract:
We prove Lech's conjecture in dimension three in characteristic p, under mild conditions on the residue field.
(slides of talk)
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Equations of Rees algebras and rational normal scrolls
Jeffrey Madsen, University of Notre Dame
Abstract:
Let I be a height two ideal in R=k[x0, x1] generated by forms of the same degree.
Such ideals arise from parametrizations of rational projective curves, in which case the bigraded coordinate ring of the graph of the
parametrization is given by the Rees algebra R[It]. The defining equations of the Rees algebra are difficult to determine,
and are only known in some special cases. I will show how one can compute all the equations of sufficiently large degree.
The construction involves a special class of Rees algebras, which are coordinate rings of rational normal scrolls.
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Generalized multiplicities and depth of blowup algebras
Jonathan Montaño, Purdue University
Abstract:
The j-multiplicity and ε-multiplicity of arbitrary ideals are called generalized multiplicities as they coincide with the Hilbert-Samuel multiplicity for m-primary ideals. Several features and applications of the Hilbert-Samuel multiplicity have been shown to extend to generalized multiplicities. In this talk, we extend the work of Sally, Rossi, Valla, Wang, Goto, Polini, and Xie, on the relation between minimal multiplicities and depth of blowup algebras over Cohen-Macaulay local rings. We define the notion of Goto-minimal j-multiplicity for ideals of maximal analytic spread and we study the interplay among this new notion, the notion of minimal j-multiplicity of Polini-Xie, and the Cohen-Macaulayness of the blowup algebras of ideals satisfying certain residual assumptions. Moreover, in joint work with Jack Jeffries, we generalize Teissier's theorem showing characterizations of the j- and ε-multiplicities of a monomial ideal as normalized volumes of certain regions.
(slides of talk)
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Supernatural bundles and resolutions of equivariant bundles on the diagonal
Steven Sam, University of California at Berkeley
Abstract:
Eisenbud and Schreyer showed that the cohomology table of a vector bundle on projective space is a non-negative linear combination of the cohomology tables of supernatural vector bundles. A natural question is whether this numerical decomposition has an algebraic interpretation. One obstruction to such an interpretation is that the coefficients in the decomposition are in general rational. We show that in some cases a certain Fourier-Mukai transform of the bundle has a filtration which realizes the numerical decomposition. This is joint work with Daniel Erman.
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Chen ranks and resonance varieties
Hal Schenck, University of Illinois at Urbana-Champaign
Abstract:
The Chen groups of a group G are the lower central series quotients of the maximal metabelian quotient of G.
Under certain conditions, we relate the ranks of the Chen groups to the first resonance variety of G, a jump locus for the cohomology of G.
In the case where G is the fundamental group of the complement of a complex hyperplane arrangement, our results positively resolve Suciu's
Chen ranks conjecture. We obtain explicit formulas for the Chen ranks of a number of groups of broad interest, including pure Artin groups associated to
Coxeter groups, and the group of basis-conjugating automorphisms of a nitely generated free group. (joint work with Dan Cohen, Lousiana State University)
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Cohomology of thickenings of projective varieties
Anurag Singh, University of Utah
Abstract:
Let X be a smooth projective subvariety of Pn over a field of characteristic zero. We discuss a version of the Kodaira vanishing theorem for thickenings of X in Pn, and a related result on the injectivity of the natural maps from Ext modules to local cohomology modules.
This is joint work with Bhatt, Blickle, Lyubeznik, and Zhang.
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On the limit of the Hilbert-Kunz Multiplicity as p → ∞
Kevin Tucker, University of Illinois at Chicago
Abstract:
In many ways, for a fixed characteristic, the Hilbert-Kunz multiplicity is a rather complicated invariant of singularities. It can take on irrational values, though an explicit example is yet unknown. It is easy to estimate in practice, and very difficult to compute and interpret precisely. In some cases, for rings arising from a reduction to positive characteristic, it is known that the Hilbert-Kunz multiplicities approach a limit as p → ∞. One can hope that these limits exist in general, that the limit may be simpler and easier to interpret than in a fixed characteristic. In this talk, I will review some of the known examples, and discuss some related limits and progress towards showing the existence of the limit Hilbert-Kunz multiplicity.
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Computing support of local cohomology modules
Wenliang Zhang, University of Nebraska
Abstract:
In this talk, I'd like to discuss an effective algorithm to calculate the support of an F-finite F-module
(in the sense of Lyubeznik) in characteristic p. Local cohomology of regular rings are primary
examples of F-finite F-modules. If time permits, I will mention some applications of it.
This is joint work in progress with Mordechai Katzman.
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