Fall 2019
November 11: Federico Castillo, University of Kansas
Title: Newton polytopes of multidegree
Abstract: The multidegree of a multiprojective variety can be seen as a polynomial encoding the
intersection of the given variety with products of linear spaces. For a product of only two projective
spaces, the multidegrees were classified (up to a scalar) by June Huh in 2012. Describing all possible
multidegrees, even up to scalar, seem an intractable problem. We will focus on a simplified problem:
we explain where this multidegrees are supported. This is joint work with Binglin Li and Naizhen
Zhang.
November 4: Jonathan Montaño, New Mexico State University
Title: Analytic spread and integral closure of decomposable modules II
October 28: Jonathan Montaño, New Mexico State University
Title: Analytic spread and integral closure of decomposable modules
October 21: Youngsu Kim, University of Arkansas
Title: Generic Links of Determinantal Varieties
Abstract: Linkage is a classical topic in algebraic geometry and commutative algebra. Fix an affine space A. We say two subschemes X, Y of A are (directly) linked if their union is a complete intersection in A while X and Y having no common component. Two linked subschemes share several properties in common. Linkage has been studied by various people, Artin-Nagata, Peskine-Szprio, Huneke-Ulrich, to name a few.
In 2014, Niu showed that if Y is a generic link of a variety X, then LCT (A, X) <= LCT (A, Y), where LCT stands for the log canonical threshold. In this talk, we show that if Y is a generic link of a determinantal variety X, then X and Y have the same log canonical threshold. This is joint work with Lance E. Miller and Wenbo Niu.
October 14: Monica A. Lewis, University of Michigan
Title: The local cohomology of a parameter ideal with respect to an arbitrary ideal
Abstract: Let S be a complete intersection presented as R/J for R a regular ring and J a parameter ideal. Let I be an ideal containing J. It is well known that the set of associated primes of H^i_I(S) can be infinite, but far less is known about the set of minimal primes. In 2017, Hochster and Núñez-Betancourt showed that if R has prime characteristic p > 0, then the finiteness of Ass H^i_I(J) implies the finiteness of Min H^{i-1}_I(S), raising the following question: is Ass H^i_I(J) always finite? We give a positive answer when i=2 but provide a counterexample when i=3. The counterexample crucially requires Ass H^2_I(S) to be infinite. The following question, to the best of our knowledge, is open: (under suitable hypotheses on R) does the finiteness of Ass H^{i-1}_I(S) imply the finiteness of Ass H^i_I(J)? When S is a domain, we give a positive answer when i=3. When S is locally factorial, we extend this to i=4. Finally, if R has prime characteristic p > 0 and S is regular, we give a complete answer by showing that Ass H^i_I(J) is finite for all values of i.
October 7: James Lewis, New Mexico State University
Title: Local cohomology: definition and basic properties
September 30: Luca Carai, New Mexico State University
Title: A generalization of Gelfand duality to compact Hausdorff spaces with continuous relations
September 23: Pat Morandi, New Mexico State University
Title: A generalization of Gelfand Duality to Completely Regular Spaces II
September 16: Pat Morandi, New Mexico State University
Title: A generalization of Gelfand Duality to Completely Regular Spaces
September 9: Guram Bezhanishvili, New Mexico State University
Title: An outline of Gelfand duality and its generalizations
