In algebraic geometry, the existence and geometry of quotient schemes is a delicate issue.  Even when quotients exist, they may not reflect enough properties of the original group action to be useful.  The machinery of geometric invariant theory is one prescription for identifying open subsets of the original scheme that admit useful quotients, but it can be shown that there are, in general, other open sets that also admit well-behaved quotients.  In this talk, we examine particular actions of diagonalizable groups on affine space and illustrate the wide variety

Let \(R\) be a Noetherian ring, \(P\) be a prime ideal of \(R\) such that \(R_P\) is Cohen-Macaulay of dimension \(h\), \(\omega\) be a finitely generated \(R\)-module such that \(\omega_P\) is a canonical module for \(R_P\), and \(W\) be a subset of \(R\) that naturally maps onto the set of nonzero elements of \(R/P\). We show that for every finitely generated $R$-module $M$, there exists \(g \in W\) such that \(H_P^j(M)_g \cong Hom(Ext_R^{h-j}(M, \omega), H_P^h(\omega))_g\), which gives the well-known local duality when we localize at \(P\).

In algebraic geometry, the existence and geometry of quotient schemes is a delicate issue.  Even when quotients exist, they may not reflect enough properties of the original group action to be useful.  The machinery of geometric invariant theory is one prescription for identifying open subsets of the original scheme that admit useful quotients, but it can be shown that there are, in general, other open sets that also admit well-behaved quotients.  In this talk, we examine particular actions of diagonalizable groups on affine space and illustrate the wide variety

The Hilbert-Samuel multiplicity of an ideal is a fundamental invariant of singularities of the ideal and is known to satisfy various convexity properties. In this talk, I will discuss more general convexity properties for ideal sequences (rather than single ideals) and provide an application to Chi Li’s normalized volume of a singularity. 

A framework was developed in a joint work with F. J. Gallego and M. Gonzalez to systematically deal with the deformation of finite morphisms, multiple scheme structures on algebraic varieties and their smoothing. There are several applications of this framework. In this talk I will talk about some of them. First is the description of some components of the moduli space of varieties of general type in all dimensions.

The log minimal model program has recently been completed for klt threefolds over excellent base schemes of residue characteristic p>5�>5.  In this talk I will survey the known results, together with some motivations and applications for working in this more general setup.

The talk will be on Zoom. Here is the link: https://umsystem.zoom.us/j/95107302505?pwd=Y1BMTWw1TDl0My9Haks0ZTFadk5hdz09

The passcode is: 967345

Let \(R\) be a Noetherian ring, \(P\) be a prime ideal of \(R\) such that \(R_P\) is Cohen-Macaulay of dimension \(h\), \(\omega\) be a finitely generated \(R\)-module such that \(\omega_P\) is a canonical module for \(R_P\), and \(W\) be a subset of \(R\) that naturally maps onto the set of nonzero elements of \(R/P\). We show that for every finitely generated $R$-module $M$, there exists \(g \in W\) such that \(H_P^j(M)_g \cong Hom(Ext_R^{h-j}(M, \omega), H_P^h(\omega))_g\), which gives the well-known local duality when we localize at \(P\).

The analytic spread of a module M is the minimal number of generators of a submodule that has the same integral closure as M. In this talk, we will present a result that expresses the analytic spread of a decomposable module in terms of the analytic spread of its component ideals. In the second part of the talk, we will show an upper bound for the symbolic analytic spread of ideals of small dimension. The latter notion is the analogue of analytic spread for symbolic powers. These results are joint work with Carles Bivià-Ausina and Hailong Dao, respectively.

In this talk, I will discuss some joint work with Alan Dills on concepts devised to describe the size of the Frobenius skew-polynomial ring over a commutative graded algebra over a field in prime characteristic. The ideas are inspired from Gelfand-Kirillov dimension theory. I will discuss what these notions are for the cusp and how to compute them.

Wilf’s conjecture establishes an inequality that relates three fundamental invariants of a numerical semigroup: the minimal number of generators (or the embedding dimension), the Frobenius number, and the number of gaps. Based on a preprint by Srinivasan and S-, the talk will discuss the past, present, and future of this conjecture.