Workshop

The talk is concerned with the structure of the module of derivations and its connections with singularities and vector fields of varieties. Modules of derivations are not well understood -- despite great advances on the Zariski-Lipman conjecture, there is still no complete characterization for when they are free. Our work is partially motivated by a question of Poincaré, who asked how to decide whether a polynomial vector field in the complex plane leaves some algebraic curve invariant. We reformulate this problem in terms of bounding from below the initial degree of the module consisting of all vector fields that leave a fixed curve invariant. This module is a quotient of the module of derivations. We also treat the case of smooth varieties of higher dimension. For plane curves of low genus we establish a correspondence between finer invariants of the module of derivations, such as its graded betti numbers, and properties of the singularities of the curve. This is joint work with Chardin, Hassanzadeh, Simis, and Ulrich.

Given a complex of R-modules M, one can construct a variety that contains homological information about M called the cohomological support variety V_R(M). These have various homological applications – for example, Pollitz showed they can be used to characterize when R is a complete intersection. In this talk, we will discuss the following realizability question: given an appropriately chosen variety V, when can V be realized as V_R(M) for some M? To study this question, we give bounds on the dimension of such varieties. This is joint work with Ben Briggs and Josh Pollitz.

We investigate the lengths of certain local cohomology modules over polynomial rings. By fixing the degree component, and using the fact that the length of an Artinian ring is the same as that of its injective hull, we transform this into a question about rings of the form $k[x_1,\dots,x_n]/(x_1^{d_1}, \dots, x_n^{d_n})$ and the annihilator of $x_1 + \dots + x_n$ therein. We in particular use refinements of functions introduced by Han and Monsky. This was motivated by questions about behavior of the length of local cohomology with support in the maximal ideal of thickenings, that is, $R/I^t$ as $t$ grows. This project is joint work with Mel Hochster.

A central question in Gorenstein liaison asks whether every Cohen-Macaulay ideal is glicci, i.e., whether every Cohen-Macaulay ideal belongs to the Gorenstein liaison class of a complete intersection. After introducing Gorenstein liaison and motivating this question, I will report on a joint work with Rajchgot and Satriano, where we show that every multiplicity-free prime ideal is glicci.

Moderator: Hema Srinivasan, University of Missouri