Local Cohomology and Singularities

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The Principal Investigator plans to study geometric problems using different techniques in commutative algebra. This research focuses on spaces given by the set of points that satisfy certain polynomial equations in many variables. Since many phenomena can be described in terms of polynomial equations, these spaces appear in many fields of science and its applications. In such spaces most points are what is called "smooth", which, roughly speaking, means that after zooming in, their vicinity looks like a linear space. For instance, in a sphere every point is smooth and, just as the Earth, from a very close view its neighborhood looks like a plane. Then, those points that are not smooth present a particular behavior and, for that reason, are called "singular points". For instance, a cone has exactly one singular point at its vertex. The set of singular points can be described in terms of the derivatives of the polynomial equations that the points in the space satisfy. For many purposes, detecting singularities is not enough, as some are worse than others. For instance, the sharper vertices of cones are considered worse. To distinguish different singularities, one needs to use more sophisticated algebraic techniques. This research project seeks to study singularities using local cohomology modules, which can be seen as algebraic objects associated to a point. This has already proven to be a powerful tool to detect different kinds of singularities. The Principal Investigator plans to use local cohomology to study measurements of how bad a singular point is. The research includes long-standing problems in the study of singularities as well as new conjectures that could have theoretical and computational consequences. The project involves graduate students in the research. The Principal Investigator seeks to study the structure of local cohomology modules and singularities in positive and mixed characteristic. One of the main problems that one encounters while working with local cohomology modules is that they are usually very large and difficult to handle. However, these modules behave as if they were finitely generated over regular local rings that contain a field. An example of a regular ring in mixed characteristic for which injective dimension behaves differently from equal characteristic was recently found. Motivated by this result, the Principal Investigator intends to explore potential counter-examples for the properties regarding associated primes and Bass numbers of local cohomology modules over regular local rings of mixed characteristic. In addition, the Principal Investigator plans to work on the following related conjecture: the support of a local cohomology module is a Zariski closed set in the spectrum of the ring. Using local cohomology over rings containing a field, Lyubeznik introduced a family of invariants now called Lyubeznik numbers. These invariants have shown several connections with the algebraic and geometric properties of a ring. This inspired an analogous definition of these numbers in mixed characteristic. The project aims to compare the Lyubeznik numbers of rings that contain fields with those that do not. In particular, the research seeks a topological or arithmetic criterion that relates both notions of Lyubeznik numbers. In addition, the project seeks to find geometric properties encoded by the Lyubeznik numbers in mixed characteristic. Lastly, the Principal Investigator plans to work on singularities in positive characteristic via the Frobenius map. In particular, he is planning to work on the ACC conjecture for F-pure thresholds and its corollaries. In addition, the Principal Investigator and a collaborator will investigate a conjectured inequality that relates the F-pure thresholds with the Hilbert-Kunz multiplicities. If this project succeeds, the conjectured relation could have several computational and geometric consequences.