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Organizing Committee
University of Minnesota
Purdue University
Syracuse University
University of Notre Dame
Purdue University
Abstract
The adoption of D-module techniques has transformed the interface between commutative algebra and algebraic geometry over the last two decades. The discovery of interactions and parallels with the Frobenius morphism has been an impetus for many new results, including new invariants attached to singularities but also D- and F-module based algorithms for computing quantities that used to be unattainable.
Our goal for this workshop is to discuss computational aspects and new challenges in singularity theory, focusing on special varieties that arise from group actions, canonical maps, or universal constructions. By bringing together geometers, algebraists, and invariant theorists, we will address problems from multiple perspectives. These will include comparisons of composition chains for D- and F-modules, the impact of group actions on singularity invariants, and the structure of differential operators on singularities in varying characteristics.
Our goal for this workshop is to discuss computational aspects and new challenges in singularity theory, focusing on special varieties that arise from group actions, canonical maps, or universal constructions. By bringing together geometers, algebraists, and invariant theorists, we will address problems from multiple perspectives. These will include comparisons of composition chains for D- and F-modules, the impact of group actions on singularity invariants, and the structure of differential operators on singularities in varying characteristics.
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Workshop Schedule
Monday, August 09, 2021
9:45 - 10:00 AM EDT
Welcome
Virtual
Brendan Hassett, ICERM/Brown University
10:00 - 10:40 AM EDT
On the Hodge filtration on local cohomology
Virtual
Speaker
Mircea Mustaţă, University of Michigan
Session Chair
Uli Walther, Purdue University
Abstract
The local cohomology sheaf of a smooth complex variety along a closed subvariety comes endowed with a Hodge filtration, via Saito's theory of mixed Hodge modules. I will discuss some properties of this filtration, based on joint work with Mihnea Popa.
10:50 - 11:00 AM EDT
Break
Coffee Break - Virtual
11:00 - 11:40 AM EDT
Extremal Singularities in Positive Characteristic
Virtual
Speaker
Karen Smith, University of Michigan
Session Chair
Uli Walther, Purdue University
Abstract
What is the most singular possible singularity? What can we say about its geometric and algebraic properties? This seemingly naive question has a sensible answer in characteristic p.
The ""F-pure threshold,"" which is an analog of the log canonical threshold, can be used to ""measure"" how bad a singularity is. The F-pure threshold is a numerical invariant of a point on (say) a hypersurface---a positive rational number that is 1 at any smooth point (or more generally, any F-pure point) but less than one in general, with ""more singular"" points having smaller F-pure thresholds. We explain a recently proved lower bound on the F-pure threshold in terms of the multiplicity of the singularity. We also show that there is a nice class of hypersurfaces--which we call ""Extremal hypersurfaces""---for which this bound is achieved. These have very nice (extreme!) geometric properties. For example, the affine cone over a non Frobenius split cubic surface of characteristic two is one example of an ""extremal singularity"". Geometrically, these are the only cubic surfaces with the property that *every* triple of coplanar lines on the surface meets in a single point (rather than a ""triangle"" as expected)--a very extreme property indeed.
11:50 AM - 1:00 PM EDT
Lunch/Free Time
Virtual
1:00 - 1:40 PM EDT
Pure subrings of polynomial rings
Virtual
Speaker
Anurag Singh, University of Utah
Session Chair
Uli Walther, Purdue University
Abstract
Let G be a linearly reductive group over a field K, with a linear action on a polynomial ring over K. Then the invariant ring is a pure subring of the polynomial ring; many key properties of classical invariant rings including finite generation and the Cohen-Macaulay property, as in the Hochster-Roberts theorem, follow from purity.
Now let A denote either a field, or the ring of integers, or a ring of p-adic integers. When is a given finitely generated A-algebra a pure subring of a polynomial ring over A? We will discuss how this can be addressed via D-modules, Group Actions, and Frobenius! The recent Computing on Singularities is joint work with Jack Jeffries.
1:50 - 2:00 PM EDT
Break
Coffee Break - Virtual
2:00 - 2:40 PM EDT
Primary decomposition with differential operators
Virtual
Speaker
Yairon Cid-Ruiz, Ghent University
Session Chair
Uli Walther, Purdue University
Abstract
We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis- Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed. This is joint work with Bernd Sturmfels.
3:00 - 5:00 PM EDT
Gathertown Afternoon Coffee
Coffee Break - Virtual
Tuesday, August 10, 2021
10:00 - 10:40 AM EDT
Non-abelian Mellin transformation
Virtual
Speaker
Botong Wang, University of Wisconsin
Session Chair
Claudiu Raicu, University of Notre Dame
Abstract
As the constructible counterpart of the Fourier-Mukai transformation, the non-abelian Mellin transformation of a constructible complex can be considered as taking the hyper-cohomology of the complex twisted by all possible local systems simultaneously. We will explain a t-exactness result about the non-abelian Mellin transformation, generalizing a theorem of Gabber-Loeser on affine torus. We will also discuss some local vanishing properties of the Sabbah's specialization functor, which is a key step in the proof of the t-exactness result.
10:50 - 11:00 AM EDT
Break
Coffee Break - Virtual
11:00 - 11:40 AM EDT
Bernstein's inequality and holonomicity for certain singular rings
Virtual
Speaker
Jack Jeffries, University of Nebraska-Lincoln
Session Chair
Claudiu Raicu, University of Notre Dame
Abstract
We prove the Bernstein inequality and develop the theory of holonomic D-modules for rings of invariants of finite groups in characteristic zero, and for strongly F-regular finitely generated graded algebras with FFRT in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic D-modules, in this context, have finite length. This is based on joint work with Josep Àlvarez Montaner, Daniel J. Hernández, Luis Núñez-Betancourt, Pedro Teixeira, and Emily E. Witt.