AEI-DFG – Classical Singularity theory meets positive characteristic methods.

• Applicants: Professor Manuel Blickle, Ph.D.; Professor Dr. Duco van Straten
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541528446

While singularities in commutative algebra and geometry are studied with different motivations and apparently distant methods, they are really the same objects. A polynomial over the integers defines a hypersurface over the complex numbers, or over a number field, or by reduction modulo a prime over a field of positive characteristic, so that in the same object complex geometry, topology, arithmetics and positive characteristic interacts. History has shown there is a fruitful transfer of information between these aspects. This project sits at the intersection of algebraic methods for the study of singularities as pursued by Blickle and Smirnov, and geometric aspects of classical singularity theory of spaces and maps as pursued by de Bobadilla and van Straten. The overarching goal of the project is to bring together various methods and approaches to gain a new perspective for addressing key questions in this field. Specifically, the following key objectives are to be addressed: 1. Find a "volume-like" invariant for the class of rational singularities: this invariant should be positive only for rational singularities and should detect non-singularity by ist maximal value. 2. Build the theory F-signature for general Cartier modules. 3. Compare Arnol’d and Nguyen classifications of hypersurface singularities in zero and positive characteristic and derive a list of adjacencies in mixed characteristic from it. 4. Exploit upper semi-continuouity properties of Frobenius type invariants to obstruct adjacencies. 5. Study the cohomological Milnor fibration for hypersurface smoothings in positive/mixed characteristic, derive upper semicontinuous invariants by deformation.6. Generalize the Steenbrink/Rapoport–Zink spectral sequence to cover models with semi-log terminal singularities, as those appearing from MMP, in order to reduce the redundant information present in semi-stable models. Explore possible applications to the Le-Ramanujam problem. 7. Generalize the cohomological counterpart of the notion of disentaglements and image Milnor number in positive/mixed characteristic, paying special attention on Mond’s conjecture.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partners: Professor Dr. Javier Fernandez de Bobadilla; Ilya Smirnov, Ph.D.