The project “Classical Singularity theory meets positive characteristic methods” by BCAM researchers Javier Fernández de Bobadilla and Ilya Smirnov has been selected in the AEI-DFG bilateral Spanish-German research project call

• Led by Javier Fernández de Bobadilla and Ilya Smirnov, the researchers will collaborate with Johannes Gutenberg University of Mainz.
• This call aims to select research projects involving both Spanish and German groups. The DFG will fund the German groups, and the AEI will fund the Spanish beneficiaries through the direct grant process under the “International Collaboration Projects” (PCI) program, scheduled for the second half of 2024.

The project “Classical Singularity theory meets positive characteristic methods” by Javier Fernández de Bobadilla (Ikerbasque Research Professor and leader of the Singularities and Algebraic Geometry group at BCAM) and Ilya Smirnov (Ikerbasque Research Fellow and Ramón y Cajal Fellow at BCAM / Singularities and Algebraic Geometry group) has been selected in the 2023 AEI-DFG bilateral Spanish-German research project call.

The two researchers will carry out cutting-edge research aimed at addressing fundamental questions in algebraic geometry, focusing on the interaction between zero, positive, and mixed characteristic methods. This ambitious project tackles multiple objectives with the potential to transform the understanding of singularities by bridging seemingly distant methods.

The main objectives of this research include:

1. Volumetric invariant for rational singularities: Develop an invariant that is positive exclusively for rational singularities and detects non-singularity through a maximum value.
2. F-signature theory for Cartier modules: Build a theoretical framework for these modules, expanding applications in positive characteristics.
3. Singularity classifications: Compare the Arnold and Nguyen classifications for hypersurface singularities in zero and positive characteristics, deriving an adjacency list in mixed characteristics.
4. Semicontinuous properties of Frobenius-type invariants: Explore these properties to obstruct the existence of singularity deformations.
5. Cohomological Milnor fibration: Study the Milnor fibers of hypersurfaces in positive and mixed characteristics to derive semicontinuous invariants that can obstruct deformations.
6. Generalization of the Steenbrink/Rapoport-Zink spectral sequence: Adapt this concept to models with semi-logarithmic terminal singularities, with possible applications to the Le-Ramanujam problem.
7. Generalization of disentanglements and image Milnor numbers: Pay special attention to Mond's conjecture, extending the study to positive and mixed characteristics.

Fernández de Bobadilla and Smirnov emphasize that this work not only provides novel tools for analyzing singularities but also opens the door to applications in areas such as stable model theory and classical problems in algebraic geometry.

This initiative is part of an international effort to deepen the understanding of fundamental mathematical structures, contributing to the advancement of mathematics.

The call under which the project is framed aims to select research projects involving both Spanish and German groups. The DFG will fund the German groups, and the AEI will fund the Spanish beneficiaries through the direct grant process under the “International Collaboration Projects” (PCI) program, scheduled for the second half of 2024.

The BCAM researchers will collaborate with Johannes Gutenberg University of Mainz.

Funded by  PCI2024-155055-2 by MCIN/AEI /10.13039/501100011033 and by the European Union NextGenerationEU/ PRTR