MTM2016-80659-P. Singularities: discriminants and valuations. Applications. (SinDisVal)

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This project represents the evolution of research developed within the research team, which shares common scientific objectives, training, and methodology. The team's consistent quality and coherence over time are evidenced by the successive results achieved in this field within research projects funded by national calls for proposals and based in Valladolid, led by A. Campillo and F. Delgado. The overall objective is to make significant progress in solving several problems with a high scientific and technical impact in the field of mathematics. These problems are: – To delve deeper into the study of valuations as a key element in addressing the singularity problem of algebraic varieties with positive characteristics, and as a source of information on the local and global geometry of algebraic varieties. To study the topology of valuation spaces, primarily in two dimensions. – To study the discriminant locus and critical points for a germ of application and their interaction with the singularities defined by the application components.

• Study of Newton's polyhedron of the discriminant in non-generic situations and its relation to important invariants of the singularity of the applications.
• Study of higher-order polars and approximate Jacobian Newton diagrams.
• Discrete invariants of plane curves in positive characteristic.
• Codes on surfaces.

Within the major problems mentioned above, some of our specific objectives, to which we hope to make contributions, are:

• Study the topological type of valuations centered on a point and check if it determines the type of homotopy of real valuations, in arbitrary dimension.
• Generalize the results of Dickenstein and Sessa and Casas-Alvero on higher-order polars for reduced k-regular curves and for quasi-ordinary singularities.
• Generalize the results already obtained by the group on Jacobian Newton polygons approximated to (1) a reduced curve, (2) a quasi-ordinary singularity, (3) a foliation (thesis problem of NE Saravia Molina) and (4) curves in positive characteristic.
• Study of Milnor's number and Lojasiewicz's exponent in positive characteristic.
• Building codes on surfaces.

The team's publications (many in prestigious journals), individual work, contact with national and international experts (facilitated, in some cases, by the awarding of this project), and a sound work and results dissemination plan will (we hope) allow us to achieve a high degree of our objectives. Felipe, García, and Teissier will work on valuations. García and Teissier will work on morphism discriminants. García and Gwozdziewicz will work on upper polars and approximate Jacobians, and doctoral candidate N.E. Saravia Molina will work on foliations. Felipe and García will work on discrete invariants (with A. Ploski collaborating). Márquez and García will work on surface codes.
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This project is the evolution of the research conducted for the research team, whose members share common scientific objectives, training and common methodology. The quality and consistency over time of the team is supported by successive research results obtained in the framework of national projects coordinated by A. Campillo and F. Delgado. The overall objective is to achieve significant advances in solving some problems that have a high scientific and technological impact in the field of Mathematics. These problems are:

• Further study of the valuations as key object to solve the problem of resolution of singularities of algebraic varieties in positive characteristic, and as a source of information on local and global geometry of algebraic varieties. Study the topology of spaces of valuations, mainly in dimension two.
• Study of the discriminant place and the critical points for a germ application and its interaction with singularities defined by the components of the map.
• Study of the Newton polyhedron of the discriminant in non-generic situations and their relationship to important invariants of the singularity of maps.
• Study of higher order polars and approximate Jacobian Newton diagrams. – Discrete invariants of plane curves in positive characteristic.
• Codes on surfaces.

Within the large problems above, some of our specific objectives, which we hope to make contributions, are:

• Study the topological type of valuations centered on a point and check if it determines the homotopy type of real valuations, in any dimension.
• To generalize the results of Dickenstein and Sessa and Casas-Alvero on higher order polars to k-regular reduced curves and to quasiordinary singularities.
• To generalize the results already obtained by the group on approximate Jacobian Newton polygons to (1) a reduced curve, (2) a quasiordinary singularity, (3) a foliation (thesis problem of Saravia Molina) and (4) curves in positive characteristic.
• Study of the Milnor number and the exponent of Lojasiewicz in positive characteristic.
• Construction of codes on surfaces.

From the publications of the team (many of them in prestigious journals), with our personal work, the contact with national and international experts (aided, if possible, by the granting of this project) and a good work plan and dissemination of results will allow a high level of achievement of our goals. De Felipe, García and Teissier will work on valuations and García and Teissier on discriminants of maps. García and Gwozdziewicz will work on higher order polars and approximate Jacobian Newton polygons and Saravia Molina on their foliations aspects. Finally de Felipe y García (in collaboration with A. Ploski) will work on discrete invariants. García y Márquez will work on codes on surfaces

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Researcher at the University of La Laguna

• Evelia Rosa García Barroso