On the Motivic Goettsche Invariants

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Algebraic geometry is the study of solutions of polynomial equations, with many applications in other parts of mathematics, physics, and engineering. Some of the well-studied objects in algebraic geometry are the projective plane (the plane with additional "points at infinity") and curves on it. A natural question is that of finding the number of curves that satisfy certain special conditions, such as passing through fixed points or having special shapes, which was the starting point of a branch of algebraic geometry called enumerative geometry. During the past twenty years, the interaction between enumerative geometry and string theory in physics has resulted in many exciting new developments and has been a central area of current research. This research project aims to define and study new invariants, in a broader sense on general algebraic surfaces, that have connections to both mathematics and physics. These invariants contain sophisticated information about the number of singular curves, the geometric properties of the surface, and the parametrization space of singular curves. The project envisages development of a theoretical framework and tools for computation, as well as establishing connections between different subjects in mathematics. It will add to the understanding of geometry of algebraic varieties and has application to the description of spacetime in string theory. This research project treats several problems in algebraic geometry, especially about the enumeration of singular curves on algebraic surfaces and its motivic generalization. Previous work has established that the numbers of varieties with certain given singularities satisfy universal formulas. This project concerns the generalization of these universal formulas in the motivic sense. The research defines a sequence of motivic invariants and studies their quantitative properties as well as their geometric interpretation. The work aims to extend the existing connections with Gromov-Witten theory, stable pairs theory, and modular forms for these new invariants, and to compare them with other motivic curve-counting invariants. One of the advantages of these new invariants is that they satisfy nice formulas and only depend on topological intersection numbers.