This project addresses the theory of rational curves on K3 surfaces, as a prototype and model for investigations of rational curves on higher-dimensional varieties of Fano and intermediate type. The key problems concern the existence of infinitely many rational curves on K3 surfaces over countable fields, techniques for the generation of such curves and computation of their numerical invariants, Brill-Noether loci and enumerative geometry of rational curves, aspects of mixed-characteristic deformation theory, Galois representations, and Brauer groups.
The term `K3 surface' was coined by A. Weil in the 1950's, and honors the seminal contributions of Kummer, Kaehler, and Kodaira to their structure. These surfaces have been central to complex geometry for decades, but recently their arithmetic properties have received increasing attention. This project addresses problems at the interface of complex, algebraic, and arithmetic geometry. In particular, what is the structure of the curves on a K3 surface? Can they be constructed explicitly? And how do they reflect symmetries of the ambient surface?
This project is at the interface of higher-dimensional algebraic geometry and number theory. Algebraic geometry is one of the pillars of modern mathematics; it is concerned with the study of systems of algebraic equations, for example linear, or quadratic equations. In number theory, the focus is on equations with coefficients in the rational numbers and on rational solutions of these equations, called rational points. There are deep analogies between rational points and rational curves, which are geometric objects, and which are visualized as spheres. How many such spheres can be put into a higher-dimensional variety? This is a challenging problem, when the ambient variety is constrained. Of particular interest to mathematical physicists are Calabi-Yau varieties, which are expected to play a role in the description of our universe. Mirror symmetry, inspired by mathematical physics, predicts that a K3 surface, the first nontrivial Calabi-Yau variety contains infinitely many rational curves. This turned out to be a difficult problem in algebraic geometry, that inspired work by many research groups. One of the main outcomes of the research on this project is a rigorous proof that general K3 surfaces contain infinitely many rational curves. Even more importantly, this question was connected to the study of hidden symmetries of K3s surfaces, Galois symmetries. This lead to new experimental investigations regarding such symmetries, which in turn stimulated the development of very efficient software (implemented on supercomputers at NYU), to compute these Galois symmetries. We plan to incorporate this software into widely accepted computational algebra packages, such as Magma. Further significant results concern the geometry of spaces of rational curves on higher-dimensional analogs of K3 surfaces. All of these results have important implications for number- theoretical questions, which have also been explored within this project. This research resulted in several high-profile publications, with new papers in the pipeline. The results were presented at international conferences and workshops and have already lead to subsequent publications by researchers in France, Germany, and the UK.
