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?
Algebraic geometry is the study of solutions of polynomial equations, known as varieties, using techniques from both algebra and geometry. In some situations, the underlying polynomials have coefficients over the integers, their congruence classes, the real numbers, or finite fields. Then it is natural to seek solutions taking values in these systems. Rational curves are solutions in rational functions in a single variable, e.g., the solution x=(t2-1)/(t2+1) y=2t/(t2+1) for the equation x2+y2=1. These are fundamental to the classification of varieties, e.g., rationally connected varieties, which admit rational curves joining any pair of points. For other classes of varieties, like K3 surfaces, properties of rational curves remain conjectural. The simplest example of K3 surfaces are quartic surfaces like x4+y4+z4=1. Most K3 surfaces admit infinitely many rational curves of increasing geometric complexity, but this remains open in general. One approach is to exhibit rational curves with infinitely many numerical invariants (homology classes) of increasing degrees. A coarser question is to describe the convex hull of the invariants that arise--this has long been known for K3 surfaces but has only recently become clear for higher-dimensional analogues through joint work with Bayer and Tschinkel. Smooth rationally connected varieties over the real numbers R have many good properties, e.g., there is a real rational curve through each real point. One subtle question is whether rationally connected real varieties without real points admit solutions over R[x,y]/(x2+y2+1), corresponding to maps from `pointless' curves of genus zero. Recent joint work with Allums clarifies the case of hypersurfaces and complete intersections, but this remains mysterious in general. In some situations, solving systems of equations is tied up with classifying them in geometric terms. In joint work with Kresch and Tschinkel, we have classified systems Q1(t; v,w,x,y,z)= Q2(t; v,w,x,y,z)=0 of homogeneous quadratic equations in v, w, x, y, and z varying in a parameter t. In geometric terms, these are fibrations in quartic del Pezzo surfaces over the projective line. This project involves understanding the interplay between classically-defined invariants of the family and the structure of degenerate fibers for special values of t. (See the attached figure for a schematic.) Our ultimate goal is to understand solutions where the variables v, w, x, y, and z are rational functions in t, especially where the coefficients are taken from the real numbers or finite fields. This project has numerous broader impacts, primarily through the development of human resources. The research group led by the PI involves numerous undergraduates, graduate students, and postdoctoral fellows. Former group members now work in academic mathematics, industry, and government agencies supporting national security. In addition, this award supports public activities like workshops and conferences, where young mathematicians have the opportunity to learn state-of-the-art techniques and present their own work through poster sessions and information presentations.
