Speaker
Rick Miranda (Colorado State University)

Title
Moduli spaces for rational elliptic surfaces (of index 1 and 2)

Abstract

Elliptic surfaces form an important class of surfaces both from the theoretical perspective (appearing in the classification of surfaces) and the practical perspective (they are fascinating to study, individually and as a class, and are amenable to many particular computations). Elliptic surfaces that are also rational are a special sub-class. The first example is to take a general pencil of plane cubics (with 9 base points) and blow up the base points to obtain an elliptic fibration; these are so-called Jacobian surfaces, since they have a section (the final exceptional curve of the sequence of blowups). Moduli spaces for rational elliptic surfaces with a section were constructed by the speaker, and further studied by Heckman and Looijenga. In general, there may not be a section, but a similar description is possible: all rational elliptic surfaces are obtained by taking a pencil of curves of degree 3k with 9 base points, each of multiplicity k. There will always be the k-fold cubic curve through the 9 points as a member, and the resulting blowup produces a rational elliptic surface with a multiple fiber of multiplicity m (called the index of the fibration). A. Zanardini has recently computed the GIT stability of such pencils for m=2; in joint work with her we have constructed a moduli space for them via toric constructions. I will try to tell this story in this lecture.