Geometry of moduli of cubic surfaces.
Rosenberg, Joel Evan
1999
Abstract
The study of cubic surfaces is a relatively old topic in algebraic geometry, dating from the discovery in 1849 of the 27 lines on a smooth cubic surface. The concept of a moduli space is rather newer, essentially dating from the work of Mumford in the 1960s. The classical results about cubic surfaces have led many people to study the moduli of cubic surfaces, and from many different viewpoints. Our approach in this work will be to look at the moduli space M of unmarked cubic surfaces in a purely algebraic way, although for convenience we will restrict our attention to varieties over <blkbd>C</blkbd> . Our goal will be to parallel, as far as possible, the rich theory of the moduli spaces of stable curves. In this light, our first topic will be to study some natural subvarieties of the fourfold M . First, we look at the boundary divisor Delta, describing the moduli of the singular cubic surfaces, and we show that Delta is isomorphic to the moduli space of six unordered marked points on a line. Next, we discuss the Eckardt divisor <italic>E</italic>, the moduli space of those cubic surfaces with automorphisms. We show that this space is very closely related to M1,3</fen> , the moduli space of genus 1 curves with 3 unordered marked points. We next make a foray into the intersection theory on the moduli space M . We carry over the notions of the tautological class kappa and the Hodge class lambda from the theory of the moduli of curves, and show that the Picard group Pic M is generated by lambda. We verify the well-known calculation of the class of the divisor Delta, and present a method for computing the class of the Eckardt divisor. Next, we will look at the technique of semistable reduction. Our new result in this topic describes the semistable reduction of a family of cubic surfaces with central fiber the cone over a cubic curve. We find that if the general member of the family does not contain the vertex of the cone, then the semistable replacement is a cyclic cubic surface, while if the general member is smooth at the vertex, the replacement is an Eckardt surface. We also use the techniques of semistable replacement to construct a family of stable cubic surfaces, distinct from that presented by Naruki, covering all of M . For our last topic, we study another divisor in M . In 1899, Hutchinson presented a way to obtain a three-parameter family of Hessians of cubic surfaces as blowups of Kummer surfaces. We show that this family consists of those Hessians containing an extra class of conic curves. Based on this, we find the invariant of a cubic surface <italic>C </italic> in pentahedral form that vanishes if its Hessian is in Hutchinson's family, and show that its degree is 32. Finally, we give an explicit map between cubic surfaces in pentahedral form and blowups of Kummer surfaces.Subjects
Cubic Surfaces Geometry Moduli Spaces Semistable Reduction
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