An interesting algebraic variety

Abstract

In this article I would like to describe the structure of the Igusa compactification of the Siegel modular space of degree two and level two. This space is of particular interest for several reasons. For one thing, it lies in the intersection of a number of different areas of mathematics--Riemann surfaces, theta functions a nd automorphic forms, algebraic geometry, and differential topology. For another, it is a special case of several general constructions--t hose of the theory of compactifications, geometric invariant theory, resolution of singularities, and symplectic geometry. Further, there are a number of different and fruitful methodsalgebraic, analytic, geometric, and topological--with which to analyze this space. (Not to be neglected is the chance to impress the reader with an object whose name has 13 words, thereby being 6 1/2 times as good as the Fermat conjecture.) The point of this article is that this space has a beautiful and intricate geometric structure, which I shall attempt to describe well enough for the reader to appreciate. I will begin by describing elliptic curves (thereby, like Proust's madeleine, taking the reader back to the halcyon days of graduate school) and then discuss our situation, which is a generalization thereof. Recall that an elliptic curve (as in figure IIa) is C/L, the complex plane C modulo a lattice L. (A lattice is the set of integral linear combinations of two non-collinear vectors based at some point of the complex plane.) Without changing the elliptic curve, we may normalize the lattice by requiring that the vectors be based at the origin, the first vector be the standard unit vector in the x-direction, and the second vector have positive y-coordinate, so that the lattice is specified by the endpoint of this vector, i.e. by a point x in S 1 = {zeC[Imz>O}