Compactifying parameter spaces

Abstract

Keynote Questions (a) (The five conic problem) Given five general plane conics C 1 , …, C 5 ⊂ ℙ 2 , how many smooth conics C ⊂ ℙ 2 are tangent to all five? (Answer on page 308.) (b) Given 11 general points p 1 ,…, p 11 ϵ ℙ 2 in the plane, how many rational quartic curves C ⊂ ℙ 2 contain them all? (Answer on page 321.) All the applications of intersection theory to enumerative geometry exploit the fact that interesting classes of algebraic varieties—lines, hypersurfaces and so on—are themselves parametrized by the points of an algebraic variety, the parameter space , and our efforts have all been toward counting intersections on these spaces. But to use intersection theory to count something, the parameter space must be projective (or at least proper) so that we have a degree map, as defined in Chapter 1. In the first case we treated in this book, that of the family of planes of a certain dimension in projective space, the natural parameter space was the Grassmannian, and the fact that it is projective is what makes the Schubert calculus so useful for enumeration. When we studied the questions about linear spaces on hypersurfaces, we were similarly concerned with parameter spaces that were projective—the Grassmannian ( k,n ) and, in connection with questions involving families of hypersurfaces, the projective space ℙ N of hypersurfaces itself. These spaces have an additional feature of importance: a universal family of the geometric objects we are studying, or (amounting to the same thing) the property of representing a functor we understand. This property is useful in many ways, first of all for understanding tangent spaces, and thus transversality questions. In many interesting cases, however, the “natural” parameter space for a problem is not projective. To use the tools of intersection theory to count something, we must add points to the parameter space to complete it to a projective (or at least proper) variety. It is customary to call these new points the boundary , although this is not a topological boundary in any ordinary sense—the boundary points may look like any other point of the space—and (more reasonably) to call the enlarged space a compactification of the original space.