On the Number of Smooth Conics Tangent to Five Fixed Conics

In section 1, we define the abelian groups Ak(X) in a manner analagous to the homology groups Hk(X) by considering cycles of subvarieties modulo rational equivalence. In section 2, we construct, for any Cartier divisor D on X and subvariety V ⊂ X, an intersection class D · [V ] ∈ Ak−1(|D|∩|V |) and state its basic properties. Bezout’s theorem follows easily. In section 3, we discuss the centerpiece problem of the paper, namely computing the number of smooth plane conics tangent to five fixed smooth conics. We will see that Bezout’s theorem is inadequate to solve the problem. In section 4, we give intersection-theoretic definitions of the Chern Classes and Segre Classes of an algebraic vector bundle, and use them to complete the computation begun in section 3. Advice to Reader: Sections 1 and 2 are primarily a resume and definitions of facts, following chapters one and two of [1]. Section 3 loosely follows pages 749-756 of [2], though we adapt the material there to Fulton’s algebraic framework. Finally, section 4 follows chapters three and four of [1]. While proofs are given, they can probably be skimmed without losing much. The most important parts of the paper are section 3 and the final pages of section 4, in which concrete calculations are made.