Kontsevich's Formula for Rational Curves from Classical and Quantum Perspectives

Abstract

Kontsevich's formula for rational plane curves is a recursive relation for the number $N_d$ of degree $d$ rational curves in $\mathbb{P}^2$ passing through $3d-1$ general points. We provide two proofs of this recursion: the first more direct and combinatoric, the second more abstract. In order to achieve this, we introduce several moduli spaces, such as the Deligne-Mumford-Knudsen spaces and the Kontsevich spaces, and exploit their properties. In particular, the boundary structure of these spaces gives rise to certain fundamental relations crucial to both proofs. For the second proof, we reconsider the objects in question from the cohomological viewpoint and generalize the numbers $N_d$ to Gromov-Witten invariants. We introduce quantum cohomology and deduce Kontsevich's formula from the associativity of the quantum product. We also adapt these steps to the case of curves in $\mathbb{P}^1\times\mathbb{P}^1$, whose bidegrees lead to slightly more complicated but analogous results.