On the enumerative geometry of moduli of stable maps

In this thesis, we explore the enumerative geometry of rational curves in the complex projective plane, focusing on the classical problem of counting rational curves of degree d passing through 3d-1 general points. Following the framework of Kock and Kontsevich, we construct the moduli spaces of n-pointed stable curves and stable maps, detailing their properties and boundary structures. The central result of this work is the rigorous derivation of Kontsevich's recursive formula, achieved by analyzing transversal intersections within the boundary divisors of the moduli space. We conclude by providing a modern cohomological interpretation of the problem, defining genus-zero Gromov-Witten invariants, and outlining the generalized recursion that computes these invariants in arbitrary projective spaces.