Geometry of moduli space of stable maps and degeneration

Abstract

We study geometry of moduli space of holomorphic maps from curves to projective scheme through various compactifications. Most famous one is moduli space of stable maps introduced by Kontsevich. When genus is one and target space is projective space, main component of moduli space of stable maps is nonsingular. Vakil and Zinger found some desingularization via modular blow ups. Kim introduced log stable maps with target expansions which gives another desingularization of moduli space of stable maps. We compare theses two desingularization. Also, Gross-Seibert and Abramovich-Chen defined logarithmic stable maps without target expansions. Using these moduli space, one can define log Gromov-Witten invariants. We prove the degeneration formula of log Gromov-Witten invariants.