Relations in the tautological ring by localization

Commun. Korean Math. Soc. 2006 Vol. 21, No. 3, 475-490 Printed September 1, 2006

Fumitoshi Sato Korea Institute for Advanced Study

Abstract : We give a way to obtain formulas for $\pi_{*} \psi^k_{n+1} $ in terms of $\psi$ and $\lambda$-classes where $ \pi :\moduli{g}{n+1} \rightarrow \moduli{g}{n} (g=0,1,2)$ by the localization theorem. By using the formulas, we obtain Kontsevich--Manin type reconstruction theorems for $\moduli{0}{n}(\CP^m), \moduli{1}{n},$ and $ \moduli{2}{n}$. We also (re)produce a lot of well-known relations in tautological rings, such as WDVV equation, the Mumford relations, the string and dilaton equations $(g=0,1,2)$ etc. and new formulas for $\pi_*(\lambda_g \psi_{n+1}^k+ \cdots + \psi_{n+1}^{g+k})$.