A compactification of the moduli space of self-maps of ${\mathbb{CP}}^1$ using stable maps

Abstract

We present a new compactification $M(d,n)$ of the moduli space of self-maps of ${\mathbb{CP}}^1$ of degree $d$ with $n$ markings. It is constructed via GIT from the stable maps moduli space $ar{M}_{0,n}({\mathbb{CP}}^1\times {\mathbb{CP}}^1,(1,d))$. We show that it is the coarse moduli space of a smooth Deligne-Mumford stack and we compute its rational Picard group. Using the recursive boundary structure inherited from the stable maps space, we give an explicit algorithm for computing top-intersection numbers of divisors on $M(d,n)$. We also study the $m$-fold iteration map $M(d,n)⇢M(d^m,n)$ and we give a geometric way to extend this rational map to parts of the boundary of $M(d,n)$.

arXiv comment90 pages, comments welcome

References in corpus (2)

• Grothendieck Duality for Deligne-Mumford Stacks
• The space of rational maps on P^1