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

TL;DR

This paper introduces a new compactification of the moduli space of degree d self-maps of the projective line with n markings, constructed via GIT from stable maps, and provides tools for intersection theory and boundary analysis.

Contribution

It constructs a new GIT-based compactification of the moduli space, analyzes its geometric properties, and develops algorithms for intersection calculations and boundary extensions.

Findings

The space is a coarse moduli of a smooth Deligne-Mumford stack.

An explicit recursive algorithm for intersection numbers is provided.

The boundary structure allows extension of iteration maps to the boundary.

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 $M(d,n) \dashrightarrow M(d^m,n)$ and we give a geometric way to extend this rational map to parts of the boundary of $M(d,n)$.