A Compactification of the Space of Holomorphic Maps from $\P^1$ into $\P^r$

Abstract

Let $M_{d}(\P^r)$ be the space of $(r+1)$-tuples $(f_0,...,f_r)$ modulo homothety, where $f_0,...,f_r$ are homogeneous polynomials of degree $d$ in two variables. Let $M_{d}^{\circ}(\P^r)$ be the open subset of $M_{d}(\P^r)$ such that $f_0,...,f_r$ have no common zeros. Then $M_{d}^{\circ}(\P^r)$ parametrizes the space of holomorphic maps of degree $d$ from $\P^1$ into $\P^r$. In general the boundary divisor $M_{d}(\P^r) \setminus M_{d}^{\circ}(\P^r)$ is not normal crossing. In this paper we will give a natural stratification of this boundary and show that we can process an iterated blow-ups along these strata (or its proper transformations) to obtain a compactification of $M_{d}^{\circ}(\P^n)$ with normal crossing divisors.