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

Details A Compactification of the Space of Holomorphic Maps from $¶^1$ into $¶^r$ A Compactification of the Space of Holomorphic Maps from $¶^1$ into $¶^r$

2005-05-23

arxiv.org/abs/math/0505482v2

Mathematics

Algebraic Geometry

25 pages, typos corrected, more precise reference added

Scientific paper

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.

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