Post-Critically Finite Maps on ${\mathbb{P}}^n$ for $n\ge 2$ are Sparse

Abstract

Let $f:{\mathbb{P}}^n\to {\mathbb{P}}^n$ be a morphism of degree $d\ge 2$. The map $f$ is said to be post-critically finite (PCF) if there exist integers $k\ge 1$ and $\ell \ge 0$ such that the critical locus ${\mathrm{Crit}}_f$ satisfies $f^{k+\ell }({\mathrm{Crit}}_f)\subseteq f^{\ell }({\mathrm{Crit}}_f)$. The smallest such $\ell$ is called the tail-length. We prove that for $d\ge 3$ and $n\ge 2$, the set of PCF maps $f$ with tail-length at most $2$ is not Zariski dense in the the parameter space of all such maps. In particular, maps with periodic critical loci, i.e., with $\ell =0$, are not Zariski dense.

32 pages

References in corpus (1)

• The Kodaira dimension of spaces of rational curves on low degree hypersurfaces