The semistable reduction problem for the space of morphisms on $\mathbb{P}^{n}$

Abstract

We restate the semistable reduction theorem from geometric invariant theory in the context of spaces of morphisms from ${\mathbb{P}}^n$ to itself. For every complete curve $C$ downstairs, we get a ${\mathbb{P}}^n$-bundle on an abstract curve $D$ mapping finite-to-one onto $C$, whose trivializations correspond to not necessarily complete curves upstairs with morphisms corresponding to identifying each fiber with the morphism the point represents. Finding a trivial bundle is equivalent to finding a complete $D$ upstairs mapping finite-to-one onto $C$; we prove that in every space of morphisms, there exists a curve $C$ for which no such $D$ exists. In the case when $D$ exists, we bound the degree of the map from $D$ to $C$ in terms of $C$ for $C$ rational and contained in the stable space.