The semistable reduction problem for the space of morphisms on ℙn

Abstract

We restate the semistable reduction theorem from geometric invariant theory in the context of spaces of morphisms from Pn to itself. For every complete curve C downstairs, we get a Pn-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.