The Semistable Reduction Problem for the Space of Morphisms on $\mathbb{P}^{n}$

TL;DR

This paper explores the semistable reduction problem for morphisms on projective space, establishing conditions under which certain bundles exist and providing bounds on degrees of maps in specific cases.

Contribution

It reformulates the semistable reduction theorem in the context of morphisms on projective space and demonstrates the existence or non-existence of certain trivial bundles.

Findings

Existence of a curve C with no corresponding D in the space of morphisms.

Bounded degree of the map from D to C when D exists and C is rational.

Identification of conditions for trivial bundle existence in the stable space.

Abstract

We restate the semistable reduction theorem from geometric invariant theory in the context of spaces of morphisms on $\mathbb{P}^{n}$. 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.