#### Vol. 6, No. 7, 2012

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

### Alon Levy

Vol. 6 (2012), No. 7, 1483–1501
##### Abstract

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

##### Keywords
semistable reduction, moduli space, dynamical system, GIT, geometric invariant theory
##### Mathematical Subject Classification 2010
Primary: 14L24
Secondary: 37P45, 37P55