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 n to itself. For every complete curve C downstairs, we get a 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
Milestones
Received: 15 June 2011
Revised: 2 August 2011
Accepted: 11 September 2011
Published: 4 December 2012
Authors
Alon Levy
Department of Mathematics
Brown University
Providence, RI 02912
United States
Department of Mathematics
Columbia University
New York, NY 10027
United States