The semistable reduction problem for the space of morphisms on ℙn

Levy, Alon

doi:10.2140/ant.2012.6.1483

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

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

Alon Levy

Vol. 6 (2012), No. 7, 1483–1501

DOI: 10.2140/ant.2012.6.1483

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

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

Vol. 6, No. 7, 2012