Vol. 6, No. 7, 2012

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 5, 995–1242
Issue 4, 749–993
Issue 3, 531–747
Issue 2, 251–530
Issue 1, 1–249

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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

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.

semistable reduction, moduli space, dynamical system, GIT, geometric invariant theory
Mathematical Subject Classification 2010
Primary: 14L24
Secondary: 37P45, 37P55
Received: 15 June 2011
Revised: 2 August 2011
Accepted: 11 September 2011
Published: 4 December 2012
Alon Levy
Department of Mathematics
Brown University
Providence, RI 02912
United States
Department of Mathematics
Columbia University
New York, NY 10027
United States