2011/04/22 by Alon Y. Levy, Alon Levy, Levy, Alon
Mathematics · #14L24 #37P45 #37P55 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #math.AG #math.DS #msc:14L24 #msc:37P45 #msc:37P55
paper · pdf · doi:10.48550/arxiv.1104.4517
17 pages
openalex publication_date 2011/04/22 · arxiv created 2011/06/08 · arxiv updated 2011/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We restate the semistable reduction theorem from geometric invariant theory in the context of spaces of morphisms on ℙn. For every complete curve C downstairs, we get a ℙn-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.