Abstract
We restate the semistable reduction theorem from geometric invariant theory in the context of spaces of morphisms from to itself. For every complete curve downstairs, we get a -bundle on an abstract curve mapping finite-to-one onto , 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 upstairs mapping finite-to-one onto ; we prove that in every space of morphisms, there exists a curve for which no such exists. In the case when exists, we bound the degree of the map from to in terms of for rational and contained in the stable space.
Citation
Alon Levy. "The semistable reduction problem for the space of morphisms on $\mathbb{P}^{n}$." Algebra Number Theory 6 (7) 1483 - 1501, 2012. https://doi.org/10.2140/ant.2012.6.1483
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