Moduli Spaces of Dynamical Systems on Pn
 Title:
 Moduli Spaces of Dynamical Systems on Pn
 Author(s):
 Levy, Alon
 Thesis Advisor(s):
 Zhang, ShouWu
 Date:
 2011
 Type:
 Theses
 Degree:
 Ph.D., Columbia University
 Department(s):
 Mathematics
 Persistent URL:
 https://doi.org/10.7916/D8QJ7Q9F
 Abstract:
 This thesis studies the space of morphisms on Pn defined by polynomials of degree d and its quotient by the conjugation action of PGL(n+1), which should be thought of as coordinate change. First, we construct the quotient using geometric invariant theory, proving that it is a geometric quotient and that the stabilizer group in PGL(n+1) of each morphism is finite and bounded in terms of n and d. We then show that when n = 1, the quotient space is rational over a field of any characteristic. We then study semistable reduction in this space. For every complete curve C in the semistable completion of the quotient space, we can find curves upstairs mapping down to it; this leads to an abstract complete curve D with a projective vector bundle parametrizing maps on the curve. The bundle is trivial iff there exists a complete curve D in the semistable space upstairs mapping down to C; we show that for every n and d we can find a C for which no such D exists. Finally, in the case where D does exist, we show that, whenever it lies in the stable space, the map from D to C is ramified only over points with unusually large stabilizer, which for a fixed rational C will bound the degree of the map from D to C.
 Subject(s):
 Mathematics
 Item views
 1260
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 Suggested Citation:
 Alon Levy, 2011, Moduli Spaces of Dynamical Systems on Pn, Columbia University Academic Commons, https://doi.org/10.7916/D8QJ7Q9F.