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Moduli Spaces of Dynamical Systems on Pn

Alon Levy

Title:
Moduli Spaces of Dynamical Systems on Pn
Author(s):
Levy, Alon
Thesis Advisor(s):
Zhang, Shou-Wu
Date:
Type:
Theses
Degree:
Ph.D., Columbia University
Department(s):
Mathematics
Persistent URL:
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
1207
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Suggested Citation:
Alon Levy, , Moduli Spaces of Dynamical Systems on Pn, Columbia University Academic Commons, .

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