Moduli Spaces of Dynamical Systems on Pn
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.
AC:P:10408
Academic Commons
Levy, Alon
Author
Zhang, Shou-Wu
Thesis advisor
Columbia University. Mathematics
Originator
Theses
English
Mathematics
text
2011
eng
2017-06-07T02:50:16Z
2017-06-07T14:51:18Z
Ph.D.
2
Mathematics
Columbia University
10.7916/D8QJ7Q9F