2011 Theses Doctoral

# 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.

## Subjects

## Files

- Levy_columbia_0054D_10226.pdf application/pdf 312 KB Download File

## More About This Work

- Academic Units
- Mathematics
- Thesis Advisors
- Zhang, Shou-Wu
- Degree
- Ph.D., Columbia University
- Published Here
- May 17, 2011