2020 Theses Doctoral

# Moduli of Surfaces and Applications to Curves

This thesis has two parts. In the first part, we construct a moduli scheme F[n] that parametrizes tuples (S_1, S_2,..., S_{n+1}, p_1, p_2,..., p_n) where S_1 is a fixed smooth surface over Spec R and S_{i+1} is the blowup of S_i at the point p_i, ∀1≤i≤n. We show this moduli scheme is smooth and projective. We prove that F[n] has smooth divisors D_{i,j}^(n), ∀1≤i<j≤n, which correspond to tuples that map p_j->p_i under the projection morphism S_j->S_i. When R=k is an algebraically closed field, we demonstrate that the Chow ring A*(F[n]) is generated by these divisors over A*(S_1^n). We end by giving a precise description of A*(F[n]) when S_1 is a complex rational surface.

In the second part of this thesis, we focus on finding a characterization of the smooth surfaces S on which a smooth very general curve of genus g embeds as an ample divisor. Our results can be summarized as follows: if the Kodaira dimension of S is κ(S)=-∞ and S is not rational, then S is birational to CxP^1. If κ(S) is 0 or 1, then such an embedding does not exist if the genus of C satisfies g≥22. If κ(S)=2 and the irregularity of S satisfies q(S)=g, then S is birational to the symmetric square Sym^2(C).

We analyze the conditions that need to be satisfied when S is a rational surface. The case in which S is of general type and q(S)=0 remains mainly open; however, we provide a partial answer to our question if S is a complete intersection.

## Subjects

## Files

- Marinescu_columbia_0054D_15851.pdf application/pdf 796 KB Download File

## More About This Work

- Academic Units
- Mathematics
- Thesis Advisors
- de Jong, Aise Johan
- Degree
- Ph.D., Columbia University
- Published Here
- July 6, 2020