On the enumerative geometry of branched covers of curves

In this thesis, we undertake two computations in enumerative geometry involving branched covers of algebraic curves. Firstly, we consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E → P1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola. Secondly, we consider the loci of curves of genus 2 and 3 admitting a d-to-1 map to a genus 1 curve. After compactifying these loci via admissible covers, we obtain formulas for their Chow classes, recovering results of Faber-Pagani and van Zelm when d = 2. The answers exhibit quasimodularity properties similar to those in the Gromov- Witten theory of a fixed genus 1 curve; we conjecture that the quasimodularity persists in higher genus, and indicate a number of possible variants.