Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Enumerating pencils with moving ramification on curves
Seminar | January 21 | 3:45-5 p.m. | 939 Evans Hall
Carl Lian, Columbia university
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 $E\to \mathbb P^1$ 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.