The fundamental group of the open symmetric product of a hyperelliptic curve

Abstract

On the second symmetric product $C^{(2)}$ of a hyperelliptic curve $C$ of genus $g$ let $L$ be the line given by the divisors on the standard linear series $g^1_2$ and for a point $b \in C$ let $C_b$ be the curve $\{(x+b) : x \in C \}$. It is proved that $\pi _1 ( C^{(2)} \setminus (L \cup C_b) )$ is the integer-valued Heisenberg group, which is the central extension of $\mathbb {Z}^{2g}$ by $\mathbb { Z}$ determined by the symplectic form on $H_1 (C , \mathbb {Z})$.