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

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 $H_1 (C , mathbb{Z})$. end{abstract}