The fundamental group of the open symmetric product of a hyperelliptic curve (Q744900)

Let \(C\) be a smooth irreducible projective hyperelliptic curve of genus \(g \geq 2\) with symmetric square \(C^{(2)}\), \(L \subset C^{(2)}\) be the unique line, which is associated with the linear series \(g^1_2\) on \(C\) and \(C_b = \{ (x+b) \in C^{(2)} \mid x \in C \}\) for some \(b \in C\). The article under review shows that the fundamental group \(\pi _1( C^{(2)} \setminus (L \cup C_b)) = H(g)\) of the complement of \(L \cup C_b\) to \(C^{(2)}\) is the integer valued Heisenberg group \(H(g)\), which is the central extension \(0 \rightarrow {\mathbb Z} \rightarrow H(g) \rightarrow {\mathbb Z}^{2g} \rightarrow 0\) of \({\mathbb Z}^{2g}\) by \({\mathbb Z}\). The generator \(\delta\) of \({\mathbb Z}\) is the class of the meridian loop around \(L\) and \(\delta ^{g-1}\) is the class of the meridian loop around \(C_b\). In the case of \(g=2\), the result is an immediate consequence of [\textit{M. V. Nori}, Ann. Sci. Éc. Norm. Supér. (4) 16, 305--344 (1983; Zbl 0527.14016)], obtaining the fundamental group of the complement of the theta divisor on a general principally polarized abelian variety of dimension \(\geq 2\). The proof is by an induction on the genus \(g \geq 2\) of \(C\), using surgery and van Kampen's Theorem.