The authors solve the Hurwitz monodromy problem for degree 4 covers. The Hurwitz space \(H_{4,g}\) of all simply branched covers of \({\mathbb{P}}^{1}\) of degree 4 and genus \(g\) is an unramified cover of the space \(P_{2g+6}\) of \((2g+6)-\)tuples of distinct points in \({\mathbb{P}}^{1}.\) They determine the monodromy of \(\pi_{1}(P_{2g+6})\) on the points of the fiber. This turns out to be the same problem as the action of \(\pi_{1}(P_{2g+6})\) on a certain local system of \(({\mathbb Z}/2)-\)vector spaces. Authors generalized their result by treating the analogous local system with \(({\mathbb Z}/N)\) coefficients, \(3 \nmid N,\) in place of \(({\mathbb Z}/2).\) This in turn allow to answer a question of Ellenberg concerning families of Galois covers of \({\mathbb{P}}^{1}\) with deck group \(({\mathbb Z}/N)^{2}: S_{3}.\)
Theorem 1. Let \(g > 1.\) Then monodromy group \(G_{2}\) of \(H_{4,g} \rightarrow P_{2g+6}\) fits into the split exact sequence \[ 1 \rightarrow \prod_{\Omega} Sp(2g+2,{\mathbb Z}/2) \rightarrow G_{2} \rightarrow PSp(2g+4,{\mathbb Z}/3) \rightarrow 1, \] where \(\Omega = {\mathbb{P}}^{2g+3}({\mathbb Z}/3))\) and \(PSp(2g+4,{\mathbb Z}/3)\) permutes the factors of the product in the obvious way.
Theorem 2. Suppose \(3 \nmid N\) and \(g \geq 0\) (\(g > 1\) if \(N\) is even). Then the monodromy group \(G_{N}\) of \(V_{N}\) fits into an exact sequence \[ 1 \rightarrow \prod_{\Omega} Sp(2g+2,{\mathbb Z}/N) \rightarrow G_{N} \rightarrow PSp(2g+4,{\mathbb Z}/3) \rightarrow 1, \] where \(\Omega\) and the action of \(PSp(2g+4,{\mathbb Z}/3)\) are as in Theorem 1.