Period matrix and the trigonal construction (Q1580295)

The trigonal construction gives a bijection between the set of tetragonal curves \(X\) of genus \(g\) and the set of pairs of trigonal curves \(Y\) of genus \(g+1\) with an unramified double covering \(\pi:Z\to Y\). The aim of this paper is to show that there exists a relation between the Teichmüller space of tetragonal curves of genus \(g\) and the Teichmüller space of trigonal curves of genus \(g+1\). In particular the author shows that the period matrix of a trigonal curve depends analytically on the period matrix of the associated curve and its representation as a four degree cover of \(\mathbb{P}^1\).