Covers of algebraic varieties III. The discriminant of a cover of degree 4 and the trigonal construction

For each Gorenstein cover $\varrho \colon X\to Y$ of degree $4$ we define a scheme $\Delta (X)$ and a generically finite map $\Delta (\varrho )\colon \Delta (X)\to Y$ of degree $3$ called the discriminant of $\varrho$. Using this construction we deal with smooth degree $4$ covers $\varrho \colon X\to {{\mathbb P}^{n}_{\mathbb{C}}}$ with $n\ge 5$. Moreover we also generalize the trigonal construction of S. Recillas.