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

Abstract

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