Singular hyperelliptic curves

Abstract

Singular curves with a morphism of degree two onto a projective line should be classified into two types according as the equipped morphism is separable or not; we call a curve with a separable one a hyperelliptic curve of separable type, and the other a hyperelliptic curve of inseparable type. We give concrete expressions of a hyperelliptic curve of separable type by means of its global “equation” and a hyperelliptic curve of inseparable type by means of its local rings. Furthermore, we discuss about Weierstrass points of a hyperelliptic curve of inseparable type.