Separable gonality of a Gorenstein curve.

*(English)*Zbl 0921.14014From the paper: An integral projective curve \(X\), which may have singular points, of arithmetic genus \(g\geq 2\) is said to be hyperelliptic if there is a finite morphism \(X\to\mathbb{P}^1\) of degree 2. However, there is a phenomenon which never happens in the case of smooth hyperelliptic curves; that is, the degree-two morphism may be inseparable. A hyperelliptic curve with this property is said to be of inseparable type. – On the other hand, any (singular or nonsingular) hyperelliptic curve is Gorenstein, i.e. the dualizing sheaf of the curve is invertible. The purpose of this short note is to give a characterization of hyperelliptic curves of inseparable type in the category of Gorenstein curves in terms of the separable gonality of a curve, which is defined to be the smallest possible degree of a finite separable morphism from the curve to the projective line.

Main result: Let \(X\) be a Gorenstein curve of genus \(g\geq 2\). Then \(k_s(X)\leq g+1\). Furthermore, equality occurs if and only if \(X\) is hyperelliptic curve of inseparable type.

Main result: Let \(X\) be a Gorenstein curve of genus \(g\geq 2\). Then \(k_s(X)\leq g+1\). Furthermore, equality occurs if and only if \(X\) is hyperelliptic curve of inseparable type.