Weierstra\ss semigroups and nodal curves of type p,q

TL;DR

This paper investigates plane curves of type p,q with only nodal singularities, constructing explicit examples and determining their Weierstraß semigroups, thereby expanding understanding of their algebraic and geometric properties.

Contribution

It provides explicit constructions of plane p,q-curves with prescribed nodes and determines their Weierstraß semigroups, revealing many new semigroups of this type.

Findings

Explicit examples of plane p,q-curves with given nodes

Determination of Weierstraß semigroups for these curves

Discovery of many new semigroups of this type

Abstract

We study plane curves of type p,q having only nodes as singularities. Every Weierstra\ss semigroup is the Weierstra\ss semigroup of such a curve at its place at infinity for properly chosen p,q. We construct plane curves of type p,q with explicitly given nodes and determine their Weierstra\ss semigroups. Many such semigroups are found.