What question did this study set out to answer?

The aim is to explore numerical semigroups of toric type derived from double coverings of curves.

April 1, 2026Open Access

The numerical semigroup of toric type at a ramification point on a double covering of a curve

Key Points

The aim is to explore numerical semigroups of toric type derived from double coverings of curves.
Investigate numerical semigroups H generated by three elements.
Analyze the relationship between ramification points and Weierstrass semigroups.
Examine embeddings of monomial curves into affine spaces derived from toric varieties.
Identified conditions under which the Weierstrass semigroup H(P) is of toric type.
Demonstrated that specific embeddings relate the numerical semigroup H to H(π(P)).
Provided examples illustrating the characteristics of these semigroups in various settings.

Abstract

Let H be a numerical semigroup of toric type, i.e., a minimal embedding of its monomial curve into the affine space is derived from an embedding of some affine toric variety by substituting monomials. In this paper we mainly treat numerical semigroups H generated by three elements. In some cases we investigate whether there is a double covering π : C → C of a curve over an algebraically closed field k of characteristic O with a ramification point P whose Weiertrass semigroup H(P) is of toric type with H(π(P) =H.

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