Projective Closure of Affine Monomial Curves II

TL;DR

This paper introduces star gluing for numerical semigroups, showing it preserves Cohen-Macaulay and Gorenstein properties in projective closures, and explores Betti sequences and Cohen-Macaulay types through gluing techniques.

Contribution

It presents a new gluing method for numerical semigroups and analyzes its impact on algebraic properties of affine and projective monomial curves.

Findings

Star gluing preserves Cohen-Macaulay and Gorenstein properties.

Conditions on Gr"{o}bner bases ensure Betti sequences are preserved.

Constructed semigroups with equal Cohen-Macaulay type for affine and projective curves.

Abstract

In this paper our aim is twofold. First, we introduce the notion of star gluing of numerical semigroups and show that arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure are preserved under this gluing operation. We then give a condition on Gr\"{o}bner basis of the defining ideal of an affine monomial curve which ensures that the Betti sequence of the affine curve is the same as the Betti sequence of its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct some numerical semigroups, using a gluing technique, such that the Cohen-Macaulay type of corresponding affine curve and its projective closure are both $n$.