On pseudo symmetric monomial curves

Abstract

Abstract. We study monomial curves, toric ideals and monomial algebras associated to 4-generated pseudo symmetric numerical semigroups. Namely, we determine indispensable binomials of these toric ideals, give a characterization for these monomial algebras to have strongly indispensable minimal graded free resolutions. We also characterize when the tangent cones of these monomial curves at the origin are Cohen-Macaulay.

Gluing semigroups and strongly indispensable free resolutions

“…But this gives a binomial in I S of S-degree less than d t = α pr m r + α qs m s , which is a contradiction. Now, There is a classification of 4-generated pseudo symmetric semigroups having SIFRE in S ¸ahin and S ¸ahin [20]. The next result reveals that none of the extensions of these semigroups have a SIFRE.…”

“…It is known that generic lattice ideals have SIFRE ([17,Theorem 4.2], [3,Theorem 4.9]). Numerical semigroups having SIFREs have been classified for some small embedding dimensions in [2,20].…”

Betti numbers for numerical semigroup rings

“…The defining equations or the Betti numbers of the tangent cone are not known for this class of semigroups. S ¸ahin and S ¸ahin ( [72]) describe the Cohen-Macaulay property of gr m K[H] in terms of Komeda's parametrization. 9.4.…”

4-generated pseudo symmetric monomial curves with not Cohen–Macaulay tangent cones

“…Since symmetric and pseudo-symmetric semigroups are maximal with respect to inclusion with fixed genus, a natural question is whether Rossi's conjecture is even true for local rings corresponding to 4-generated pseudo-symmetric semigroups. In [15], we showed that if α 2 ≤ α 21 + 1 , then the tangent cone is Cohen-Macaulay, and hence, the Hilbert function of the local ring is nondecreasing. In this paper, we focus on the open case of 4-generated pseudo symmetric monomial curves with α 2 > α 21 + 1 .…”