Quadratic numerical semigroups and the Koszul property

Abstract

Let $H$ be a numerical semigroup. We give effective bounds for the multiplicity $e(H)$ when the associated graded ring $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadrics. We classify Koszul complete intersection semigroups in terms of gluings. Furthermore, for several classes of numerical semigroups considered in the literature (arithmetic, compound, special almost complete intersections, $3$-semigroups, symmetric or pseudo-symmetric $4$-semigroups) we classify those which are Koszul.Comment: 24 pages, v3: bibliography updated and minor edits. v2: minor editing changes, added Remark 1.14 and reference to previous work of Rossi and Valla. To appear in the Kyoto Journal of Mathematic

On the Cohen-Macaulay Property for Quadratic Tangent Cones

“…Herzog ([18]) and A. Garcia ([17]), and also our results from [19], we show in Proposition 5 that if emb dim(H) < 5, then gr m K[H] is Cohen-Macaulay. It requires a bit more work to prove in Theorem 8 that if emb dim(H) = 5, then gr m K[H] is not Cohen-Macaulay precisely when H is generated as 8, 4u , 4u + 2u , 4u + 2u + u , 6u + 7u + 4u − 8 , or 8, 4u , 4u + 2u , 4u + 2u + 3u , 6u + 9u + 4u − 8 , with u, u , u positive integers and u > 1 is odd.…”

“…For further reference we first recall from our joint work with J. Herzog [19] some restrictions that we found on the multiplicity of a quadratic numerical semigroup. [19] for related properties.…”

“…Let L be a numerical semigroup, an odd integer in L and H = 2L, . By [19,Definition 2.2], the semigroup H is called a quadratic gluing of L. It is proved in [19,Corollary 2.7] that L and H are quadratic, Koszul, respectively G-quadratic, at the same time. It is also known by Delorme's work [12] that if L is a complete intersection (CI), then so is H. We refer to Section 2 in [19] for more details about the CI property for quadratic numerical semigroups.…”

“…In recent work ( [19]), J. Herzog and the author gave effective bounds for the multiplicity of a numerical semigroup H such that gr m K[H] is defined by quadrics. The motivation for the current paper came from the puzzling observation that all such numerical semigroups that we had obtained by blind computer search have the property that gr m K[H] is Cohen-Macaulay.…”

On the multiplicity of tangent cones of monomial curves

“…That is, if gr m (R) is a Cohen-Macaulay almost complete intersection, it may happen that e(gr m (R)) < d c − (d − 1)d c−2 . This is the case for instance for the quadratic semigroup Λ = 11,13,14,15,19 , see also [20,Remark 1.10].…”