Normality of ideals beyond the standard graded setting: families from numerical semigroup rings

@misc{matsuoka2026normality,
  title = {{Normality of ideals beyond the standard graded setting: families from numerical semigroup rings}},
  author = {Naoyuki Matsuoka},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18087},
  eprint = {2609.18087},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18087},
}

TY  - UNPB
TI  - Normality of ideals beyond the standard graded setting: families from numerical semigroup rings
AU  - Naoyuki Matsuoka
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18087
UR  - https://arxiv.org/abs/2609.18087
AB  - Let k be an arbitrary field and let S=k[x,y,z] be the polynomial ring with three variables x,y,z. We study integrally closed (x,y,z)-primary ideals of S that are homogeneous for a positive weighted grading but need not be homogeneous for the standard grading. From a numerical semigroup H of embedding dimension three, we obtain such ideals as inverse images I_h=φ_H^−1(t^hk[t]∩ k[H]). If ℓ is the least degree of a defining relation of k[H], then I_ℓ is monomial, whereas I_ℓ+1 has a binomial generator in the cases considered here. We determine I_ℓ+1 for numerical semigroups of embedding dimension three and multiplicity three or four. The six-generated cases arising in multiplicity three and in the symmetric multiplicity-four case form two explicit families. For every ideal I in these families, we prove that its Rees algebra is a Cohen--Macaulay normal domain. In the non-symmetric multiplicity-four case, I_ℓ+1 is seven-generated; for H=⟨4,9,15⟩, we prove that its Rees algebra is again a Cohen--Macaulay normal domain.
ER  -