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

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 $I_h={\phi }_H^{-1}(t^{h}k[t]\cap k[H])$. If $\mathrm{\ell }$ is the least degree of a defining relation of $k[H]$, then $I_{\mathrm{\ell }}$ is monomial, whereas $I_{\mathrm{\ell }+1}$ has a binomial generator in the cases considered here. We determine $I_{\mathrm{\ell }+1}$ 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, $I_{\mathrm{\ell }+1}$ is seven-generated; for $H=⟨4,9,15⟩$, we prove that its Rees algebra is again a Cohen–Macaulay normal domain.