Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

Let a,b and n>1 be three positive integers such that a and ∑j=0n−1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of N generated by {∑j=0n−1bj}∪{∑j=0n−1bj+a∑j=0i−2bj∣i=2,…,n} is determinantal. Moreover, we prove that for n>3, the ideal I has a unique minimal system of generators if and only if a<b−1.

Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

Let $a, b$ and $n &gt; 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-1} b^j\} \cup \{\sum_{j=0}^{n-1} b^j + a\, \sum_{j=0}^{i-2} b^j \mid i = 2, \ldots, n\}$ is determinantal. Moreover, we prove that for $n &gt; 3$, the ideal $I$ has a unique minimal system of generators if and only if $a &lt; b-1$.

Nearly Gorenstein vs Almost Gorenstein Affine Monomial Curves

We extend some results on almost Gorenstein affine monomial curves to the nearly Gorenstein case. In particular, we prove that the Cohen–Macaulay type of a nearly Gorenstein monomial curve in $${\mathbb {A}}^4$$ A 4 is at most 3, answering a question of Stamate in this particular case. Moreover, we prove that, if $${\mathcal {C}}$$ C is a nearly Gorenstein affine monomial curve that is not Gorenstein and $$n_1, \dots , n_{\nu }$$ n 1 , ⋯ , n ν are the minimal generators of the associated numerical semigroup, the elements of $$\{n_1, \dots , \widehat{n_i}, \dots , n_{\nu }\}$$ { n 1 , ⋯ , n i ^ , ⋯ , n ν } are relatively coprime for every i.

CRITICAL BINOMIAL IDEALS OF NORTHCOTT TYPE

In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.