Critical binomial ideals of Norhtcott 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 $x_i^{c_i}-x_1^{u_{i1}}\cdots x_n^{u_{1n}}\in k[x_1,\dots ,x_n]$ with $u_{ii}=0$, $i\in \{1,\dots ,n\}$. We prove that, the monomial curves in that family are set-theoretic complete intersection. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Fr"obenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.