# Critical binomial ideals of Norhtcott type

## TL;DR

This paper investigates a family of binomial ideals defining monomial curves in affine space, proving they are set-theoretic complete intersections and computing key invariants for irreducible cases.

## Contribution

It introduces a new class of binomial ideals related to monomial curves, proves their set-theoretic complete intersection property, and provides a method to generate high-height semigroup ideals.

## Key findings

- Monomial curves in the family are set-theoretic complete intersections.
- Computed invariants like genus, type, and Frobenius number for irreducible cases.
- Method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

## Abstract

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 $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}} \in k[x_1, \ldots, x_n]$ with $u_{ii} = 0$, $i\in \{ 1, \ldots, n\}$. We prove that, the monomial curves in that family are set-theoretic complete 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.