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.

Guest–Ohnita and Crawford have shown the path-connectedness of the space of harmonic maps from ${{S}^{2}}$ to $\text{C}{{P}^{n}}$ of a fixed degree and energy. It is well known that the $\partial$ transform is defined on this space. In this paper, we will show that the space is decomposed into mutually disjoint connected subspaces on which $\partial$ is homeomorphic.