The strongly robust simplicial complex of monomial curves

Abstract

To every simple toric ideal $I_T$ one can associate the strongly robust simplicial complex ${\Delta }_T$, which determines the strongly robust property for all ideals that have $I_T$ as their bouquet ideal. We show that for the simple toric ideals of monomial curves in ${\mathbb{A}}^s$, the strongly robust simplicial complex ${\Delta }_T$ is either $\{\varnothing \}$ or contains exactly one 0-dimensional face. In the case of monomial curves in ${\mathbb{A}}^3$, the strongly robust simplicial complex ${\Delta }_T$ contains one 0-dimensional face if and only if the toric ideal $I_T$ is a complete intersection ideal with exactly two Betti degrees. Finally, we provide a construction to produce infinitely many strongly robust ideals with bouquet ideal the ideal of a monomial curve and show that they are all produced this way.

17 pages