The strongly robust simplicial complex of monomial curves

Abstract

AbstractTo every simple toric ideal $$I_T$$ I T one can associate the strongly robust simplicial complex $$\Delta _T$$ Δ T , which determines the strongly robust property for all ideals that have $$I_T$$ I T as their bouquet ideal. We show that for the simple toric ideals of monomial curves in $$\mathbb {A}^{s}$$ A s , the strongly robust simplicial complex $$\Delta _T$$ Δ T is either $$\{\emptyset \}$$ { ∅ } or contains exactly one 0-dimensional face. In the case of monomial curves in $$\mathbb {A}^{3}$$ A 3 , the strongly robust simplicial complex $$\Delta _T$$ Δ T contains one 0-dimensional face if and only if the toric ideal $$I_T$$ 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.