Projective toric varieties of codimension 2 with maximal Castelnuovo–Mumford regularity

Abstract

The Eisenbud–Goto conjecture states that reg X ≤ deg X − codim X + 1 for a nondegenerate irreducible projective variety X over an algebraically closed field. While this conjecture is known to be false in general, it has been proven in several special cases, including when X is a projective toric variety of codimension 2. We classify the projective toric varieties of codimension 2 having maximal regularity, that is, for which equality holds in the Eisenbud–Goto bound. We also give combinatorial characterizations of the arithmetically Cohen–Macaulay toric varieties of maximal regularity in characteristic 0.