Castelnuovo–Mumford regularity of projective monomial varieties of codimension two

Let K be an algebraically closed field, and let be a projective monomial variety of codimension two with n≥2, i.e., a projective toric variety of codimension two whose homogeneous coordinate ring is a simplicial semigroup ring. We give an explicit formula for the Castelnuovo–Mumford regularity of V, , in terms of the reduced Gröbner basis of I(V) with respect to the reverse lexicographic order. As a consequence, we show that , where degV is the degree of V, and characterize when equality holds.