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 simplicialsemigroup ring. We give an explicit formula for the Castelnuovo–Mumford regularity of , , in terms of the reduced Gröbner basis of with respect to the reverse lexicographic order. As a consequence, we show that , where is the degree of , and characterize when equality holds.