The regularity of a toric variety

We give a method for computing the degrees of the minimal syzygies of a toric variety by means of combinatorial techniques. Indeed, we complete the explicit description of the minimal free resolution of the associated semigroup algebra, using the simplicial representation of Koszul homology which appeared in A. Campillo and C. Marijuán (1991, Sém. Théor. Nombres Bordeaux3, 249–260). As an application, we obtain an algorithm for computing the Castelnuovo–Mumford regularity of a projective toric variety. This regularity is explicitly bounded by means of the semigroup generators which parametrize the variety