Castelnuovo-Mumford Regularity of Projective Monomial Curves via Sumsets

Let A = {a(0), ... , (an-1)} be a finite set of n = 4 non-negative relatively prime integers, such that 0 = a(0) < a(1) < . . . < a(n-1) = d. The s-fold sumset of A is the set sA of integers that contains all the sums of s elements in A. On the other hand, given an infinite field k, one can associate with A the projective monomial curve CA parametrized by A, C-A = {(v(d) : u(a)1v(d-a)1 : . . .: ua(n-2)v(d-a)n-2 : u(d)) | (u : v) ? P-k(1)} ? Pn-1 (k) . The exponents in the previous parametrization of CA define a homogeneous semigroup S ? N-2. We provide several results relating the Castelnuovo-Mumford regularity of C-A to the behavior of the sumsets of A and to the combinatorics of the semigroup S that reveal a new interplay between commutative algebra and additive number theory. Mathematics Subject Classification. Primary 13D02; Secondary 13D45, 11B13, 14H45, 20M50.