Maximal and Cohen–Macaulay projective monomial curves

In this paper we continue the investigation of Cohen–Macaulay projective monomial curves begun in [Les Reid, Leslie G. Roberts, Non-Cohen–Macaulay projective monomial curves, J. Algebra 291 (2005) 171–186]. In the process we introduce maximal curves. Cohen–Macaulay curves are maximal, but not conversely. We show that the number of all curves of degree d that are Cohen–Macaulay grows exponentially, but not as fast as the total number of curves, and also that maximal curves of degree d with sufficiently large embedding dimension relative to d are Cohen–Macaulay.