Maximal and Cohen–Macaulay projective monomial curves

Abstract

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.

The fiber cone of a monomial ideal in two variables

“…The product of two convex ideals is not necessarily convex. Let I = (x 12 , x 8 y, x 5 y 2 , x 2 y 3 , y 4 ) and J = (x 13 , x 7 y 2 , x 3 y 6 , xy 12 , y 20 ). Then G(IJ) = {x 25 , x 21 y, x 18 y 2 , x 15 y 3 , x 12 y 4 , x 9 y 5 , x 7 y 6 , x 5 y 9 , x 3 y 10 , xy 16 , y 24 }.…”

“…This is the first non-trivial case to be considered and it turns out that even this case is not so easy to deal with. There exist several nice and interesting criteria for a projective monomial curve in P n to be arithmetically Cohen-Macaulay, see for example the paper of Cavaliere and Niesi [4], the paper by Reid and Roberts [12] and that of Molinelli, Patil and Tamone [11]. A Gröbner basis criterion is given by Kamoi [10].…”

“…(13)Since ic + (a + b)(k − i) = deg u > deg u ′ = jc + (a + b)(k − j) and c > a + b, it follows that i > j. By comparing the exponents of y in case(12), we get a(k − i) > jc + b(k − j) which is not possible. Now we consider the case (13).…”

On the fiber cone of monomial ideals

“…Note that such rings may be viewed as the coordinate ring of a projective monomial curve. Projective monomial curves have been studied in many papers, see for example [1], [3], [9], [10] and [11]. For the symmetric ideal I, the fiber cone F (I) is the coordinate ring of a projective monomial curve only if a + b = c. Yet, also when a + b = c, it is shown in Corollary 3.4 that F (I) is Cohen-Macaulay, if and only if µ(J) ≤ 3.…”

The Cohen-Macaulay property of affine semigroup rings in dimension 2

“…The criterion of theorem 1.2 is straightforward to check for a single ring, but does not lend itself to analyzing classes of rings. Reid and Roberts [8] introduce a related notion of a maximal projective monomial curve in order to demonstrate a large class of CM curves. The special case of projective monomial curves continues to be studied.…”