Let $o\left(n\right)$ be the greatest odd integer less than or equal to $n$. In this paper we provide explicit formulae to compute $\mathbb{N}$-graded Betti numbers of the circulant graphs $C_{2n}(1,2,3,5,...,o\left(n\right))$. We do this by showing that this graph is the product (or join) of the cycle $C_n$ by itself, and computing Betti numbers of $C_n*C_n$. We also discuss whether such a graph (more generally, $G*H$) is well-covered, Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum, or $S_2$.

Nous montrons dans cet article des bornes pour la régularité de Castelnuovo-Mumford d’un schéma admettant des singularités, en fonction des degrés des équations définissant le schéma, de sa dimension et de la dimension de son lieu singulier. Dans le cas où les singularités sont isolées, nous améliorons la borne fournie par Chardin et Ulrich et dans le cas général, nous établissons une borne doublement exponentielle en la dimension du lieu singulier.

The Castelnuovo-Mumford regularity is one of the most important invariants in studying the minimal free resolution of the defining ideals of the projective varieties. There are some bounds on the Castelnuovo-Mumford regularity of the projective variety in terms of the other basic invariants such as dimension, codimension and degree. This paper studies a bound on the regularity conjectured by Hoa, and shows this bound and extremal examples in the case of divisors on rational normal scrolls.