On the regularity of configurations of \(\mathbb F_q\)-rational points in projective space (Q357879)

The authors consider the number \(s = s(n,q)\) defined as the smallest integer such that for any given \(n\) distinct \(\mathbb{F}_q\)-rational points \(P_1,\ldots,P_{n-1}\) in \(\mathbb{P}^{n-1}\) there exists a hypersurface \(H\) defined over \(\mathbb{F}_q\) such that \(P_1,\ldots,P_{n-1} \in H\) and \(P_n \notin H\). Equivalently, \(s(n,q)\) is the maximal Castelnuovo-Mumford regularity \(r_{\mathfrak{X}}\) of a subset \(\mathfrak{X}\) of some projective space \(\mathbb{P}^k(\mathbb{F}_q)\) with \(|\mathfrak{X}| = n\). Systems \(\mathfrak{X} \subseteq \mathbb{P}^k(\mathbb{F}_q)\) play a role, for example, in algebraic coding theory. Starting from the inequality \(s(n,q) \leq s(n+1,q) \leq s(n,q) + 1\) for all \(n \geq 1\) (cf. [\textit{M. Kreuzer} and \textit{R. Waldi}, Commun. Algebra 25, No. 9, 2919--2929 (1997; Zbl 0883.13014)]), one can extend \(s\) to a step function \(s(x,q)\), for real numbers \(x \geq 1\). The authors study jump discontinuities of such a function. For example, they prove that \(x=i+1\) is a discontinuity point for \(i = 1,\ldots,q-1\). When \(q \geq 3\), they prove the existence of discontinuities in some semiopen intervals, but they cannot locate their exact position. Also, they provide upper and lower bounds for the function \(s(n,q)\) for values of \(n\) in some ranges depending on \(q\). Finally, they show that \(s(n,q)\) can also be viewed as the index of stability \(s(Q_n)\) of some Cohen-Macaulay \(\mathbb{F}_q\)-algebra \(Q_n\). Thus, the results obtained for \(s(n,q)\) can be applied to \(S(Q_n)\) as well.