Liaison and Cohen-Macaulayness conditions (Q545668)

Let \(K\) be an algebraically closed field, \(\mathbb{P}_{K}^{n}\) the projective space of dimension \(n\) over \(K\), \(C\) a projective curve in \( \)\(\mathbb{P}_{K}^{n}\), i.e. \(C\) is an equidimensional projective subscheme of dimension 1, hence a locally Cohen- Macaulay projective subscheme of dimension 1. There are positive integers \(\beta_{1}\leq\dots \leq\beta_{n-1}\) and a complete intersection \(Y\) of type \((\beta_{1},\dots ,\beta_{n-1})\) such that \(Y\) contains \(C\). Let \(C'\) be the curve algebraically linked to \(C\) by \(Y\) and \(\rho C\) be the regularity of the Hilbert function of \(C\). By the work of \textit{F. Cioffi, M. G. Marinari} and \textit{L. Ramella} [Collect. Math. 60, No. 1, 89--100 (2009; Zbl 1188.14020)], it is proved that if \(C\) is a space curve such that \(\text{reg}(C)=\text{reg}(Y)-1>\rho C+1\) then \(C\) is a plane curve, hence arithmetically Cohen- Macaulay, here \(\text{reg}(*)\) denotes the Castelnuovo-Mumford regularity. The authors show \(\text{reg}(Y)-\text{reg}(C)\geq0\) when \(\text{reg}(C)>\rho C+1\) and give some necessary or sufficient conditions for \(C'\) being connected by means of the regularity of \(C\) (Theorem 3.1, Proposition 3.6) By the work of \textit{E. D. Davis} [in: Curves Semin. Queen's, Vol. VII, Queen's Pap. Pure Appl. Math. 85, Exposé F, 14 p. (1990; Zbl 0736.14011)], when \(\text{deg}(Y)-\text{deg}(C)\leq5\), a smooth integral space curve \(C\) with the same postulation as an arithmetically normal curve is necessarily arithmetically Cohen-Macaulay. Here \(\text{deg}(Y)-\text{deg}(C)\) measures the difference between \(C\) and the complete intersection \(Y\) in term of degree of curves. The authors prove the analogous result of Davis with more general hypotheses that \(C\) is any space curve with \(\text{reg}C>\rho C+1\) and \(\text{deg}(Y)-\text{deg}(C)\leq5\) (Theorem 4.2). They further show the result holds for non-degenerate equidimensional and locally Cohen-Macaulay subschemes \(X\) of codimension two with the same postulation of an arithmetically Cohen-Macaulay subscheme and such that \(\text{deg}(Y)-\text{deg}(C)\leq4\), where \(Y\) is a complete intersection containing \(X\) (Theorem 4.9). The authors also show that for curves \(C\) in \(\mathbb{P}_{K}^{n}\) where \(n\geq4\) is arithmetically Cohen-Macaulay if \(\text{reg}(C)>\rho C+1\) and \(\text{deg}(Y)-\text{deg}(C)\leq3\) (Proposition 4.6).