Arithmetically Gorenstein curves on arithmetically Cohen-Macaulay surfaces (Q1396461)

The aim of this paper is to prove the following theorem: Let \(\Sigma\subset\mathbb{P}^N\) be a smooth connected arithmetically Cohen-Macaulay surface. Then there are at most finitely many complete linear systems on \(\Sigma\), not of the type \(|kH-K|\) \((H\) hyperplane section and \(K\) canonical divisor on \(\Sigma)\), containing integral arithmetically Gorenstein curves. This result is proved in \(\S 1\) by two fundamental steps. The first one (lemma 1.4) translates the existence of subcanonical curves on a smooth connected arithmetically Cohen-Macaulay surface in terms of certain divisors called \textit{lone} and \textit{minimal}. This is similar to results obtained in case of subcanonical surfaces in \(\mathbb{P}^N\) [see \textit{G. Casnati}, \textit{A. Dolcetti} and \textit{P. Ellia}, Rev. Roum. Math. Pures Appl. 40, 289-300 (1995; Zbl 0876.14021), \textit{P. Ellia} and \textit{R. Hartshorne} in: Commutative algebra and algebraic geometry, Proc. Ferrara meeting Honor Mario Fiorentini, Lect. Notes Pure Appl. Math. 206, 53-79 (1999; Zbl 0958.14018)]. The second step (lemma 1.5) consists in proving a finiteness result (up to linear equivalence) for some particular minimal divisors on a regular surface: This is closely related for ideas and methods to a result proved in case of arithmetically Cohen-Macaulay surfaces by \textit{J. O. Kleppe}, \textit{J. C. Migliore}, \textit{R. Miro-Roig}, \textit{U. Nagel} and \textit{C. Peterson} [``Gorenstein liaison, complete intersection liaison invariants and unobstructedness'', Mem. Am. Math. Soc. 732 (2001; Zbl 1006.14018)]. In \(\S 2\) the cases of rational and \(K3\) arithmetically Gorenstein surfaces are considered as examples of explicit description of the possible arithmetically Gorenstein curves on an arithmetically Cohen-Macaulay surface (see 2.6 and 2.10). In both cases the goal is fulfilled by studying the lone divisors on them.