On self-linkage of extremal curves (Q6110320)

A closed subscheme \(C \subset \mathbb P^n_k\) of codimension two is self-linked if there is a complete intersection \(X\) of two hypersurfaces that algebraically link \(C\) to itself. \textit{Rao} characterized when an arithmetically Cohen-Macaulay (ACM) subscheme \(C\) is self-linked in terms of its free resolution [\textit{A. P. Rao}, Duke Math. J. 49, 251--273 (1982; Zbl 0499.14014)] assuming char \(k \neq 2\). This result was extended by \textit{Casnati and Catanese} to the non-ACM case [\textit{G. Casnati} and \textit{F. Catanese}, Asian J. Math. 6, No. 4, 731--742 (2002; Zbl 1065.14063)]. Much less is known when char \(k = 2\) beyond the the result of \textit{Migliore}, which says that a double line \(C \subset \mathbb P^3_k\) is self-linked if and only if char \(k=2\) [\textit{J. Migliore}, Trans. Am. Math. Soc. 294, 177--185 (1986; Zbl 0596.14019)]. Here the author extends Migliore's result, proving that a connected extremal curve is self-linked if and only if char \(k=2\). The connectedness condition is necessary because complete intersections are connected. An extremal curve is one whose Rao function achieves the upper bounds proved by \textit{M. Martin-Deschamps} and \textit{D. Perrin} [C. R. Acad. Sci., Paris, Sér. I 317, No. 12, 1159--1162 (1993; Zbl 0796.14029)]; they can be characterized as the degree \(d\) curves that contain a degree \(d-1\) subcurve [\textit{S. Nollet}, Manuscr. Math. 94, No. 3, 303--317 (1997; Zbl 0918.14014)]. Extremal curves form an irreducible component of the Hilbert scheme which is typically generically non-reduced [\textit{M. Martin-Deschamps} and \textit{D. Perrin}, Ann. Sci. Éc. Norm. Supér. (4) 29, No. 6, 757--785 (1996; Zbl 0892.14005)]. It would be interesting to better understand the codimension two subschemes of \(\mathbb P^n\) that are self-linked in characteristic two.