Postulation of canonical curves in \({\mathbb{P}}^ 3\)
A non-singular curve \(Y\subset {\mathbb{P}}^ 3\) of genus g is said to be ''canonical'' if its plane section is a canonical divisor, i.e. if the canonical bundle \(K_ Y\simeq {\mathcal O}_ Y(1)\). In other words, \(Y\subset {\mathbb{P}}^ 3\) is the projection of its canonical image in \({\mathbb{P}}^{g-1}\). Thus the canonical curves of given genus form a single irreducible family. Now recall that a curve \(Y\subset {\mathbb{P}}^ 3\) is said to have ''maximal rank'' if the restriction maps: \(H^ 0({\mathcal O}_{{\mathbb{P}}^ 3}(n))\to H^ 0({\mathcal O}_ Y(n))\) have maximal rank for all n. Given any family of curves in \({\mathbb{P}}^ 3\), it is a natural question whether its general member has maximal rank. While one usually expects the answer to be ''yes'', the case of canonical curves is somewhat intriguing in that in the first two nontrivial cases, namely those of genus 5 and 6, the answer turns out to be ''NO'' [proofs were given by \textit{L. Gruson} and \textit{C. Peskine} in Algebr. Geom., Proc., Tromsø Symp. 1977, Lect. Notes Math. 687, 31-59 (1978; Zbl 0412.14011) and by \textit{R. Hartshorne} and \textit{I. Sols} in J. Reine Angew. Math. 325, 145-152 (1981; Zbl 0448.14004)]. Recently the question was taken up by \textit{E. Ballico} [Ark. Mat. 22, 139-151 (1984; Zbl 0546.14018)], who showed that a general canonical curve of genus \(g=7, 8, 9\) or 11 has maximal rank. In this note we prove the following theorem: A general canonical curve of genus \(g\geq 7\) in \({\mathbb{P}}^ 3\) has maximal rank.
scientific article; zbMATH DE number 3920660