On postulation of curves in \({\mathbb{P}}^4\)
We say that a curve \(C\subset {\mathbb{P}}^ n\) has maximal rank if for every \(k\geq 0\) the restriction map \(H^ 0({\mathbb{P}}^ n,{\mathcal O}_{{\mathbb{P}}^ n}(k))\to H^ 0(C,{\mathcal O}_ C(k))\) is either injective or surjective. The maximal rank conjecture states that a general embedding of a general smooth curve of genus g has maximal rank. In this paper we prove this conjecture for non special embeddings in \({\mathbb{P}}^ 4\). The proof uses the method of \textit{R. Hartshorne} and \textit{A. Hirschowitz} [``Droites en position generale dans \({\mathbb{P}}^ n\)'' in Algebraic Geometry, Proc. int. Conf., La Rábida/Spain 1981, Lect. Notes Math. 961, 169-188 (1982; Zbl 0555.14011)] and \textit{A. Hirschowitz} [Acta Math. 146, 209-230 (1981; Zbl 0475.14027)] to show inductively the existence of a reducible curve with maximal rank.
non special embeddings in projective 4-space
scientific article; zbMATH DE number 3849400