On postulation of curves: Embeddings by complete linear series
A closed subscheme C in \({\mathbb{P}}^ n\) is said to be of maximal rank if for every \(k\geq 0\) the restriction map \(r_ C(k):H^ 0({\mathbb{P}}^ n,{\mathcal O}_{{\mathbb{P}}^ n}(k))\to H^ 0(C,{\mathcal O}_ C(k))\) has maximal rank. Here we prove the following theorem. Let d,g be positive integers satisfying \(d\geq g+1+\sqrt{g};\) let X be a general smooth, complete curve of genus g; then the general embedding of X into \({\mathbb{P}}^{d-g}\) by a non-special complete linear system has maximal rank. First we prove the theorem for a suitable reducible curve Y in \({\mathbb{P}}^{d-g}\) and then we apply deformation theory [\textit{E. Sernesi}, ''On the existence of certain families of curves'', Invent. Math. 75, 25-57 (1984)] to show that Y can be smoothed.
scientific article; zbMATH DE number 3851255