Postulation and gonality for projective curves
A curve Z in \({\mathbb{P}}^ n\) enjoys the maximal rank property if the natural restriction map \(H^ 0({\mathbb{P}}^ n,{\mathcal O}_{{\mathbb{P}}^ n}(k))\to H^ 0(Z,{\mathcal O}_ Z(k))\) has maximal rank for every positive integer k. A smooth projective curve \(Y\subseteq {\mathbb{P}}^ n\) is said to be canonical if the hyperplane sections of Y are canonical. The author relates these properties to intrinsic properties of the curve, proving that for \(g>n\), \(n>3\) the general trigonal (bielliptic) canonical curve of genus g in \({\mathbb{P}}^ n\) has maximal rank. He also proves that for all \(n,d,g\in {\mathbb{N}},\) \(g>n>3,\) \(d>2g,\) a general embedding of degree d in \({\mathbb{P}}^ n\) of a general hyperelliptic curve of genus g has maximal rank. Both results are proven by degeneration arguments.
scientific article; zbMATH DE number 4132332
scientific article