On projections of ruled and Veronese surfaces
A closed subscheme \(X\subset {\mathbb{P}}^ n\) is said to be of maximal rank if for every \(h\geq 1\) the restriction homomorphism \(H^ 0({\mathcal O}_{{\mathbb{P}}^ n}(h))\to H^ 0({\mathcal O}_ X(h))\) is injective or surjective. In many previous papers the authors gave contributions to answer the following question (Hartshorne projection conjecture). Let \(X\subset {\mathbb{P}}^ n\) be a generic projection of a projectively normal curve of \({\mathbb{P}}^ N\), \(3\leq n\leq N\); is X of maximal rank? In particular, after the authors' previous paper [Boll. Unione Mat. Ital., VI. Ser., D, Algebra Geom. 5, No.1, 103-133 (1986; Zbl 0631.14028)] the problem was still open for canonical curves with non-general moduli. The authors prove the following result in the complex case. Let C be a (linearly normal) trigonal canonical curve of genus g in \({\mathbb{P}}^{g- 1}\). Then the general projection of C into \({\mathbb{P}}^ k\) is of maximal rank for every k satisfying \(g\geq k+2\geq 7\) and \(\binom{k+3}{3}\geq 6g-8\). - This is a corollary of the following theorem. Let S be a smooth ruled surface of degree d in \({\mathbb{P}}^{d+1}\). Then the general projection of S into \({\mathbb{P}}^ k\) is of maximal rank for every k satisfying \(d\geq k\geq 5\) and \(\binom{k+3}{3}\geq 6d+4\). By the same method, consisting in degenerating the surface to a suitable union of planes, the authors also prove a similar result for the d-uple Veronese embedding of \({\mathbb{P}}^ 2\).
scientific article; zbMATH DE number 4105121
scientific article