A sharp Castelnuovo bound for smooth surfaces
Let \(X\subset {\mathbb{P}}^ r\) be a smooth irreducible complex projective surface of degree d, not contained in any hyperplane. The main result is the following: The hypersurfaces of degree \(d+2-r\) or greater cut out a complete linear series on X. - This result improves Pinkham's earlier result that hypersurfaces of degree \(\geq d-2\) (resp. \(\geq d-1)\) cut out a complete linear series when \(r\geq 5\) (resp. \(r=4).\) As corollary of the main result, it follows that the ideal sheaf \(I_{X/{\mathbb{P}}^ r}\) is \((d+3-r)\)-regular. In particular, the homogeneous ideal of X is generated by forms of degree at most \(d+3-r.\) To prove the main result, the author projects to \({\mathbb{P}}^ 3\) first, then applies a construction of Gruson and Peskine to reduce to checking that suitable cohomology groups of certain vector bundles on \({\mathbb{P}}^ 3\) vanish. Here, regularity considerations and an Eagon-Northcott complex provide the keys.
scientific article; zbMATH DE number 4053773