A new proof of the explicit Noether-Lefschetz theorem (Q1121326)

Let \(Y:=\{algebraic\) surfaces of degree d in \({\mathbb{P}}_ 3\}\) and \(\Sigma_ d:=\{S\in Y| \quad S\quad is\quad\) smooth and Pic(S) is not generated by the hyperplane bundle\(\}\). Previously, the author proved the explicit Noether-Lefschetz theorem [J. Differ. Geom. 20, 279-289 (1984; Zbl 0559.14009)]: For \(d\geq 3\), every component of \(\Sigma_ d\) has codimension \(\geq d-3\) in Y. Here the author gives a new and short proof of this result as a consequence of some vanishing theorem for Koszul cohomology on \({\mathbb{P}}_ n\), the proof of which is given in this paper.