An analogue of Max Noether's theorem (Q1073859)

It is a general hope that basic properties of complete intersections remain true for dependency loci of sections of an ample vector bundle. Along this line the author proves the following generalisation of Max Noether's theorem [see \textit{P. Deligne}, Sém. Géom. algébr. 1967- 1968, SGA 7 II, Lect. Notes Math. 340, exposé XIX, 328-340 (1973; Zbl 0269.14019)]: if E is a (-2)-regular rank r vector bundle on \(P^ n\), (n\(\geq 3\), \(r\geq n-2)\), \(T\subset H^ 0(E)\) is a generic \((r+3-n)\)- subspace and X is the locus of those \(x\in P^ n\) where the map \(T\to E(x)\) has rank at most \(r+2-n\), then Pic(X) is either \({\mathbb{Z}}\), generated by a hyperplane section or \({\mathbb{Z}}\oplus {\mathbb{Z}}\) (according to whether \(r+3-n=1\) or \(r+3-n\geq 2)\) unless E is one of the following ''exceptional'' vector bundles \(E={\mathcal O}_{{\mathbb{P}}^ 3}(2)\), \({\mathcal O}_{{\mathbb{P}}^ 3}(3)\), \({\mathcal O}_{{\mathbb{P}}^ 4}(2)\oplus {\mathcal O}_{{\mathbb{P}}^ 4}(2)\) (these are precisely the exceptions which occur in the classical Max Noether theorem 1) As an application the author computes Pic(X) for certain generic weighted complete intersection surfaces, thus generalizing a result of \textit{J. Steenbrink} [Algebr. geom., Proc. internat. Conf., La Rábida/Spain 1981, Lect. Notes Math. 961, 302-313 (1982; Zbl 0507.14025)].