Linear systems cut out by quadric on projections of varieties (Q1066207)

Let X be an algebraic variety embedded in \({\mathbb{P}}^ n\) over an algebraically closed field. Let L(X)(k) denote the linear system cut out on X by the hypersurfaces of degree k. Then L(X)(k) is complete for k large enough. The problem arises: Given a particular embedding of X, how large can k be before L(X)(k) is complete? In the case where X is a curve, the problem was studied by Castelnuovo and more recently by \textit{L. Gruson, R. Lazarsfeld} and \textit{C. Peskine} [Invent. Math. 72, 491-506 (1983; Zbl 0565.14014)]. The paper under review investigates the case of a projection \(\pi_ p(X)\) from a point \(p\not\in X\), where X is a projective normal variety, i.e. L(X)(k) is complete for all \(k\geq 1\). The author proves that if X is a variety defined locally by the \(2\times 2\) minors of a matrix M with linear entries and if p does not lie on a secant line or a tangent space to X, then \(L(\pi_ p(X))(k)\) is complete for all \(k\geq 2\).