On the failure cycles for the quadratic normality of a projective variety (Q1919900)

Let \(L\) be a very ample line bundle on an algebraic surface \(X\), and let \(X\to |L|^*\) be the corresponding embedding. Assume that \(L\) is \textit{not} quadratically normal, i.e. that the map \(H^0(X,L)^{\otimes 2}\to H^0(X,L^2)\) is not surjective. Let \(f\) be the codimension of its image. Let \(G\) be the Grassmannian of codimension \(2\) linear subspaces of \(|L|^*\), and let \(B{\i}G\) be the set of linear subspaces whose intersection with \(X\) has positive dimension. If \(h^1(X,{\mathcal O}_X)<f+\text{codim}(B)-1\), there exists an element \(\Lambda\) of \(G-B\) such that the restriction of \(L\) to the \(0\)-dimensional scheme \(X\cap\Lambda\) is not quadratically normal. Analogous results were proved for curves by \textit{M. Green} and \textit{R. Lazarsfeld} [Invent. Math. 83, 73-90 (1986; Zbl 0594.14010)]. The proof uses an idea of Lazarsfeld: Start from a certain exact complex \(C^\bullet\) on \(X\times G\), calculate all terms \(E_1^{pq}=R^q(\text{pr}_2)_*(C^p)\) of the associated spectral sequence, and use the fact that this spectral sequence abuts to \(0\).