This paper is motivated by a conjecture that surfaces of general type with \(K^2<4\chi\) admit pencils of curves of relatively low genus. This in turn is related to the conjecture that if a linearly general set \(\Sigma\) of \(d\ge 2n+2p+1\) points in \(\mathbb{P}^n\), uniform for quadrics, imposes at most \(2n+p\) conditions on quadrics \((p\le n-2)\) then the intersection of these quadrics contains a curve. It is proved that these conditions imply that \(\Sigma\) is contained in a \(p\)-dimensional rational normal scroll, and that if \(p\le 2\) the conjecture holds.
The proofs employ traditional techniques with a strong geometric flavour: indeed the author attributes the cases \(p=1, 2\) to Castelnuovo and Fano. A final paragraph, again studying surfaces by their sections, considers directly relations with quadrics to low rank and associated bundles: This leads to similar numerical conditions.
[For the entire collection see Zbl 0683.00009.]