Components of maximal dimension in the Noether-Lefschetz locus
Let Y be the Hilbert scheme of surfaces of degree d in \({\mathbb{P}}^ 3\) and let \(\Sigma_ d\) be the subscheme of Y parametrizing smooth surfaces S such that Pic(S) is not generated by \({\mathcal O}_ S(1)\). For any irreducible component \(\Sigma\) of \(\Sigma_ d\), it is known that \(d-3\leq co\dim_ Y\Sigma \leq h^{2,0}(S)=\left( \begin{matrix} d-1\\ 3\end{matrix} \right)\). The main result of this paper states that, if \(co\dim (\Sigma)=d-3\geq 2\), then \(\Sigma\) is the family of surfaces containing a line. The proof is based on a result of the author [J. Differ. Geom. 27, No.1, 155-159 (1988; see the review after the following one)] and a technique of \textit{G. Gotzmann} [Math. Z. 158, 61-70 (1978; Zbl 0352.13009)].
scientific article; zbMATH DE number 4103221