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On the existence of components of the Noether-Lefschetz locus with given codimension. (English) Zbl 0769.14018

Let \(\mathbb{P}^ 3\) be the projective space of dimension 3 over the complex numbers. For \(d\geq 4\) we denote by \(\mathbb{P}^ N=\mathbb{P}^{{d+3\choose 2}- 1}\) the projective space whose points correspond to surfaces of degree \(d\) in \(\mathbb{P}^ 3\) and by \(S(d)\subseteq\mathbb{P}^ N\) the open subset consisting of points corresponding to smooth surfaces. By the Noether- Lefschetz theorem, there is a countable set of proper irreducible closed subvarieties of \(S(d)\) such that for every point \(s\) outside the union of these subvarieties, the corresponding surface \(S\) has \(\text{Pic} S\cong\mathbb{Z}\) generated by \({\mathcal O}_ S(1)\). The union of the mentioned subvarieties, i.e., the locus of surfaces with Picard group different from \(\mathbb{Z}\), is called the Noether-Lefschetz locus and denoted NL\((d)\).
It is known that the codimension \(c\) of a component of the Noether- Lefschetz locus NL\((d)\) satisfies \(d-3\leq c\leq{d-1\choose 3}\). We prove that for \(d\geq 47\) and for every integer \(c\in\bigl[{9\over 2}d^{{3\over 2}},{d-1\choose 3}\bigr]\) there exists a component of NL\((d)\) with codimension \(c\). This is done with families of surfaces of degree \(d\) in \(\mathbb{P}^ 3\) containing a curve lying on a cubic or on a quartic surface or a curve with general moduli. Moreover we produce an explicit example, for every \(d\geq 4\), of components of maximum codimension \(d-1\choose 3\), thus giving a new proof of the fact that these components are dense in the locus of smooth surfaces (density theorem).

MSC:

14M07 Low codimension problems in algebraic geometry
14N05 Projective techniques in algebraic geometry
14J25 Special surfaces
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