On the existence of components of the Noether-Lefschetz locus with given codimension

Abstract

It is known that the codimension c of a component of the Noether-Lefschetz locus NL(d) satisfies\(d - 3 \leqslant c \leqslant \left( \begin{gathered} d - 1 \hfill \\ 3 \hfill \\ \end{gathered} \right)\). We prove that ford≥47 and for every integer\(c \in \left[ {\tfrac{9}{2}d^{\tfrac{3}{2}} ,\left( {\begin{array}{*{20}c} {d - 1} \\ 3 \\ \end{array} } \right)} \right]\) there exists a component of NL(d) with codimension c. This is done with families of surfaces of degree d in ℙ3 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 everyd≥4, of components of maximum codimension\(\left( {\begin{array}{*{20}c} {d - 1} \\ 3 \\ \end{array} } \right)\), thus giving a new proof of the fact that these components are dense in the locus of smooth surfaces (density theorem).