Closed families of smooth space curves
An algebraic family \(\{Y_{\lambda}:\) \(\Lambda\in \Lambda\}\) of subvarieties \(Y_{\lambda}\) of a variety X parametrized by a variety \(\Lambda\) is called effectively parametrized, if for all \(\lambda_ 0\in \Lambda\) the set \(\{\lambda \in \Lambda:\quad Y_{\lambda} = Y_{\lambda_ 0}\}\) is finite, it is called closed if \(\Lambda\) is complete. It is the aim of the paper to give upper bounds for the dimension of \(\Lambda\) for such a family. The first result is that if all the varieties \(Y_{\lambda}\) are smooth, then \(\dim (\Lambda)\leq (\dim Y_{\lambda}+1)co\dim (Y_{\lambda},X).\) This estimate is best possible in general as the example of linear subspaces in \({\mathbb{P}}_ n\) shows. The method of proof is to analyze the infinitesimal situation of subvarieties all passing through a fixed point P of X and to study families of curvilinear schemes supported at P. In the case of curves in \({\mathbb{P}}_ n\) the above estimate can be improved to give the following main result of the paper: If the curves \(Y_{\lambda}\) of the family are nondegenerate and nonsingular in \({\mathbb{P}}_ n\) then \(\dim(\Lambda)\leq n- 1\). In fact the result is slightly more general. Examples show that the estimate is almost best possible.
scientific article; zbMATH DE number 3950710