Complete intersection subvarieties of general hypersurfaces
A complete intersection curve \(C\) contained in a general hypersurface \(X\) of degree \(d>n\geq 1\) in \(\mathbb{P}^{n+1}\) is called nontrivial, if the equation of \(X\) is not part of a regular sequence defining \(C\), i.e. if \(C\) is not cut out by \(n-1\) hypersurfaces on \(X\). The main result of this paper is a classification of the possible types (i.e. the degrees of the equations defining \(C\)) of nontrivial complete intersection curves contained in a general hypersurface \(X\) of degree \(d\geq 6\). Furthermore, the author shows that (1) there are no nontrivial complete intersection surfaces on \(X\), (2) for \(d>2n-1\), there are no nontrivial complete intersection curves on \(X\), (3) for \(d>\frac32n+ \frac12\), all nontrivial complete intersection curves on \(X\) are lines, and (4) for \(d>n+2\), all nontrivial complete intersection curves on \(X\) are lines or conics. The proof is divided into two parts. In the first part, results about the regularity and the components of Hilbert schemes of projective spaces are used to reduce the questions to a certain inequality for Hilbert functions of complete intersections. Then the author gives a rather long and tedious inductive proof of that inequality.
nontrivial complete intersection curves contained in a general hypersurface
scientific article; zbMATH DE number 1025786