Complete intersections on general hypersurfaces

TL;DR

This paper investigates the conditions under which complete intersections of a given codimension can be contained in a general hypersurface in projective space, providing a complete characterization for certain codimension ranges.

Contribution

The authors give a complete answer to when complete intersections of codimension r can lie on a generic hypersurface in projective space for the case 2r ≤ n+2, based on degrees.

Findings

Complete characterization for when complete intersections lie on general hypersurfaces for 2r ≤ n+2.

Conditions depend on degrees of hypersurfaces and generators.

Provides a comprehensive answer to a classical geometric problem.

Abstract

We ask when certain complete intersections of codimension $r$ can lie on a generic hypersurface in $\PP^n$. We give a complete answer to this question when $2r \leq n+2$ in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection.