Projective varieties with many degenerate subvarieties

Abstract

We study the problem of classifying the irreducible projective varieties $X$ of dimension $n\ge 2$ in $\Bbb P^N$ which contain an algebraic family $\Cal F$ of dimension $h+1$ ($h<n$) of subvarieties $Y$ of dimension $n-h$, each one contained in a $\Bbb P^{N-h-1}$. We prove that one of the following happens: (i) there exists an integer $r$, $r<N-n$ such that $X$ is contained in a variety $V_r$ of dimension at most $N-r$ containing a family of dimension $h+1$ of subvarieties of dimension $N-h-r$, each one contained in a linear space of dimension $N-h-1$; (ii) The degree of $Y$ is bounded by a function of $h$ and $N-n$ (in this case $X$ is called of isolated type). Successively we study some special cases; in particular we give a complete classification of surfaces in $\Bbb P^5$ containing a family of dimension $2$ of curves of $\Bbb P^3$.