A note on scrolls of smallest embedded codimension

Abstract

Let $M$ be a submanifold of ${\Bbb P}^N$ of dimension $n>2$. Suppose that $(M,{\Cal O}_M(1))\cong{\Bbb P}({\Cal E}),{\Cal O}(1))$ for some vector bundle ${\Cal E}$ on a surface $S$. Then $N\ge 2n-1$ by Barth-Lefschetz Theorem. We are interested in the case $N=2n-1$. In 1994 Ionescu and Toma gave a classification of the cases where $S$ is not of general type. Here we propose a conjecture concerning this remaining case, which is verified for $n\le 1100$ by a computer programm.