Codimension two nonsingular subvarieties of quadrics: scrolls and classification in degree d≤10

Abstract

Let $X$ be a codimension two nonsingular subvariety of a nonsingular quadric $L^2$ of dimension $n\ge 5$. We classify such subvarieties when they are scrolls. We also classify them when the degree $d\le 10$. Both results were known when $n=4$.