Codimension two nonsingular subvarieties of quadrics: scrolls and   classification in degree $d\leq 10$
  Let $X$ be a codimension two nonsingular subvariety of a nonsingular quadric $\Q{n}$ of dimension $n\geq 5$. We classify such subvarieties when they are scrolls. We also classify them when the degree $d\leq 10$. Both results were known when $n=4$.   Keywords: Classification; liaison; low codimension; low degree; quadric; scroll; vector bundle 
Mark Andrea A. de Cataldo (Washington U. in St. Louis)