A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics

We prove that there are only finitely many families of codimension two nonsingular subvarieties of quadrics $\Q{n}$ which are not of general type, for $n=5$ and $n\geq 7$. We prove a similar statement also for the case of higher codimension. The case $n=6$ has been recently settled by Fania-Ottaviani. Keywords: Codimension two, Grassmannians, Lifting, Low codimension, Not of General Type, Polynomial Bound, Quadrics