On the extension of varieties defined by quadratic equations

One says that a smooth projective variety $V\subset\mathbf P^n$ extends $m$ steps nontrivially if there exists a projective variety $W\subset\mathbf P^{n+m}$ such that $V=W\cap\mathbf P^n$, where $W$ is not a cone, is nonsingular along $V$, and is transversal to $\mathbf P^n$.
In the paper it is proved, in particular, that if $V$ is given by quadratic equations, $\operatorname{dim}V\geqslant2$ and $h^1(V,\mathscr T_V(-1))=m<n$, then the variety $V$ extends nontrivially at most $m$ steps, and this bound is attained for certain varieties.
Bibliography: 16 titles.