Geodesic
On the extension of varieties defined by quadratic equations
Sbornik. Mathematics, Tome 63 (1989) no. 2, pp. 305-317 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, then the variety $V$ extends nontrivially at most $m$ steps, and this bound is attained for certain varieties. Bibliography: 16 titles. 
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S. M. L'vovskii. On the extension of varieties defined by quadratic equations. Sbornik. Mathematics, Tome 63 (1989) no. 2, pp. 305-317. http://geodesic.mathdoc.fr/item/SM_1989_63_2_a2/

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