On the extension of varieties defined by quadratic equations

S. M. L'vovskii

Sbornik. Mathematics, Tome 63 (1989) no. 2, pp. 305-317

Résumé

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.