The extension of varieties defined by quadratic equations

We say that a smooth projective variety $V\subset\mathbb{P}^n$ extends $m$ steps nontrivially if there exists a projective variety $W\subset\mathbb{P}^{n+m}$ such that $V=W\cap\mathbb{P}^n$, where $W$ is not a cone, is nonsingular along $V$, and is transversal to $\mathbb{P}^n$. Following S. Lvovski's work, we describe some classes of varieties which have nontrivial extension and give some estimates for the number of steps $m$ as well.