Smoothable zero dimensional schemes and special projections of algebraic varieties

We study the degrees of generators of the ideal of a projected Veronese variety $v_2({\mathbb{P}}^3)\subset {\mathbb{P}}^9$ to ${\mathbb{P}}^6$ depending on the center of projection. This is related to the geometry of zero dimensional schemes of length $8$ in ${\mathbb{A}}^4$, Cremona transforms of ${\mathbb{P}}^6$, and the geometry of Tonoli Calabi-Yau threefolds of degree $17$ in ${\mathbb{P}}^6$.