What is the condition on a set of 14 points for them to uniquely determine a quadric in $\mathbb{P}^4$? Is being in linear general position enough to guarantee uniqueness? If not, what is the alternative condition? In what precise sense of "general position" must the points be in general position?
What if I know that the quadric is non-singular? If I pick 14 points from the intersection of two non-singular quadrics in $\mathbb{P}^4$, can they be in general position?
Thank you.