A quartic surface in $\mathbb{P}^3$ is said to be a "symmetroid" if its equation is obtained as the determinant of a 4x4 symmetric matrix of linear forms. It is well known that the general symmetroid has 10 nodes and the family of such surfaces seem to have the same dimension as the moduli space of 10 points in $\mathbb{P}^3$.


QUESTION: is any general set of 10 points in $\mathbb{P}^3$ the singular locus of a symmetroid? Or better, how can one detect whether such a point set comes from a symmetroid?



Quartic symmetroids and 10-points sets
The answer is *no*. 


In fact, let $\Gamma \subset \mathbb{P}^3$ be the set of $10$ nodes of a general quartic symmetroid. Then the Gale transform $\Gamma' \subset \mathbb{P}^5$ of $\Gamma$ must be a zero dimensional scheme of length $10$ given by the simple intersection points of two Veronese surfaces, and this condition is not satisfied by a collection of $10$ general points.


Another characterization is the following: the set $\Gamma$ arises as a linear section of the secant variety $\textrm{Sec}(C\_6)$, where $C\_6 \subset \mathbb{P}^6$ is a rational normal curve. Note that $\textrm{Sec}(C\_6)$ has precisely degree $10$, since $10$ is the number of nodes of a general projection of $C\_6$ to $\mathbb{P}^2$.


For (many) more details, see


**(1)** D. Eisenbud, S. Popescu: [The projective geometry of the Gale transform](http://www.sciencedirect.com/science/article/pii/S0021869399979401), *Journal of Algebra* **230** (2000), 127-173,


in particular Example 6.3 p. 153. 


**(2)** D. Eisenbud, K. Hulek and S. Popescu: [A note on the intersection of Veronese surfaces](http://link.springer.com/chapter/10.1007%2F978-94-007-1092-4_7), *Commutative Algebra, Singularities and Computer Algebra*, Volume **115** of the series NATO Science Series (2003), pp 127-139,


in particular Section 3. 


