The dimension of the intersection of all Hassett divisors

We prove that the dimension of the intersection Z Z of all Hassett divisors of special cubic fourfolds is sixteen. We do this by studying which subsets of the natural numbers N N can be obtained as the image of a positive-definite integral quadratic form and what the minimal possible rank of such a form is. In particular, for the subset of N N consisting of all possible discriminants of special cubic fourfolds, we show this rank is four and that this is the codimension of Z Z in C C , the twenty-dimensional moduli space of cubic fourfolds.