The dimension of the intersection of all Hassett divisors

Abstract We prove that the dimension of the intersection $${\mathcal {Z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> of all Hassett divisors of special cubic fourfolds is sixteen. We do this by studying which subsets of the natural numbers $$\mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> 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 $$\mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> consisting of all possible discriminants of special cubic fourfolds, we show this rank is four and that this is the codimension of $${\mathcal {Z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> in $${\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> , the twenty-dimensional moduli space of cubic fourfolds.