The connection between quadratic forms and Hassett-maximal cubic fourfolds

Let S be a set of natural numbers. Is it possible to find an integral n-ary quadratic form with f(Z^n)=S?
In this lecture we will discuss integral quadratic forms and questions of representability of natural numbers by quadratic forms. We will define "formally-supported sets" and present some types of questions that should be asked about such sets and forms. Mainly, we will be concerned with one specific example of a formally-supported subset, and using its properties to compute the dimension of the space of all multi-Hassett cubic fourfolds.