Algebraic boundary of matrices of nonnegative rank at most three

Understanding the boundary of the set of matrices of nonnegative rank at most r is important for applications in nonconvex optimization. The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Gröbner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.