Toric degenerations of Cox rings of blow-ups of projective $3$-space

We study combinatorial and geometric properties of toric degenerations of Cox rings of blow-ups of $\mathbb{P}^3$ at points in general positions. We focus in particular on Ehrarht-type formulas for the multigraded Hilbert functions of these spaces. From our computations, it follows that the presentation ideal of the Cox ring of the blow-up of $P^3$ at seven points is quadratically generated, as conjectured by Lesieutre and Park. The talk is based on recent work with Mara Belotti.