Toric topology of the Grassmannian of planes in $\mathbb{C}^5$ and the del Pezzo surface of degree $5$

We determine the integral homology of the orbit space of a maximal compact torus action on the Grassmannian $\operatorname{Gr}(2,\mathbb C^5)$. This problem has been also studied by Buchstaber and Terzić via purely topological methods. Here, we propose an alternative approach via the well-known Geometric Invariant Theory of the algebraic torus action on this Grassmannian.