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We determine the integral homology of the orbit space of a maximal compacttorus action on the Grassmannian Gr(2,5). Our approach uses the well-knownGeometric Invariant Theory of the maximal algebraic torus action on thisGrassmannian.

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