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To achieve this, we follow an idea of Mukai and explore a special instance of Gale duality, namely, a correspondence between configurations of $n+4$ points in the projective spaces $\\mathbb{P}^n$ and $\\mathbb{P}^2$. We first prove that the blowup $X$ of $\\mathbb{P}^n$ at $n+4$ general points is isomorphic to a certain Gieseker moduli space of rank $2$ vector bundles on the surface $S$ obtained by blowing up $\\mathbb{P}^2$ at the $n+4$ Gale dual points. 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