Reconstructing the Grassmannian of lines from Kapranov's tilting bundle

Abstract

Let $E$ be the tilting bundle on the Grassmannian $\text{Gr}(n,r)$ of $r$-dimensional quotients of $\Bbbk^n$ constructed by Kapranov. Buchweitz, Leuschke and Van den Bergh introduced a quiver $Q$ and a surjective $\Bbbk$-algebra homomorphism $\Phi\colon\Bbbk Q\rightarrow A=\text{End}(E)$, together with a recipe on how the kernel may be computed. In this paper, for the case $r=2$ we give a new, direct proof that $\Phi$ is surjective and then complete the picture by calculating the ideal of relations explicitly. As an application, we then use this presentation to show that $\text{Gr}(n,2)$ is isomorphic to a fine moduli space of certain stable $A$-modules, just as $\mathbb{P}^n$ can be recovered from the endomorphism algebra of Beilinson's tilting bundle $\bigoplus_{0\leq i\leq n}\mathcal{O}_{\mathbb{P}^n}(i)$.