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 ${\mathbb{k}}^n$ constructed by Kapranov. Buchweitz, Leuschke and Van den Bergh introduced a quiver $Q$ and a surjective $\mathbb{k}$-algebra homomorphism $\Phi :\mathbb{k}Q\to A=\text{End}(E)$, 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 ${⨁}_{0\le i\le n}{\mathcal{O}}_{{\mathbb{P}}^n}(i)$.

44 pages

References in corpus (2)

• Pieri resolutions for classical groups
• Quiver flag varieties and multigraded linear series