Reconstructing the Grassmannian of lines from Kapranov's tilting bundle

TL;DR

This paper provides a new proof and explicit relations for the algebra associated with Kapranov's tilting bundle on Grassmannians of lines, and shows how to recover the Grassmannian as a moduli space of modules.

Contribution

It offers a direct proof of surjectivity and computes the relations for the algebra, enabling the reconstruction of the Grassmannian of lines from its tilting bundle.

Findings

Explicit relations for the algebra of the tilting bundle on Gr(n,2)

Grassmannian of lines is isomorphic to a moduli space of stable modules

New proof of algebra surjectivity for the case r=2

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 $\bigoplus_{0\leq i\leq n}\mathcal{O}_{\mathbb{P}^n}(i)$.