Quiver Grassmannians associated with string modules

TL;DR

This paper introduces a new technique for computing the Euler characteristic of quiver Grassmannians linked to orientable string modules, with explicit calculations for specific affine quivers, enhancing understanding of their geometric properties.

Contribution

The paper develops a novel method to compute Euler characteristics of quiver Grassmannians associated with orientable string modules, providing explicit results for affine quivers of type A_{p,1}.

Findings

Computed Euler characteristics for various affine quivers.

Provided an alternative proof for a known result when p=1.

Enhanced understanding of the geometry of quiver Grassmannians.

Abstract

We provide a technique to compute the Euler characteristic of a class of projective varieties called quiver Grassmannians. This technique applies to quiver Grassmannians associated with "orientable string modules". As an application we explicitly compute the Euler characteristic of quiver Grassmannians associated with indecomposable preprojective, preinjective and regular homogeneous representations of an affine quiver of type $\tilde{A}_{p,1}$. For $p=1$, this approach provides another proof of a result due to P. Caldero and A. Zelevinsky in \cite{CZ}.