  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}.

Combinatorics;Representation Theory