Let's put m>n two nonnegative integers and $Gr:=Grass(n,k^m)$ the grassmanian of the subspaces of dimension $n$ in $k^m$. We have a natural immersion $Gr \subset P({\Lambda}^{n} k^m)$ and I call $C(Gr) \subset {\Lambda}^{n} k^m$ the affine cone above $Gr$. If we blow-up $C(Gr)$, we always obtain something smotth, but my question is: Is the blow-up a crepant resolution of the singularities of $C(Gr)$?
In fact the answer is no when n=1 or n=m-1, but what happens in the other cases ?

Let's put $m>n$ two nonnegative integers and $Gr:=Grass(n,k^m)$ the grassmanian of the subspaces of dimension $n$ in $k^m$. We have a natural immersion $Gr \subset P({\Lambda}^{n} k^m)$ and I call $C(Gr) \subset {\Lambda}^{n} k^m$ the affine cone above $Gr$. If we blow-up $C(Gr)$, we always obtain something smooth, but my question is: Is the blow-up a crepant resolution of the singularities of $C(Gr)$?
In fact the answer is no when $n=1$ or $n=m-1$, but what happens in the other cases ?

Let's put m>n two nonnegative integers and Gr:=Grass(n,k^m) the grassmanian of the subspaces of dimension n in k^m. We have a natural immersion Gr \subset P({\Lambda}^{n} k^m) and I call C(Gr) \subset {\Lambda}^{n} k^m the affine cone above Gr. If we blow-up C(Gr), we always obtain something smotth, but my question is: Is the blow-up a crepant resolution of the singularities of C(Gr)?
In fact the answer is no when n=1 or n=m-1, but what happens in the other cases ?

Let's put m>n two nonnegative integers and $Gr:=Grass(n,k^m)$ the grassmanian of the subspaces of dimension $n$ in $k^m$. We have a natural immersion $Gr \subset P({\Lambda}^{n} k^m)$ and I call $C(Gr) \subset {\Lambda}^{n} k^m$ the affine cone above $Gr$. If we blow-up $C(Gr)$, we always obtain something smotth, but my question is: Is the blow-up a crepant resolution of the singularities of $C(Gr)$?
In fact the answer is no when n=1 or n=m-1, but what happens in the other cases ?