Does anyone know what the blow-up of the Grassmannian at a point looks like? Consider $G=Gr(r,n)$ and $V\in G$. I want to understand more explicitly what $Bl\_V(G)$ should mean.


Of course for affine space $\mathbb{A}^n$, the blow-up at the origin is a subset of $\mathbb{A}^n\times \mathbb{P}^{n-1}$ defined by $B=\{(x,L)\in \mathbb{A}^n\times \mathbb{P}^{n-1}| x\in L\}$. Similarly we can view the blow-up of a point in projective space $\mathbb{P}^n$ as a subvariety of $\mathbb{P}^n\times \mathbb{P}^{n-1}$.


If we take the viewpoint that the blow-up at a point should be the projectivization of the tangent space, and use the fact that $T\_VG = Hom(V,\mathbb{C}^n/V)$, we probably want to describe $Bl\_V(G)$ as a subset of $\mathbb{P}(T\_VG )\times G$ given by some convenient set of equations. Of course we can always look locally and describe the blow-up that way, but I want something more global. In particular, the map from $Bl\_V(G)$ to $G$ should be an isomorphism away from $V$, and only blow-up $V$.


Any references on the subject would also be appreciated.


I posted this question on the Stack Exchange but I didn't receive any complete answers so I was hoping someone here had come across the answer.



Blowing-up the Grassmannian at a point
Let me discuss a more general question of blowing up a subscheme $Z$ in $X$. Imagine that there is a resolution
$$
F \stackrel{s}\to E \to I\_Z \to 0
$$
of the ideal of $Z$ by vector bundles $E$ and $F$. Then there is an embedding
$$
Bl\_Z(X) \to P\_X(E^\*)
$$
(it is induced by a surjection of graded algebras $\oplus S^kE \to \oplus I^k$). Moreover, the morphism $s$ gives a global section of the vector bundle $p^\*F^\* \otimes O(1)$ on $P\_X(E^\*)$, where $p:P\_X(E^\*) \to X$ is the natural projection. It is easy to see that $Bl\_Z(X)$ is in the zero locus of this section. In fact, if $Z$ is a locally complete intersection in $X$ then the blowup coincides with this zero locus. So, the genral answer for the blowup is the zero locus of $s$, considered as a global section of $p^\*F^\*\otimes O(1)$ in $P\_X(E^\*)$.


Coming back to the Grassmannian example, a point $P$ can be realized as the zero locus of a section of the vector bundle $(U^\*)^{\oplus n-r}$, where $U$ is the tautological vector bundle. Consequently, its ideal has a Koszul resolution
$$
\Lambda^2(U^{\oplus n-r}) \to U^{\oplus n-r} \to I\_P \to 0.
$$
This gives you explicit $E$, $F$, and $s$ in this case.


