Blowing-up the Grassmannian at a point

Question

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.

Answer 1

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.