# is the exceptional divisor of an equivariant blow-up linearized?

I hope someone can help me with this.

Let $X$ be a smooth projective variety, say over the complex numbers and let $Z$ be a closed subvariety of $X$. Assume that $X$ is acted upon by a finite group $G$ and that $Z$ is stable under the action of $G$.

Consider the blow-up $Y$ of $X$ along $Z$. The action of $G$ canonically extends to $Y$. Let $E$ be the exceptional divisor.

Question: Is sheaf $\mathcal{O}_E(1)$ G-linearized? If so, how does one prove it?

thanks!

By definition $Y = Proj_X(A)$, where $A = \oplus I^k$ and $I$ is the ideal of $Z$. Clearly, $I$ has a $G$-equivariant structire, so the same is true for $A$. This gives an action of $G$ on $Y$. By Serre's Theorem $coh(Y)$ is the quotient category of the category of graded $A$-modules. Analogously, $coh(Y)^G$ is the quotient of the category of graded $G$-equivariant $A$-modules. Clearly, $A(1)$ (the shift of grading) is $G$-equivariant graded $A$-module, so is the corresponding sheaf $O_Y(1)$. Analogously, $O_Y(2)$ is $G$-equivariant, as well as the map $O_Y(2) \to O_Y(1)$ (it is induced by embeddings $I^{k+1} \to I^k$). Thus $$O_E(1) = O_Y(1)/O_Y(2)$$ is $G$-equivariant.