Musings

This past week, I’ve been working on computing blowups. Classically, the blowup of $\mb A^n$ is the algebraic subset of $\mb A^n\times \mb P^{n-1}$ given by $\{(a_1, ..., a_n, [p_1, ..., p_n]) | a_ip_i=a_jp_j\}$; now, we blowup along a sheaf of ideals $\ms I$ by forming the sheaf of graded rings $\ms B = \mc O_X\oplus \ms I\oplus \ms I^2\oplus \cdots$. In the affine case this is called the blowup algebra. We then take relative $\proj$ of this sheaf of graded rings, which yields the blowup. In particular, this week, I was trying to compute the blowup of the affine cone $x^2-y^2-z^2$ at the origin (to resolve that singularity). I found this shockingly difficult, mostly due to my inability to work with the graded ring and the proj construction.