Blowups and Fiber Products

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.

The first breakthrough came in terms of Eisenbud’s alternate characterization of the blowup algebra. This made the blowup algebra much easier to work with; there’s something about viewing a finitely generated algebra as the quotient of a polynomial ring that makes computations much easier. This allowed me to dehomogenize the $\proj$ and work with coordinates. My main goal was to prove that the blowup was nonsingular, i.e. to resolve the singularity. At first I thought that I would need to blow up twice to resolve this singularity, because I was obtaining that my blowup was singular. Then I realized that the scheme I was blowing up was non-reduced! This came as a bit of a shock; I was blowing up $V(x, y)$ in $\spec k[x, y, z]/(x^2-y^2-z^2)$, and needed to blow up $V(x, y, z)$. I’m interested in why the first gives a scheme with nilpotents and the second doesn’t; I understand on an algebraic level, but not on a geometric level. Once I had blown up the cone with respect to the correct subscheme I was able to show easily that it was nonsingular.

One of the things that lead me to this realization was studying various fibrations of the cone. I found this incredibly fruitful, because I find it can be difficult to work with higher dimensional schemes, while I’m very comfortable with curves. I also find it useful to get more intuition for what the scheme might look like, to fiber it over $\mb P^1$ or something. These computations (one of which can be found here) also demonstrated that the preimage of the point with respect to which I was blowing up was not a divisor; this surprised me, but I think it stems from the fact that the surface is not smooth. This made me wonder if there were singular points that blew up to divisors, or if there was some unifying theory about the dimension of the preimage of the blown-up point.

I’ve also become interested in the theory of degeneration of a family of schemes; for example, consider the classical $\spec(k[x, y, t] / (xy-t))\to\spec( k[t])$ given by the morphism $k[t]\to k[x, y, t]/(xy-t), t\mapsto t$; or even, the cone $\spec(k[x, y, z]/(x^2-y^2-z^2))\to \spec(k[x])$ by $k[x]\to k[x, y, z]/(x^2-y^2-z^2), x\mapsto x$. How can studying the singular fibers lead to understanding of the nonsingular fibers, and vice versa? What about families (such as these) where all the fibers are isomorphic except for the singular fibers?

This week was fun, especially the end. I did a lot of computations in coordinates, a lot of local things. I was struck by how much of algebraic geometry, especially doing out examples, is just commutative algebra + formalism. Next week, as I focus on toric varieties, I hope to perform some more global computations.