# Blow-up of a three-dimensional variety at a node

Let $$X$$ be a three-dimensional variety over $$\mathbb{C}$$ with a nodal singularity at a point, say $$P$$. Is the exceptional divisor of the blow-up of $$X$$ at $$P$$ isomorphic to a smooth quadric in $$\mathbb{P}^3$$? I read this statement in an article, but am not able to find a proof.

• For me this follows quickly from the definitions of "nodal singularity" and "blow-up". Are you using different definitions? I would say a scheme has a nodal singularity if it is locally isomorphic to the vanishing locus of a single equation whose lowest-degree term is a nondegenerate quadratic form. Feb 11, 2021 at 0:14
• @WillSawin It is the same definition. Can you suggest a reference or write down the idea in the answer.
– Jana
Feb 11, 2021 at 0:25
• In local analytic coordinates, the variaty has the form $x^2+y^2+z^2+w^4=0$ in a 4-dimensional polydisc. Blowup the origin and take the strict transform, the exceptional divisor is given by the same equation in $\mathbb P^3$, which is a smooth quadric. Feb 11, 2021 at 3:16
• @AGlearner: $w^4$ should be $w^2$. Feb 11, 2021 at 4:09
• Oops! Sorry for the typo. Feb 11, 2021 at 4:13

There is no need to take analytic coordinates.

There is a general principle for blowing up the solution set of a hypersurface $$f(x_1,\dots,x_n)=0$$ at a point $$x_1,....,x_n=0$$.

Recall that blowing up involves introducing projective coordinates $$(y_1: \dots :y_n)$$ in addition to our original ones, satisfying the relations $$x_i y_j = y_i x_j$$, and taking the closure of our original variety, minus the origin.

This closure will satisfy the equation $$f(x_1,\dots,x_n)=0$$. However, this equation is not very useful, as it vanishes when the $$x_i$$ vanish. To imrprove this equation, we can multiply it by $$\frac{y_1}{x_1}$$, and then clear the denominator using $$y_1 x_j = x_1 y_j$$. In other words, we take each monomial in the $$x_i$$s and replace one of the $$x$$s by a $$y$$. This produces an honest polynomial as long as the constant term vanishes. In fact, we can repeat it $$k$$ times, replacing $$k$$ $$x_i$$s in each monomial with a $$y_i$$, as long as the monomials in $$f$$ have degree at least $$k$$.

When we do this, we obtain an equation satisfied by the blow-up. As soon as we do this the maximal number of times, we obtain an equation with an $$n$$-dimensional solution set with an $$n-1$$-dimensional space of solutions over $$0$$, hence one in which the nonzero solutions are dense, which therefore must be the blowup.

The exceptional divisor is then calculated by setting the $$x_i$$s to $$0$$. In other words, we take the lowest-degree terms in the $$x_i$$ and then swap the $$x_i$$s for $$y_i$$s.

So if the lowest degree terms form a nondegenerate quadratic form, the exceptional divisor is the solution set of a nondegenerate quadratic form - a smooth hypersurface.