# Isolating a component by blowup

In Rydh and Skjelnes's The space of generically étale families, they construct the principal component of the Hilbert scheme of points on a smooth surface $S$. Given a family of length $n$ subschemes $F$ over $B$, we get a morphism from $B$ to $Sym^n(S)$. They cook up a sheaf of ideals $I$ on $Sym^n(R)$ so that $I$ always pulls back to the discriminant ideal of $F$ over $B$. By the universal property of blow-ups, we see that the map from $B$ factors through the blowup of $Sym^n(R)$ along $I$. They show that this blowup is the principal component of the Hilbert scheme of $n$ points on $S$.

This suggests the following mechanism for isolating a component of a scheme in general. Suppose $X$ is a scheme with many components, and we're after a particular component $X_0$. Suppose we are able to produce an ideal $I$ whose closed subscheme is the union of all the components other than $X_0$, and that these components intersect $X_0$ simply. Then $X_0$ will be the blowup of $X$ along $I$.