Key examples

We now give a recipe for computing the blow up of an affine variety at a point. Let $X=Z(g_{1},g_{2},\ldots ,g_{t})\subseteq \mathbb {A} ^{n}$$X=Z(g_{1},g_{2},\ldots ,g_{t})\subseteq \mathbb {A} ^{n}$ be an affine variety such that $\mathbf {0} \in X$$\mathbf {0} \in X$. Then by functoriality of blow ups we have $\operatorname {Bl} _{\mathbf {0} }X\subseteq \operatorname {Bl} _{\mathbf {0} }\mathbb {C} ^{n}\subseteq \mathbb {C} ^{n}\times \mathbb {P} ^{n-1}$$\operatorname {Bl} _{\mathbf {0} }X\subseteq \operatorname {Bl} _{\mathbf {0} }\mathbb {C} ^{n}\subseteq \mathbb {C} ^{n}\times \mathbb {P} ^{n-1}$. For each $1\leq i\leq t$$1\leq i\leq t$ let us write $g_{i}^{\mathrm {hom} }$$g_{i}^{\mathrm {hom} }$ for the lowest degree homogeneous part of $g_{i}$$g_{i}$ which is not identically $0$$0$ and set $d_{i}=\deg(g_{i}^{\mathrm {hom} })$$d_{i}=\deg(g_{i}^{\mathrm {hom} })$. Consider a formal replacement procedure, where we obtain a polynomial ${\tilde {g_{i}}}\in \mathbb {C}$${\tilde {g_{i}}}\in \mathbb {C}$ from $g_{i}$$g_{i}$ by replacing exactly $d_{i}$$d_{i}$ of the variables $x_{j}$$x_{j}$ in each monomial of $g_{i}$$g_{i}$ with the corresponding variables $X_{j}$$X_{j}$ (for monomials of degree greater than $d_{i}$$d_{i}$ the procedure involves a choice, but this is will not be important for our purposes). Note that the polynomial we obtain in this way is homogeneous of degree $d_{i}$$d_{i}$ in the $X$$X$-variables. We now claim that

$$\operatorname {Bl} _{\mathbf {0} }X=\{((x_{1},\ldots ,x_{n}),[X_{1}:\ldots :X_{n}])\in \operatorname {Bl} _{\mathbf {0} }\mathbb {C} ^{n}\colon \,{\tilde {g}}_{i}(\mathbf {x} ,\mathbf {X} )=0\;\forall \,1\leq i\leq t\}.$$

$$\operatorname {Bl} _{\mathbf {0} }X=\{((x_{1},\ldots ,x_{n}),[X_{1}:\ldots :X_{n}])\in \operatorname {Bl} _{\mathbf {0} }\mathbb {C} ^{n}\colon \,{\tilde {g}}_{i}(\mathbf {x} ,\mathbf {X} )=0\;\forall \,1\leq i\leq t\}.$$

In particular, the exceptional divisor of $\operatorname {Bl} _{\mathbf {0} }X$$\operatorname {Bl} _{\mathbf {0} }X$ is given by

$$Z(\{g_{i}^{\mathrm {hom} }(X_{1},\ldots ,X_{n})\colon 1\leq i\leq t\})\subseteq \mathbb {P} ^{n-1}.$$

$$Z(\{g_{i}^{\mathrm {hom} }(X_{1},\ldots ,X_{n})\colon 1\leq i\leq t\})\subseteq \mathbb {P} ^{n-1}.$$

Exercise 2.5

Prove the above claim.

 

Exercise 2.6

Compute the blow ups at the origin of the following curves in $\mathbb {C} ^{2}$$\mathbb {C} ^{2}$: $\{x_{1}x_{2}=0\}$$\{x_{1}x_{2}=0\}$, $\{x_{2}^{2}-x_{1}^{3}-x_{1}^{2}=0\}$$\{x_{2}^{2}-x_{1}^{3}-x_{1}^{2}=0\}$, $\{x_{2}^{2}-x_{1}^{3}=0\}$$\{x_{2}^{2}-x_{1}^{3}=0\}$, $\{x_{2}^{2}-x_{1}^{4}=0\}$$\{x_{2}^{2}-x_{1}^{4}=0\}$. Draw their real cartoons and investigate their exceptional divisors.

 

Observe that the exceptional divisor in each of the above examples consists of precisely one point for each tangent direction to the curve at $\mathbf {0}$$\mathbf {0}$. In fact, by blowing up we have resolved the singularities in the first three curves, but not in the fourth one. We have however made the singularity milder -- the two irreducible components no longer have a common tangent at their point of intersection. So we might hope that performing a second blow up will separate them.

Exercise 2.7

Desingularize the curve $\{x_{2}^{2}-x_{1}^{4}=0\}$$\{x_{2}^{2}-x_{1}^{4}=0\}$ by applying two repeated blow ups and obtain the following sequence of diagrams.

 

In fact, we could have accomplished the same effect in one go. Note that the problematic tangent direction is the one corresponding to the $x_{1}$$x_1$-axis. So instead of blowing up the reduced origin we may consider blowing up the thickened point, which carries infinitesimal information in the $x_{1}$$x_1$-direction.

Exercise 2.8

Calculate explicitly $\operatorname {Bl} _{(x_{1}^{2},x_{2})}X$$\operatorname {Bl} _{(x_{1}^{2},x_{2})}X$, where $X$$X$ is the curve $Z(x_{2}^{2}~-~x_{1}^{4})$$Z(x_{2}^{2}~-~x_{1}^{4})$.

 

The last exercise indicates that it really is the ideal in which we are blowing up which is important and not just its reduced vanishing locus. In other words, we need the language of schemes to define the blow up in full generality and this is what we do next.