We work over the complex numbers. Let $X$ be the blow-up of $\mathbb{P}^3$ along a curve of genus $10$, which is the complete intersection of two cubics surfaces. The variety $X$ lives in $\mathbb{P}^1\times \mathbb{P}^3$, and the closure of its effective cone is generated by
$$\overline{Eff}(X)=\mathbb{R}_+E+\mathbb{R}_+H_1$$,
where $E=3H_2-H_1$ is the exceptional divisor and $H_1,H_2$ are respectively the pullback of the hyperplane section in $\mathbb{P}^1,\mathbb{P}^3$.
I have two questions:
- What is the exceptional divisor? I'm blowing-up a curve in $\mathbb{P}^3$, so it will be a surface, so my guess is that $E$ its an Hirzebruch surface. It is easy to see/well known/how can I find which one?
- I've tried to compute the top self intersection of $E$, but I think it's wong becuase it is very high (I obtain $E^3=-54$)
My computations: I know $E=-H_1+3H_2$, so
$$E^3=(-H_1+3H_2)^3=-H_1^3+27H_2^3+9H_1^2H_2-27H_1H_2^2.$$
Now $H_1^3=0$, $H_2^3=1$, $H_1^2H_2=0$ and $H_1H_2^2=3$, so I end up with $E^3=-54$. I don't know, it seems quite high as an intersection number…am I doing something wrong?
Best Answer
If you blowup a smooth subvariety $Z$ in a smooth variety $Y$, the exceptional divisor is isomorphic to the projectivization of the normal bundle $$ E \cong \mathbb{P}_Z(\mathcal{N}_{Z/Y}). $$ In your case, $Z$ is a complete intersection, hence the normal bundle is the direct sum of two line bundles; explicitly $$ \mathcal{N}_{Z/\mathbb{P}^3} \cong (\mathcal{O}_{\mathbb{P}^3}(3) \oplus \mathcal{O}_{\mathbb{P}^3}(3))\vert_Z $$ and since the projectivization is insensitive to line bundle twists, we have in this case $$ E \cong Z \times \mathbb{P}^1. $$