What is the name of these elliptic surfaces $E(n)$?

Question

I am referring to the elliptic surfaces $E(n)$, with fibration over $\mathrm{C}\mathbb{P}^1$. They are common in 4-manifold theory and complex geometry. See for example Chapter 7 in Akbulut`s "4-manifolds". Sometimes I read that they are called rational elliptic surfaces. Is this the whole name or is there a more specific name?

Answer 1

The elliptic surface $E(1)$ is a rational elliptic surface. It is a blowup of $\mathbb{CP}^2$ at nine points, and the projection to $\mathbb{CP}^1$ is constructed by using a pencil of cubics through the nine points of $\mathbb{CP}^2$; see here for example.

For $n > 0$, the elliptic surface $E(n)$ is defined to be the fiber sum of $n$ copies of $E(1)$. They all have singular fibers, and are therefore Kähler. Moreover, they have $b^+(E(n)) = 2n - 1$. As $b^+(X) = 2h^{2,0}(X) + 1$ for Kähler surfaces, we see that $h^{2,0}(E(n)) = n - 1$. So for $n > 1$, the canonical bundle $K_{E(n)}$ admits sections and therefore $\kappa(E(n)) \neq -\infty$; in particular, these elliptic surfaces are not rational.

For $n = 2$, the corresponding elliptic surface is an elliptic $K3$ surface.

Beyond that, I am not aware of any particular name for these elliptic surfaces. For more information about them, see the book $4$-Manifolds and Kirby Calculus by Gompf and Stipsicz. They are first introduced in chapter $3$.