# Mirror symmetry for blowups of the projective plane

Let $S$ be a blowup of the projective plane $\mathbb{CP}^2$ at $n$ points. When $n\le 9$, Auroux, Katzarkov and Orlov showed that them a mirror Landau-Ginzburg model is given by a certain elliptic fibration $W_n:M_n\to\mathbb{C}$ with $n+3$ singular fibers. More precisely they showed a version of homological mirror symmetry for them.

My question is, what is the mirror of a blowup of the projective plane $\mathbb{CP}^2$ at $n$ points for $n> 9$? Is it still an elliptic fibration $W_n:M_n\to\mathbb{C}$ with $n+3$ singular fibers?

Any comments are more than welcome!

It depends on the positions of the points that you blow up. If you blow up respectively $p,q$ and $r$ points on the 3 irreducible components of the toric divisor $D\subset\mathbb{CP}^2$, with $p,q,r\geq3$, then the answer is yes. Note that you have to blow up in an iterative way so that the exceptional divisor consists of $p+q+r-3$ $(-2)$-curves and three $(-1)$-curves. In this case your $M_n$ with $n=p+q+r$ is the partial compactification of the Milnor fiber of the hypersurface simple elliptic or cusp singularity $T_{p,q,r}\subset\mathbb{C}^3$. More precisely, you can construct a Lefschetz fibration $\pi_{p,q,r}:T_{p,q,r}\rightarrow\mathbb{C}$ with precisely $p+q+r+3$ singular fibers, and the smooth fiber of $\pi$ is a torus with 3 points removed. To get the elliptic fibration $W_n:M_n\rightarrow\mathbb{C}$, you can do a fiberwise compactification to $\pi_{p,q,r}$ by adding 3 points to each fiber. As in the del Pezzo case, the fiberwise compactification of $\pi_{p,q,r}$ corresponds on the mirror rational surface to smoothing the strict transform of the toric divisor $D$ to a smooth elliptic curve. This implies that both $\pi_{p,q,r}$ and $W_n$ are mirrors of the same rational surface. One can prove a triviality result concerning the deformation of the Fukaya-Seidel category $\mathcal{F}(\pi_{p,q,r})$ under this fiberwise compactification, so in particular there is a quasi-equivalence $\mathcal{F}(\pi_{p,q,r})\cong\mathcal{F}(W_n)$ between $A_\infty$-categories.