Exotic surfaces

Although exotic blow-ups of the projective plane at n points have been constructed for every \(n \ge 2\), the only examples known by means of rational blowdowns satisfy \(n \ge 5\). It has been an intriguing problem whether it is possible to decrease n. In this paper, we construct the first exotic \({\mathbb {C}}{\mathbb {P}}^2 \# 4 \overline{{\mathbb {C}}{\mathbb {P}}^2}\) with this technique. We also construct exotic \(3{\mathbb {C}}{\mathbb {P}}^2 \# b^- \overline{{\mathbb {C}}{\mathbb {P}}^2}\) for \(b^-=9,8,7\). All of them are minimal and symplectic, as they are produced from projective surfaces W with Wahl singularities and \(K_W\) big and nef. In more generality, we elaborate on the problem of finding exotic

$$\begin{aligned} (2\chi ({\mathcal {O}}_W)-1) {\mathbb {C}}{\mathbb {P}}^2 \# (10\chi ({\mathcal {O}}_W)-K^2_W-1) \overline{{\mathbb {C}}{\mathbb {P}}^2} \end{aligned}$$

from these Kollár–Shepherd-Barron–Alexeev surfaces W, obtaining explicit geometric obstructions on the corresponding configurations of rational curves.