We construct potentially new manifolds homeomorphic but not diffeomorphic to $\mathbb{CP}^{2} \# 8 \overline{\mathbb{CP}^{2}}$ and $\mathbb{CP}^{2} \# 9 \overline{\mathbb{CP}^{2}}$ via rational blowdown surgery along certain $4$-valent plumbing graphs. This way all the graph classes from \cite{weighted} have a representative which admits a rational blowdown leading to an exotic manifold. We emphasize the simplicity of the constructions which boils down to finding a good configuration of complex lines and quadrics in $\mathbb{CP}^{2}$, and deciding which intersections to blow up.

中文翻译：

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来自 $$\mathbb{CP}^{2}$$ 中的线和二次曲面的小异国情调 $$4$$-流形

我们构造了潜在的新流形同胚但不是微分同胚 $\mathbb{CP}^{2} \# 8 \overline{\mathbb{CP}^{2}}$ 和 $\mathbb{CP}^{2} \# 9 \overline{\mathbb{CP}^{2}}$ 通过合理的排污手术沿着某些 $4$-valent 管道图。这样来自 \cite{weighted} 的所有图类都有一个代表，它承认理性的排污导致奇异的流形。我们强调构造的简单性，归结为在 $\mathbb{CP}^{2}$ 中找到复杂线和二次曲线的良好配置，并决定炸毁哪些交点。