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Exotic smooth structures on $3\mathbf{CP}^2 # n\overline{\mathbf{CP}}^2$
We construct exotic $3\mathbf {CP}^2 \# 10 \overline {\mathbf {CP}}^2$ and $3\mathbf {CP}^2 \# 12{\overline {\mathbf {CP}}}^2$ as a corollary of recent results of I. Dolgachev and C. Werner concerning a numerical Godeaux surface. We also construct another exotic $3\mathbf {CP}^2 \# 12 {\overline {\mathbf {CP}}}^2$ using the surgery techniques …