# Small exotic 4-manifolds from lines and quadrics in $\mathbb{CP}^{2}$

## TL;DR

This paper constructs new exotic smooth structures on certain complex projective plane blowups using rational blowdown surgery along specific plumbing graphs, based on configurations of lines and quadrics.

## Contribution

It introduces simple constructions of exotic 4-manifolds from configurations of lines and quadrics in , expanding the class of known exotic structures.

## Key findings

- Constructed potentially new manifolds homeomorphic but not diffeomorphic to  and .
- All graph classes from previous work have representatives admitting rational blowdown.
- Simplified construction method based on configurations of lines and quadrics in .

## Abstract

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.