Small symplectic $4$-manifolds via contact gluing and some applications

We introduce a streamlined procedure for constructing small symplectic $4$-manifolds via contact gluing, based on a technique invented by David Gay around 2000. We give several applications of this procedure, which include results concerning embeddings of singular Lagrangian $R{P}^2$s, or embeddings of lens spaces as a hypersurface of contact type, in small rational surfaces such as $C{P}^2\mathrm{\#}\overline{C{P}^2}$ and $S^2\times S^2$, as well as results on the uniqueness or classification of $Q$-homology ball symplectic fillings. Further work on the classification of singular Lagrangian $R{P}^2$s is suggested. Moreover, our investigation on the $S^1$-invariant contact structures suggests an interesting and fairly strong upper bound for the self-intersection of a rational unicuspidal curve with one Puiseux pair $(p,q)$ in any algebraic surface (the bound depends only on the values $p,q$), and for the symplectic version, we prove the existence of an optimal" symplectic rational unicuspidal curve in a rational $4$-manifold which realizes the upper bound for any given Puiseux pair $(p,q)$. Our results also suggest a revisit of the symplectic divisorial capping" problem first considered by Li and Mak. Further applications of the techniques developed in this paper hinge upon better understandings on the tightness and fillability criterions of $S^1$-invariant contact structures as well as their (small) symplectic fillings.