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 produced results concerning embeddings of singular Lagrangian $RP^2$s, or embeddings of lens spaces as a hypersurface of contact type, in small rational surfaces such as $CP^2\#\overline{CP^2}$ and $S^2\times S^2$, as well as results on the uniqueness or classification of certain symplectic fillings by a $Q$-homology $4$-ball. Further work on the classification of singular Lagrangian $RP^2$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 for the tightness and fillability criterions of $S^1$-invariant contact structures as well as their (small) symplectic fillings.