Symplectic configurations: a homological and computer-aided approach

Motivated by and extending the technical results in our earlier work on symplectic Calabi-Yau $4$-manifolds, a general and systematic approach for studying certain unions of symplectic embedded surfaces in a rational $4$-manifold $X=CP^2\# N\overline{CP^2}$ is formulated, which may find applications in a broader range of problems. A distinct feature of this method is that it is computer-aided. We address several fundamental theoretical questions concerning the computational aspect. On the other hand, we also establish a symplectic analog of Cremona transformations from algebraic geometry, which is another fundamental feature and a main technical tool of this method. For an illustration, we give a new proof that a certain line arrangement in $CP^2$, called Fano planes, cannot exist in the symplectic category. The nonexistence of Fano planes in the algebraic category follows from a theorem of Hirzebruch, while in the topological category, including the symplectic category, it was first proved by Ruberman and Starkston. Our proof for the symplectic category is independent to both, and is by combining the Cremona transformation technique with Gromov's theory of pseudoholomorphic curves.