Symplectic log Kodaira dimension $-\infty$, Hirzebruch--Jung strings and weighted projective planes

Abstract

We study symplectic minimal resolutions of weighted projective planes $\mathbb{CP}(a,b,c)$ from the perspective of disconnected symplectic divisors with symplectic log Kodaira dimension $-\infty$. Building on the techniques developed in our previous work for connected divisors, we introduce the notion of exceptional gaps between distinct connected components of the divisor and use it to establish a Torelli-type theorem for certain configurations of three Hirzebruch--Jung strings. Motivated by Daigle--Russell's study of affine rulings on complete normal rational surfaces in algebraic context, we also establish a weighted version of Gromov--McDuff's characterization of symplectic $\mathbb{CP}^2$ by showing the existence of symplectic affine rulings implies certain divisor configuration to arise from the minimal resolution of $\mathbb{CP}(a,b,c)$.