We study webs of 5-branes with 7-branes in Type IIB string theory from a geometric perspective. Mathematically, a web of 5-branes with 7-branes is a tropical curve in \(\mathbb {R}^2\) with focus-focus singularities introduced. To any such a web W, we attach a log Calabi–Yau surface (Y, D) with a line bundle L. We then describe supersymmetric webs, which are webs defining 5d superconformal field theories (SCFTs), in terms of the geometry of (Y, D, L). We also introduce particular supersymmetric webs called “consistent webs", and show that any 5d SCFT defined by a supersymmetric web can be obtained from a consistent web by adding free hypermultiplets. Using birational geometry of degenerations of log Calabi–Yau surfaces, we provide an algorithm to test the consistency of a web in terms of its dual polygon. Moreover, for a consistent web W, we provide an algebro-geometric construction of the mirror \(\mathcal {X}^{\textrm{can}}\) to (Y, D, L), as a non-toric canonical 3-fold singularity, and show that M-theory on \(\mathcal {X}^{\textrm{can}}\) engineers the same 5d SCFT as W. We also explain how to derive explicit equations for \(\mathcal {X}^{\textrm{can}}\) using scattering diagrams, encoding disk worldsheet instantons in the A-model, or equivalently the BPS states of an auxiliary rank one 4d \(\mathcal {N}=2\) theory.
