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Abstract 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$$\mathbb {R}^2$$ $R^2$with focus-focus singularities introduced. To any such a webW, we attach a log Calabi–Yau surface (Y, D) with a line bundleL. 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 webW, we provide an algebro-geometric construction of the mirror$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$to (Y, D, L), as a non-toric canonical 3-fold singularity, and show that M-theory on$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$engineers the same 5d SCFT asW. We also explain how to derive explicit equations for$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$using scattering diagrams, encoding disk worldsheet instantons in the A-model, or equivalently the BPS states of an auxiliary rank one 4d$$\mathcal {N}=2$$ $N=2$theory. 

Abstract We prove that every irreducible component of the coarse Kollár-Shepherd-Barron and Alexeev (KSBA) moduli space of stable log Calabi–Yau surfaces admits a finite cover by a projective toric variety. This verifies a conjecture of Hacking–Keel–Yu. The proof combines tools from log smooth deformation theory, the minimal model program, punctured log Gromov–Witten theory, and mirror symmetry. 

Abstract We define kappa classes on moduli spaces of Kollár-Shepherd-Barron-Alexeev (KSBA)-stable varieties and pairs, generalizing the Miller–Morita–Mumford classes on moduli of curves, and computing them in some cases where the virtual fundamental class is known to exist, including Burniat and Campedelli surfaces. For Campedelli surfaces, an intermediate step is finding the Chow (same as cohomology) ring of the GIT quotient$$(\mathbb {P}^2)^7//SL(3)$$. 

We compute Coxeter diagrams of several “large” reflective two-elementary even hyperbolic lattices and their maximal parabolic subdiagrams, and give some applications of these results to the theory of K3 surfaces and hyperkahler manifolds. 

We describe a geometric, stable pair compactification of the moduli space of Enriques surfaces with a numerical polarization of degree 2, and identify it with a semitoroidal compactification of the period space. 

Abstract. We describe explicitly the geometric compactifications, obtained by adding slc surfaces X with ample canonical class, for two connected components in the moduli space of surfaces of general type: Campedelli surfaces with \pi_1(X) = Z^3_2 and Burniat surfaces with K^2 = 6. 

We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable KSBA pairs over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities. 

Abstract The flex divisor$$R_{\textrm flex}$$ of a primitively polarized K3 surface $(X,L)$ is, generically, the set of all points $$x\in X$$ for which there exists a pencil $$V\subset |L|$$ whose base locus is $$\{x\}$$. We show that if $L^2=2d$ then $$R_{\textrm flex}\in |n_dL|$$ with $$ \begin{align*} &n_d= \frac{(2d)!(2d+1)!}{d!^2(d+1)!^2} =(2d+1)C(d)^2,\end{align*}$$where $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $$F_{2d}$$ of polarized K3 surfaces.