A Proof of Looijenga’s Conjecture via Integral-affine Geometry

A cusp singularity (V , p) is the germ of an isolated, normal surface singularity such that the exceptional divisor of the minimal resolution π : V → V is a cycle of smooth rational curves meeting transversely: π−1(p) = D = D1 + · · ·+Dn. The analytic germ of a cusp singularity is uniquely determined by the self-intersections D2 i of the components of D. Cusp singularities come in naturally dual pairs (V , p) and (V ′, p′), whose exceptional divisors D and D′ are called dual cycles. For every pair of dual cusps, Inoue [Ino77] constructs an associated Hirzebruch-Inoue surface—a smooth, non-algebraic, complex surface whose only curves are the components of two disjoint cycles D and D′. Contracting D and D′ produces a surface with two dual cusp singularities and no algebraic curves. By working out the deformation theory of the contracted Hirzebruch-Inoue surface, Looijenga [Loo81] proved that if the cusp with cycle D′ is smoothable, then there exists an anticanonical pair (Y,D)—a smooth rational surface Y with an anticanonical divisor D ∈ | −KY | whose components have the appropriate self-intersections. Conversely, Looijenga conjectured that the existence of such an anticanonical pair (Y,D) implies the smoothability of the cusp with cycle D′. Recently, work of Gross, Hacking, and Keel proved Looijenga’s conjecture using methods from mirror symmetry [GHK11]. In this paper, we provide an alternative proof of Looijenga’s conjecture. In the first section, we review foundational material on cusp singularities, Hirzebruch-Inoue surfaces, anticanonical pairs, and discuss the main result of Friedman-Miranda [FM83]: The cusp D′ is smoothable if there exists a simple normal crossings surface X0 = ⋃ Vi satisfying certain combinatorial conditions. We begin the second section by defining an integral-affine structure on the dual complex Γ(X0) of such a surface. The existence of this integral-affine structure is not needed in the proof of Looijenga’s conjecture, but motivates the construction in the fourth section. Then, we associate an integral-affine surface, called the pseudo-fan, to any anticanonical pair (V,D) and describe two surgeries on the pseudo-fan of (V,D) that correspond to blowing up a point on a component of D and smoothing a node of D. In the third section, we define two surgeries on an integral-affine surface (S, P,A) with a locally polygonal boundary ∂S = P and singularities A—an internal blow-up and a node smoothing. Both surgeries appear in Symington’s work [Sym03] on almost toric fibrations of four-dimensional