The moduli space of Harnack curves in toric surfaces

In 2006, Kenyon and Okounkov Kenyon and Okounkov [12] computed the moduli space of Harnack curves of degree d in ${\mathbb {C}\mathbb {P}}^2$ . We generalise their construction to any projective toric surface and show that the moduli space ${\mathcal {H}_\Delta }$ of Harnack curves with Newton polygon $\Delta $ is diffeomorphic to ${\mathbb {R}}^{m-3}\times {\mathbb {R}}_{\geq 0}^{n+g-m}$ , where $\Delta $ has m edges, g interior lattice points and n boundary lattice points. This solves a conjecture of Crétois and Lang. The main result uses abstract tropical curves to construct a compactification of this moduli space where additional points correspond to collections of curves that can be patchworked together to produce a curve in ${\mathcal {H}_\Delta }$ . This compactification has a natural stratification with the same poset as the secondary polytope of $\Delta $ .

Freeform machining of ophthalmic toric lens mould using fast tool servo-assisted ultra-precision diamond turning process

This research aims to establish a methodology for machining of toric lenses, using fast tool servo-assisted single point diamond turning and to assess the generated surface for its characteristics. Using the established mathematical model, toric surface is explained to understand the geometry and to generate the parameters required for fast tool servo machining. A toric surface with a major diameter of 18.93 mm and a minor diameter of 15.12 mm has been cut on the intelligent ultra-precision turning machine (iUPTM). The surface profile and surface roughness were measured. After analysing the measurement data of the machined surface, on two perpendicular axes of the toric lens, form accuracy of 0.49 µm peak-to-valley (PV), and surface roughness of 12 nm in Ra, 4–8 nm in Sa are obtained. From the experimental results obtained, it can be concluded that the proposed method is a reasonable alternative for manufacturing toric lens mould.

The dualizing sheaf on first-order deformations of toric surface singularities

We explicitly describe infinitesimal deformations of cyclic quotient singularities that satisfy one of the deformation conditions introduced by Wahl, Kollár–Shepherd-Barron (KSB) and Viehweg. The conclusion is that in many cases these three notions are different from each other. In particular, we see that while the KSB and the Viehweg versions of the moduli space of surfaces of general type have the same underlying reduced subscheme, their infinitesimal structures are different.