Log Canonical Models and Positive Geometries
arXiv:2607.28368
The paper shows that for many varieties, explicit coordinates for their log canonical models can be obtained from canonical forms of positive geometries, and uses this to compute equations for examples such as hyperplane arrangement complements and moduli of marked cubic del Pezzo surfaces.
Abstract
Constructing log canonical compactifications of open varieties is a central problem in birational geometry. Finding a natural coordinate system and obtaining the equations of these models is difficult in general. We show that for a large class of varieties explicit coordinates for the log canonical model are provided by canonical forms of positive geometries, and use this to compute the equations of these models. Our theory applies, for instance, to complements of hyperplane arrangements, cubic surfaces with lines removed, and the moduli space of marked cubic del Pezzo surfaces.
23 pages, 3 figures, comments welcome!