Log Canonical Models and Positive Geometries

2026/07/30 by Benjamin Hollering, Dmitrii Pavlov, Elizabeth Pratt
Mathematics · #math.AG #math.CO #msc:14E25. #msc:14Q15 #msc:05E14 #msc:14E30

paper · pdf

22 pages, 3 figures, comments welcome!

arxiv created 2026/07/30 · arxiv updated 2026/07/31

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.

Citations

• Geometry of Adjoint Hypersurfaces for Polytopes
• Vanishing theorems for combinatorial geometries
• The Segre Determinant
• What is Positive Geometry?
• Positive geometries and canonical forms via mixed Hodge theory
• Wondertopes
• Positive del Pezzo Geometry
• A proof of a conjecture by Monin and Rana on equations defining M_0,n
• Compactifications of moduli of points and lines in the projective plane
• Projective geometry of Wachspress coordinates
• Initial degenerations of Grassmannians
• Existence of minimal models for varieties of log general type
• Compactifications of subvarieties of tori
• Matroid polytopes, nested sets and Bergman fans
• The Tropical Grassmannian
• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I

Related

• Failure of Boundedness for Generalised Log Canonical Surfaces
• Geography of log models:theory and applications
• The solvable monodromy extension property and varieties of log general type
• Log canonical models of foliated surfaces
• Log-canonical models of singular pairs and its applications