Log Canonical Models and Positive Geometries

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.

Disclosure

“ed to X (3, 6). A direct computation with many points reveals that the generic fiber in this case has cardinality 2. An example of two points whose generalized Parke–Taylor forms all agree can be found in [34, Example 6.4]. AI Use. We used GPT-5.6 Sol and Claude Opus 5 to assist in finding and parsing prior work. We used Claude Opus 5 to generate a mock review which caught many typos and notational inconsistencies, and fixed a gap in the proof of Lemma 2.12. We also used Claude Sonn”

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Classification

Suggesting mathematical examples or conjectures

Multiplier

6

Verified

2026-08-23T19:14:09.696731Z

Structural counts