The Birational Geometry of Ceva's Theorem

TL;DR

This paper explores Ceva's theorem and its higher-dimensional generalizations through algebraic and projective geometry, relating it to del Pezzo surfaces and matrix completion problems, with accessible explanations for non-specialists.

Contribution

It connects Ceva's theorem to algebraic surfaces and projective geometry, providing new proofs and higher-dimensional analogues using matrix completion techniques.

Findings

Relates Ceva's theorem to del Pezzo surfaces of degree six.

Recasts higher-dimensional Ceva analogues as matrix completion problems.

Provides accessible explanations for algebraic geometry students.

Abstract

In this article we study Ceva's theorem and its higher-dimensional extensions from the perspective of algebraic and projective geometry. First, we situate the theorem within the study of algebraic surfaces by relating it to the defining equation of a del Pezzo surface of degree six inside the product of three projective lines. Second, by interpreting (higher-dimensional analogues of) Ceva's theorem in terms of projections from projective spaces, we recast these results as matrix completion problems. We use these ideas to offer proofs of some higher-dimensional analogues of Ceva's theorem. This article is written with a nonspecialist audience in mind and we hope that some useful context is provided in the form of remarks in the sections on surfaces for students of algebraic geometry.