Positive Charts of Toric Varieties

TL;DR

This paper introduces positive affine charts for smooth projective toric varieties that include nonnegative points, linking them to the nef cone and algebraic moment maps, advancing the understanding of positive geometry.

Contribution

It constructs positive charts containing nonnegative points, relates them to smooth subcones of the nef cone, and connects to algebraic moment maps within toric varieties.

Findings

Positive charts contain nonnegative points.

Positive charts correspond to smooth subcones of the nef cone.

Associated algebraic moment maps identify critical points of monomial functions.

Abstract

We construct affine charts of a smooth projective toric variety which contain its nonnegative points, and which admit a closed embedding into the total coordinate space of Cox's quotient construction. We show that such positive charts arise from smooth subcones of the nef cone. To each positive chart we associate an algebraic moment map, the fibers of which are the critical points of a monomial function in Cox coordinates. This work provides a toric framework for the theory of $u$-equations in positive geometry.