Positive Charts of Toric Varieties

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.