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Work with closed points over an algebraically closed field $k$. For $p\in \mathrm{Spec}k[C\cap M]$, the exponents $m$ for which ${\chi }^m(p)\ne 0$ form a semigroup face: a product is nonzero exactly when both factors are nonzero. In a saturated polyhedral monoid these exponents are precisely the toric lattice points of one face of $C$.

The nonzero values define a homomorphism from the group generated by that face to $k^{\ast }$. Extending it to the full toric lattice shows that points with the same support face differ by the action of the dense algebraic torus. They therefore form a single orbit.

For ${\mathbb{A}}^2$, the possibilities are both coordinates nonzero, one specified coordinate nonzero and the other zero, or both zero. These are the dense algebraic torus, the two punctured axes and the origin. The closed-point/field convention avoids confusing these geometric orbits with arbitrary scheme points or rational-point descent.^[ref]

// Target: mathematical-optimization.bigb

For a field $k$ and an $(n+1)$-dimensional vector space $V$, the projective space $\mathbb{P}(V)$ consists of its one-dimensional linear subspaces. Choosing a basis writes its points as homogeneous coordinates $[x_0:\cdots :x_n]$, where the coordinate vector is nonzero and simultaneous multiplication by any nonzero scalar leaves the point unchanged.

The affine chart $x_i\ne 0$ has coordinates $x_j/x_i$ for $j\ne i$, identifying it with affine space ${\mathbb{A}}^n$. These $n+1$ charts cover projective space. For the real projective line, $[x:1]$ gives every real affine coordinate, and the remaining point $[1:0]$ records the direction at infinity. On the real projective plane, parallel affine lines meet on the chosen projective line at infinity.

In algebraic geometry, the scheme ${\mathbb{P}}_k^n=\mathrm{Proj}k[x_0,\dots ,x_n]$ also includes nonclosed scheme points. Its $k$-rational points have the preceding description by one-dimensional linear subspaces. This distinction matters over a general field. ^[ref]