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Toric geometry studies algebraic varieties with an action of an algebraic torus having a dense open orbit isomorphic to that algebraic torus. For normal algebraic varieties with a split algebraic torus, toric fans encode the geometry using integral toric cones and their intersections.

The simplest chart is the affine line: the multiplicative algebraic torus acts by scaling, with one open orbit and the origin as a fixed point. The projective line adds another fixed point at infinity. Its toric fan has the zero toric cone and the positive and negative half-lines in the real extension of the rank-one toric lattice.

The dictionary turns geometric questions into finite arithmetic data: smoothness criterion for a toric variety is checked by primitive toric ray generators extending to a lattice basis, properness by a complete fan, and line bundles by compatible integral linear functions on toric cones. The normal variety and split algebraic torus conventions matter; an arbitrary algebraic variety with a dense algebraic torus need not be described by the same ordinary toric fan construction.^[ref]

Now take one equation for two unknowns:The linear system $x_1+2x_2=3$ has an entire affine line of exact solutions. Since $A$ has full row rank, its Moore-Penrose inverse isThe kernel is $\mathrm{span}\{(-2,1{)}^T\}$, and all exact solutions areBecause $x_{\ast }$ is orthogonal to $(-2,1{)}^T$,The Euclidean norm is therefore smallest exactly at $t=0$. The zero residual alone cannot select a solution; the minimum-norm rule supplies the additional criterion.

This selection depends on the specified inner product. Rescaling one unknown changes what counts as a small Euclidean norm, so units and weighting matter in an application. A minimum-norm answer is mathematically canonical for the chosen geometry, rather than an automatic guarantee of physical plausibility.

The Zariski closure of a subset $S$ of an algebraic variety or a scheme is the smallest Zariski-closed subset containing $S$. For $S\subseteq k^n$, define the ideal $I(S)$ of all polynomials vanishing on $S$. ThenThis description follows because an intersection of polynomial zero sets is again the common zero set of their equations.^[ref]

Every infinite subset of the complex affine line is Zariski dense: a nonzero one-variable polynomial has only finitely many roots of a polynomial. Thus $S=\{1,1/2,1/3,\dots \}$ has Zariski closure all of $\mathbb{C}$, although its closure in the ordinary Euclidean topology is only $S\cup \{0\}$. In algebraic geometry, taking a closure retains all polynomial consequences, rather than all ordinary sequential limits.