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A lattice basis $v_1,\dots ,v_n$ of a toric lattice $N$ expresses each element uniquely as ${\sum }_i{a}_i{v}_i$ with $a_i\in \mathbb{Z}$. Relative to a fixed integral basis, the new vectors form a lattice basis exactly when the matrix of their coordinates has determinant $\pm 1$.

Being a real basis is weaker. The vectors $(1,0),(1,2)$ are linearly independent over the real numbers, but their integer combinations have even second coordinate and do not fill ${\mathbb{Z}}^2$. Their determinant is two.

A primitive lattice vector is nonzero and generates a saturated sublattice. In coordinates its entries have greatest common divisor one. The Euclidean algorithm completes it to a lattice basis. The nonzero condition excludes the zero vector, whose zero subgroup is saturated but which cannot be a basis vector.^[ref]

In the toric fan construction, a toric cone is a rational polyhedral convex conewith $v_i$ in a toric lattice $N$, satisfying $\sigma \cap (-\sigma )=\{0\}$. The last condition is strong convexity: the toric cone contains no nonzero linear subspace. The zero toric cone is allowed and corresponds to the dense algebraic torus chart.

A simplicial polyhedral cone has linearly independent primitive toric ray generators. For its affine toric variety to be smooth, those generators must extend to a lattice basis. The toric cone generated by $(1,0),(1,2)$ in ${\mathbb{Z}}^2$ is simplicial but not smooth, because the determinant is two rather than $\pm 1$. Thus real linear independence alone does not decide smoothness.^[ref]

David A. Cox coauthored a systematic text on toric varieties with John B. Little and Hal Schenck. The subject connects algebraic geometry with the geometry of polyhedra: convex cones and a fan in toric geometry encode affine pieces and their gluings. The book develops this combinatorial description alongside divisors, homogeneous coordinates and line bundles.^[ref]

Fix a toric lattice $N$ and $N_{\mathbb{R}}=N\otimes \mathbb{R}$. A toric fan $\Sigma$ is a finite collection of rational strongly convex cones in toric geometry such that every face of a polyhedral cone of a member belongs to $\Sigma$, and the intersection of two members is a face of each.

The union of its toric cones is the support of the toric fan. A complete fan has support all of $N_{\mathbb{R}}$. Finiteness gives a finite-type toric variety; an infinite fan in toric geometry keeps the same intersection axioms but need not give a quasi-compact space.

For $N=\mathbb{Z}$, the collection $\{0,{\mathbb{R}}_{\ge 0},{\mathbb{R}}_{\le 0}\}$ is the toric fan of the projective line. Its two affine charts glue by $t\leftrightarrow t^{-1}$ on the dense algebraic torus. The positive half-line with its zero face instead gives the affine line.^[ref]

Hal Schenck coauthored a text on toric varieties with David A. Cox and John B. Little. It develops the bridge between the geometry of polyhedra and algebraic geometry, including the construction of varieties from a fan in toric geometry and the study of their divisors and line bundles. Combinatorial data thereby become tools for geometric calculations.^[ref]

For an integer $k\ge 0$, the Hirzebruch surface is the ruled surfaceIts smooth complete toric toric fan has toric rays $(1,0),(0,1),(-1,k),(0,-1)$. The ruling has projective lines as fibers. Its distinguished negative section of a Hirzebruch surface has self-intersection $-k$; together with a fiber class of a Hirzebruch surface $F$ it gives the relations $F^2=0$ and $S\cdot F=1$.

For $k=0$ the bundle is trivial and ${\mathbb{F}}_0\cong {\mathbb{P}}^1\times {\mathbb{P}}^1$. Inserting the sum of two adjacent toric rays in the toric fan of the projective plane gives ${\mathbb{F}}_1$, the blowup at a point; its exceptional curve is the section of self-intersection minus one. The nonnegative indexing convention avoids calling a positive-self-intersection section negative when changing the sign of the bundle parameter.^[ref]

John B. Little coauthored a text on toric varieties with David A. Cox and Hal Schenck. Its organizing idea is that cones, a fan in toric geometry and lattice data describe substantial parts of the geometry. This provides explicit examples and calculations for algebraic varieties that can otherwise be difficult to visualize or compute with.^[ref]

For a toric divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$ on a normal split toric variety, define the rational polyhedronIts integral points index the torus character basis of global sections of the divisorial sheaf ${\mathcal{O}}_X(D)$. If $D$ is a Cartier divisor, this is an invertible sheaf. For a general Weil divisor it is a rank-one reflexive sheaf.

The polyhedron can be unbounded. For affine space ${\mathbb{A}}^1$ and $D=0$, its inequality is $m\ge 0$, and the integral points give the basis $1,t,t^2,\dots$ of $k[t]$. If the toric fan is complete, the recession toric cone is zero, so a nonempty $P_D$ is bounded. Even then its vertices need not be integral for a non-Cartier Weil divisor. ^[ref]

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 normal split toric variety $X_{\Sigma }$ of dimension $n$, the orbit-cone correspondence assigns an orbit $O(\sigma )$ of the algebraic torus to each toric cone $\sigma \in \Sigma$, withThe zero toric cone gives the dense algebraic torus; a ray of a fan gives a codimension-one orbit. The torus-fixed points correspond exactly to full-dimensional toric cones. A maximal toric cone need not be full-dimensional: the toric fan containing only the zero toric cone gives the algebraic torus itself and has no fixed point when $n>0$.

For affine space ${\mathbb{A}}^2$, the first-quadrant toric fan has a two-dimensional zero-cone orbit $(k^{\ast }{)}^2$, two one-dimensional toric ray orbits on the punctured axes, and one zero-dimensional orbit at the origin. The face-to-orbit reversal also explains the formula for an orbit closure in a toric variety. ^[ref]

A normal affine toric variety $U_{\sigma }$ is smooth exactly when the primitive toric ray generators of $\sigma$ extend to a lattice basis of the ambient toric lattice $N$. In that case a choice of basis givesFor a two-dimensional toric cone in a rank-two toric lattice with primitive generators $u,v$, the criterion becomes $|\det (u,v)|=1$. In higher rank, a rectangular coordinate matrix has no such single determinant test; the generators must span a saturated subgroup of the correct rank. For example $(1,0),(1,2)$ span a real basis but have determinant two, giving a singular chart. This distinguishes smoothness from simpliciality.^[ref]

A star subdivision inserts the toric ray through a nonzero toric lattice vector $v$ lying in the support of a toric fan. For every toric cone containing $v$, replace it by the toric cones generated by $v$ together with its faces that do not contain $v$, retaining all resulting faces and all unaffected toric cones. The support is unchanged, and the induced toric morphism is proper and birational.

For the first quadrant in ${\mathbb{Z}}^2$, inserting $(1,1)$ splits it into the toric cones generated by $(1,0),(1,1)$ and by $(1,1),(0,1)$. Both determinants are one. This is the toric fan of the blowup of affine space at the origin for ${\mathbb{A}}^2$, with a new invariant exceptional projective line. General star subdivisions need not be blowups of smooth centers; the toric blowup along an orbit closure states the special smooth ray-sum case.^[ref]

Every toric ray $\rho$ of a normal toric variety determines a torus-invariant prime Weil divisor $D_{\rho }$. An invariant toric toric divisor is an integer combination $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$.

Such a toric divisor is Cartier exactly when each toric cone $\sigma$ admits a algebraic torus character $m_{\sigma }\in M$ satisfyingThen the corresponding local algebraic torus character sections trivialize its divisor line bundle. Without this integral local condition, the toric divisor gives a divisorial rank-one reflexive sheaf and need not define a line bundle. The positive class generator of $\mathbb{P}(1,1,2)$, for example, has Cartier index two. Thus an arbitrary combination of invariant prime toric divisors must not be silently identified with invertible-sheaf data.^[ref]

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]

A toric surface is a two-dimensional normal split toric variety, described by a toric fan in a rank-two toric lattice. A full-dimensional toric cone is bounded by two primitive toric rays. Its affine chart is smooth exactly when their generators form a lattice basis, equivalently their determinant has absolute value one.

Lower-dimensional toric cones also occur. The zero toric cone has chart ${\mathbb{G}}_m^2$, while a single primitive toric ray with its zero face gives ${\mathbb{A}}^1\times {\mathbb{G}}_m$. Thus consecutive pairs of toric rays describe full-dimensional charts, not every toric surface chart. A complete fan fills the real plane and glues these charts to a proper surface.^[ref]

A normal split toric variety is an algebraic variety containing a split algebraic torus $T$ as a dense open subset, with the multiplication action of $T$ extended to the whole algebraic variety. The toric fan construction gluesfor toric cones of a toric fan, along the open charts corresponding to common faces. Gordan lemma makes each chart finitely generated, and saturation of the toric lattice monoid makes the chart a normal variety.

The first-quadrant toric cone gives affine space ${\mathbb{A}}^n$; the zero toric cone gives the algebraic torus ${\mathbb{G}}_m^n$. The two half-lines in rank one give the projective line. These examples show that a toric algebraic variety need not be complete or have any fixed points.

Some authors allow nonnormal toric algebraic varieties as well. Such algebraic varieties need not arise from ordinary toric fans; the restriction to normal varieties is essential in the stated toric fan classification.^[ref]

A torus-fixed point of a toric variety is a point left unchanged by every element of its acting algebraic torus. In a normal split toric variety $X_{\Sigma }$ of dimension $n$, the orbit-cone correspondence givesConsequently fixed points correspond exactly to full-dimensional cones. Maximality under containment is not enough if a maximal cone has smaller dimension.

For ${\mathbb{A}}^2$ with coordinatewise scaling, the origin is fixed. Points on a punctured coordinate axis have one-dimensional orbits, while points with both coordinates nonzero lie in the dense two-dimensional orbit. For the positive-dimensional torus itself, multiplication has no fixed point.

The fan consisting only of the two positive coordinate rays and zero describes ${\mathbb{A}}^2\setminus \{0\}$. Both rays are maximal cones, but neither is full-dimensional, so there are no fixed points. This example isolates the dimension condition from the informal idea of a maximal face.^[ref]