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An affine algebraic group is a linearly reductive algebraic group if every finite-dimensional rational representation is completely reducible; equivalently the functor of invariants is exact. It is a geometrically reductive algebraic group if, whenever $0\ne v\in V^G$, there is a homogeneous invariant polynomial $F\in k[V{]}^G$ of positive degree with $F(v)\ne 0$. Linear reductivity permits degree one; geometric reductivity allows higher degree.

Over the algebraically closed field, an algebraic torus is $T\cong ({\mathbb{G}}_m{)}^r$. Its coordinate ring is the Laurent polynomial ring. A rational action writes the coaction of a vector as a finite sum $\delta (v)={\sum }_{\chi }{v}_{\chi }\otimes t^{\chi }$. The coaction identity and linear independence of Laurent monomials give $\delta (v_{\chi })=v_{\chi }\otimes t^{\chi }$ and $v={\sum }_{\chi }{v}_{\chi }$. ThusEach weight space has scalar action, so any vector-space complement within each weight space is invariant. This proves complete reducibility, and hence tori are linearly reductive in every characteristic. The argument uses characters themselves, not their Lie algebra differentials, which could coincide modulo $p$.

We now prove geometric reductivity of $G=S{L}_2(k)$ by determinant separation for positive-characteristic SL2. Fix a rational $G$-module $V$ and a nonzero fixed vector $v$. For the diagonal torus, choose a weight-zero functional $l\in V^{\ast }$ with $l(v)=1$, using the torus decomposition just proved. Define matrix coefficientsThese are regular functions on $G$, and $\Phi (hw)(g)=\Phi (w)(h^{-1}g)$ makes $\Phi$ equivariant for left translations. Moreover $\Phi (w)(gt)=\Phi (w)(g)$ because $l$ is torus-invariant. Thus $\Phi$ maps into $k[G{]}^T$ for the right torus action, and $\Phi (v)=1$.

Write a matrix's two columns as $a=(a_0,a_1)$ and $b=(b_0,b_1)$, with $\Delta =a_0{b}_1-a_1{b}_0=1$. Right torus multiplication scales these columns by $t$ and $t^{-1}$. Every right-torus-invariant regular function has a weight-zero polynomial lift, hence a sum of polynomials of bidegrees $(j,j)$ in $(a,b)$. To see that a weight-zero lift exists, take any lift and retain its zero-weight component; the ideal $(\Delta -1)$ is torus-stable and the quotient map preserves weights. Multiply each piece by ${\Delta }^{n-j}$ to put the entire sum in bidegree $(n,n)$ without changing its restriction to $G$.

Let $P_n$ be the space of degree-$n$ polynomials in one column. Restriction gives an injective equivariant map $P_n\otimes P_n\to k[G{]}^T$. Indeed if a bidegree-$(n,n)$ polynomial vanishes on determinant-one matrices, scale the first column of any invertible matrix to make its determinant one. Homogeneity then makes the polynomial vanish on every invertible matrix, a dense open subset, so it is zero. Denote the image by $R_n$. These spaces increase because multiplication by $\Delta$ realizes $R_{n-1}\subset R_n$, and their union is $k[G{]}^T$. The finite-dimensional image $\Phi (V)$ therefore lies in one $R_n$, and in any sufficiently larger filtration piece.

Choose $q=p^s$ with $q-1$ at least that bound, and set $n=q-1$. The identity $(1+z{)}^q=1+z^q$ in characteristic $p$ givesThe invariant tensortherefore defines an invertible equivariant linear map $D:P_n^{\ast }\to P_n$: its matrix is anti-diagonal with all entries nonzero. Interpret any tensor $B\in P_n\otimes P_n$ as a map $P_n^{\ast }\to P_n$ and send it to $B{D}^{-1}\in \mathrm{End}(P_n)$. Equivariance of $D$ makes this identification intertwine the diagonal tensor action with conjugation on endomorphisms. Crucially, $D$ restricts to the constant function $1$ on $G$, and is sent to the identity endomorphism.

Under the resulting equivariant linear map $V\to \mathrm{End}(P_n)$, the vector $v$ is sent to $I$. Since $\dim P_n=n+1=q$, the endomorphism determinant is a homogeneous polynomial of positive degree $q$, invariant under conjugation. Pull it back to $V$. The resulting polynomial satisfiesThis constructs the required invariant for every fixed nonzero vector and proves $S{L}_2(k)$ is geometrically reductive, including characteristic two. No averaging by a group order or division by $q$ is used; those operations would fail in the field's characteristic.

Write $k={\overline{\mathbb{F}}}_q$, and let $F_0$ be the entrywise $q$th-power Frobenius endomorphism of an algebraic group on the general linear group $G={\mathrm{GL}}_2(k)$. The algebraic closure is visible in the original PDF; it is lost in the converted TeX. Let $T_0$ be the diagonal maximal algebraic torus and $B_0$ the upper triangular Borel subgroup.

Choose $\beta \in k$ with $\beta -{\beta }^q=1$. Such a $\beta$ exists because $k$ is an algebraic closure, and $\beta \notin {\mathbb{F}}_q$. PutTransporting the standard ${\mathbb{F}}_q$-model through the algebraic group automorphism $\mathrm{Int}(h_{\pm })$ gives a rational structure on an algebraic group over the same field ${\mathbb{F}}_q$. In particular these are genuine Frobenius endomorphisms of an algebraic group, and their fixed-point groups areThe auxiliary $\beta$ need not be rational for the original structure: it changes the identification with the geometric group, not the field over which the transported model is defined.

For either sign, take the rational maximal torus and Borel subgroupIndeed $F_{\pm }(h_{\pm }H{h}_{\pm }^{-1})=h_{\pm }{F}_0(H)h_{\pm }^{-1}$ for $H=T_0,B_0$, so both subgroups are $F_{\pm }$-stable. Conjugacy also preserves maximality of the algebraic torus and the property of being a Borel subgroup.

Finally,Neither $c_{+}$ nor $c_{-}$ is central, so neither endomorphism is $F_0$. If $F_{+}=F_{-}$, surjectivity of $F_0$ on $G(k)$ would imply that $c_{-}^{-1}{c}_{+}$ is central. Butis not scalar in any characteristic. Thus the two nonstandard rational structures have distinct Frobenius endomorphisms, including when $q=2$.

Choose a rational Borel subgroup $B=TU$ with rational maximal algebraic torus $T$, and use the Lang map $\mathcal{L}(g)=g^{-1}F(g)$. Conjugating the pair $({}^{g}B,F({}^{g}B))$ by $g^{-1}$ gives $(B,{}^{\mathcal{L}(g)}B)$. Part (a) therefore givesMoreover $\mathcal{L}(gb)=b^{-1}\mathcal{L}(g)F(b)$, and $F(B)=B$, so this inverse image is stable under right multiplication by $B$. The flag variety identification consequently restricts to the required isomorphism

The second printed isomorphism is false as written: it omits a unipotent quotient. Take $G={\mathrm{SL}}_2({\overline{\mathbb{F}}}_q)$ with standard Frobenius endomorphism of an algebraic group and $w=1$. Then $X(1)$ consists of the $q+1$ rational lines in ${\mathbb{F}}_q^2$, so has dimension zero. Every Borel subgroup $B^{\prime }$ has one-dimensional unipotent radical $U^{\prime }$. The Lang map is a finite surjective étale morphism, so ${\mathcal{L}}^{-1}(U^{\prime })$ has dimension one. Quotienting by the finite group $T^{\prime F}$ preserves that dimension. Thus no choices of $T^{\prime }$ and $B^{\prime }$ can make the asserted two varieties isomorphic in this example.

Here is the corrected construction, including the missing quotient. Choose $x$ by the Lang theorem for algebraic groups so that $x^{-1}F(x)=\dot{w}$. DefineThen $F(T^{\prime })=x\dot{w}T{\dot{w}}^{-1}{x}^{-1}=T^{\prime }$, so $T^{\prime }$ is a rational maximal torus contained in $B^{\prime }$. Put $J=U_x\cap U^{\prime }=F^{-1}(U^{\prime })\cap U^{\prime }$. The correct unipotent quotient in a Deligne-Lusztig variety isThe semidirect product denotes the subgroup $J{T}^{\prime F}$ acting on the right. Both $U_x$ and $U^{\prime }$ are normalized by $T^{\prime }$, so the action is well defined. Alsofor $h\in {\mathcal{L}}^{-1}(U^{\prime })$, showing directly that the map lands in the Deligne-Lusztig variety.

To prove surjectivity and identify its fibres, use the intermediate varietyThe right $U$-action follows from the twisted-translation identity for the Lang map. Right multiplication by $T^{F_w}$ also acts, and $Y(\dot{w})/T^{F_w}\cong X(w)$. Indeed if $\mathcal{L}(g)\in B\dot{w}B$, write it as $u_{1}t\dot{w}{u}_2$. Replacing $g$ by $ga$, with $a\in T$, changes its torus factor to $a^{-1}t{F}_w(a)$. Surjectivity of the twisted Lang map on the connected algebraic torus makes this factor one. Once the factor is one, two representatives of the same flag differ by $ua$ with $u\in U$ and $a\in T^{F_w}$. This proves both surjectivity and the asserted finite covering, including its group of fibres.

Now map $Z={\mathcal{L}}^{-1}(U^{\prime })$ to $Y(\dot{w})$ by $h\mapsto hxU$. Given $\mathcal{L}(g)=u_1\dot{w}{u}_2$, replace $g$ by $g{u}_1$. ThenThus $h=g{u}_1{x}^{-1}$ belongs to $Z$ and maps to the original point $gU$. This proves surjectivity. Two preimages have the form $h$ and $hk$, where $k\in U_x$. Since $F(k)\in U^{\prime }$ and $\mathcal{L}(h)\in U^{\prime }$, the identityshows that $hk\in Z$ precisely when $k\in U^{\prime }$, hence precisely when $k\in J$. Therefore the fibres are exactly the right $J$-orbits.

These are isomorphisms of geometric quotients, not just bijections on points. The groups $U_x,U^{\prime }$ contain root subgroups for the same maximal algebraic torus, so their intersection $J$ is smooth. The map $Z\to Y(\dot{w})$ is a smooth surjection with fibre $J$: the differential of the Lang map is invertible, and the kernel of the differential of $h\mapsto hxU$ on $Z$ is the translate of $\mathrm{Lie}(U_x)\cap \mathrm{Lie}(U^{\prime })=\mathrm{Lie}(J)$. This gives $Z/J\cong Y(\dot{w})$. Finally $x{T}^{F_w}{x}^{-1}=T^{\prime F}$, so quotienting the remaining finite torus action gives the displayed corrected formula.

For $w=1$, $J=U_x$ is exactly the affine-line factor in the counterexample. More generally $Z/T^{\prime F}\to X(w)$ retains unipotent fibres of dimension $\dim J$; it becomes the printed isomorphism only when this intersection is trivial. The missing factor is also recorded in the quotient construction in Tiep's lectures, proof of Theorem 5.22.

The orbit-cone correspondence is an inclusion-reversing bijectionIf $n=\mathrm{rank}N$, thenThe affine chart and orbit closure in a toric variety areIn particular, the zero cone corresponds to the dense algebraic torus, while maximal cones correspond to torus-fixed points.

The primitive ray generators of $\sigma$ areand $|\det (v_0,v_3)|=3$. They do not form a lattice basis, so the smoothness criterion for a toric variety shows that $U_{\sigma }$ is singular. The presentation from part (i) gives the same conclusion by the Jacobian criterion: all derivatives of $B^3-AC$ vanish at $A=B=C=0$.

Insert the primitive raysLet $\Sigma$ consist of the conesand all their faces. Sinceevery cone of $\Sigma$ is smooth. Its support is $\sigma$, so this fan subdivision induces a proper birational toric morphismThe source is smooth, and $p$ is an isomorphism over the dense algebraic torus; it is therefore a toric resolution of singularities.

The images of the three rays of $\sigma$ areNo cone of the complete fan of ${\mathbb{P}}^2$ contains all three rays, so $\hat{\phi }(\sigma )$ is not contained in one target cone. Thus $\hat{\phi }$ is not a morphism of fans and does not define a regular toric morphism on all of $U_{\sigma }$.

On the dense algebraic torus, the pullbacks of the two standard target characters areUnder the standard torus chart $[u:v:1]\in {\mathbb{P}}^2$, the resulting rational map of toric varieties from a lattice homomorphism isIts homogeneous coordinates vanish simultaneously exactly at the origin. Hence its indeterminacy locus is $\{0\}$.

The Cox construction uses coordinates $x_1,\dots ,x_4$ of degreesIts irrelevant locus isso put $U={\mathbb{A}}^4\setminus (V(x_1,x_3)\cup V(x_2,x_4))$. The algebraic torus $G=({\mathbb{G}}_m{)}^2$ acts byEvery point of $U$ has a nonzero coordinate from each opposing pair, so the action has the expected closed orbits and trivial stabilizers. The geometric quotient is

For a singular projective Weierstrass cubic, its smooth locus of a variety contains the point at infinity and carries the chord-and-tangent group law. The singular point is excluded. Over an algebraic closure, a node gives the multiplicative algebraic group and a cusp gives the additive group. A node whose branches are not defined over the base field gives a nonsplit one-dimensional algebraic torus. This group is distinct from an elliptic curve, whose projective model is smooth.

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]

For the polynomial Cox ring $S=k[z_{\rho }]$ of a normal split toric variety whose toric rays span the ambient real toric lattice space, the toric irrelevant ideal isThe Cox construction uses the open set $\mathrm{Spec}S\setminus V(B_{\Sigma })$. For the projective line, the two maximal toric cones give generators $z_0,z_1$, so the excluded locus is the origin. For affine space, one maximal toric cone contains every toric ray; the empty product is one, hence $B=(1)$ and nothing is excluded.

The ray-span/no-torus-factor hypothesis belongs to this polynomial presentation. A pure algebraic torus has no toric rays and requires its algebraic torus coordinates in addition to the displayed polynomial data.^[ref]

// Target: geometry-and-topology.bigb

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]

The action of the algebraic torus on a toric variety is by automorphisms, and automorphisms preserve regular local rings. The smooth and singular loci are therefore unions of torus orbits.