Geometrically Integral & Normal Varieties

• Geometrically integral, normal varieties are defined as k-schemes remaining reduced and irreducible when base changed to a separable closure.
• Proper polyhedral divisors encode the combinatorial structure of affine varieties with effective torus actions through graded k-algebras.
• Galois semilinear actions enable the descent of non-split torus varieties, linking toric geometry with deeper arithmetic properties.

A geometrically integral geometrically normal variety—a central object in algebraic geometry—refers to a $k$-scheme $X$ for which, after base change to a separable closure $\bar{k}$, the fiber $X \times_k \bar{k}$ is both integral (i.e., reduced and irreducible) and normal (i.e., regular in codimension one and integrally closed in its function field). For affine $X = \mathrm{Spec}\,A$, this property translates to the coordinate ring $A \otimes_k \bar{k}$ being simultaneously an integral domain and a normal domain. The recent algebro-geometric combinatorial classification of such affine varieties, specifically those equipped with an effective action of an algebraic torus over arbitrary fields, utilizes the formalism of proper polyhedral divisors endowed with Galois semilinear action (Martinez-Nunez, 28 Jun 2025).

1. Foundational Definitions

Let $k$ be an arbitrary field and $X$ a $k$-scheme. $X$ is geometrically integral if $X \times_k \bar{k}$ is irreducible and reduced, and geometrically normal if $X \times_k \bar{k}$ is normal. In terms of coordinate rings: $X$ is geometrically integral (resp. normal) over $k$ iff $A \otimes_k \bar{k}$ is an integral domain (resp. a normal domain), where $A$ is the coordinate ring of $X = \mathrm{Spec}\,A$.

2. Proper Polyhedral Divisors

Let $N \cong \mathbb{Z}^n$ be a lattice with dual $M = \mathrm{Hom}_\mathbb{Z}(N, \mathbb{Z})$, $\omega \subset N_\mathbb{Q}$ a pointed rational cone, and $Y$ a normal (semi-)projective $k$-variety. A proper polyhedral divisor (pp-divisor) on $Y$ with tail-cone $\omega$ is a $\mathbb{Q}$-Cartier divisor given by

$\mathcal{D} = \sum_D \Delta_D \otimes D,$

where $D$ runs over prime Cartier divisors on $Y$ and $\Delta_D \in \mathrm{Pol}_\omega^+(N_\mathbb{Q})$ are $\omega$-tailed polyhedra.

For every $m \in \operatorname{relint}(\omega^\vee) \cap M$, the divisor

$$\mathcal{D}(m) := \sum_D h_{\Delta_D}(m)\cdot D \in \mathrm{CaDiv}_\mathbb{Q}(Y)$$

is big: a suitable multiple has a non-empty affine locus. For $m \in \omega^\vee \cap M$, $\mathcal{D}(m)$ is semiample: some multiple is base-point-free. The support function $$h_\Delta(m) = \min_{v \in \Delta} \langle m, v \rangle$$. For $Y$ a curve, notation simplifies to $\mathcal{D} = \sum_P \Delta_P \cdot P$.

Properness guarantees that the $M$-graded $k$-algebra

$$A[Y, \mathcal{D}] := \bigoplus_{m \in \omega^\vee \cap M} H^0(Y, \mathcal{O}_Y(\mathcal{D}(m)))$$

is both finitely generated and normal.

3. Galois Semilinear Actions

Given a finite Galois extension $L/k$ with Galois group $\Gamma = \mathrm{Gal}(L/k)$, a $\Gamma$-semilinear action on a pp-divisor $\mathcal{D}$ over $L$ consists of:

• A $\Gamma$-semilinear action on $Y$, with isomorphisms $\psi_\gamma: \gamma^* Y \to Y$ for $\gamma \in \Gamma$; these cover $$\mathrm{Spec}(\gamma): \mathrm{Spec}\,L \to \mathrm{Spec}\,L$$ and satisfy the cocycle condition $$\psi_{\gamma_1\gamma_2} = \psi_{\gamma_1} \circ \gamma_1^*(\psi_{\gamma_2})$$.
• Linear automorphisms $F_\gamma: N \to N$ preserving $\omega$.
• A "plurifunction" $$f_\gamma \in L(N, Y)^\times = N \otimes_\mathbb{Z} L(Y)^\times$$ whose divisor $$\mathrm{div}(f_\gamma) \in \mathrm{CaDiv}_L(Y, \omega)$$ corrects the pullback: $$\psi_\gamma^*(\mathcal{D}) \leq F_\gamma(\mathcal{D}) + \mathrm{div}(f_\gamma).$$
• The compatibility (cocycle) requirement for $m \in M$: $$f_{\gamma_1\gamma_2}(m) = f_{\gamma_1}(m) \cdot \gamma_1^*\big(f_{\gamma_2}(F_{\gamma_1}^*(m))\big).$$

4. Classification Theorems

4.1 Split Torus Actions

Let $T = \mathrm{Spec}\,k[M]$ be a split torus over $k$. The Altmann–Hausen theorem (over $k$) establishes:

• If $$\mathcal{D} \in \mathrm{PPDiv}_\mathbb{Q}(Y, \omega)$$ is a pp-divisor on a geometrically integral, geometrically normal semiprojective $Y$, then

$X(\mathcal{D}) = \mathrm{Spec}\,A[Y, \mathcal{D}]$

is a geometrically integral, geometrically normal affine $k$-variety with effective $T$-action.

• Every geometrically integral, geometrically normal affine $k$-variety $X$ with effective split-$T$ action arises via $X \cong X(\mathcal{D})$ for a unique pp-divisor $\mathcal{D}$ (up to trivial "plurifunction") on a semiprojective $Y$.

4.2 Galois Descent for Non-Split Torus

Given $L/k$ Galois, split torus $T_L$ over $L$, and $k$-form $T$ over $k$, there is an equivalence of categories between:

• Geometrically integral, geometrically normal affine $k$-varieties with effective $T$-action ($T_L$ splits over $L$).
• $\Gamma$-semilinear pp-divisors $(\mathcal{D}, \{\psi_\gamma, F_\gamma, f_\gamma\})$ on $Y/L$.

In detail:

• A $\Gamma$-semilinear object $(\mathcal{D}, g)$ defines $X = X(\mathcal{D})$ over $L$ with a semilinear $T_L$-action; descent yields $X/k$ with $T$-action and $X_L \cong X(\mathcal{D})$.
• Conversely, any $X/k$ with $T$-action splits over $L$; Altmann–Hausen theory applies to provide a pp-divisor $\mathcal{D}$ over $L$; the Galois action translates to a semilinear action on $\mathcal{D}$.

Key diagnoses: normality of $X(\mathcal{D})$ is equivalent to bigness and semiampleness of $\mathcal{D}(m)$; integrality corresponds to pointedness of cones.

5. Proof Outline and Structural Insights

The classification proceeds in three principal steps:

1. Over an algebraically closed field, normal affine $T$-varieties are classified by pp-divisors $\mathcal{D}$ on semiprojective $Y$ via $X = \mathrm{Spec}\,A[Y, \mathcal{D}]$.
2. For split $T$ over arbitrary $k$, arguments are unchanged and do not require passage to the algebraic closure.
3. For non-split $T$ and Galois descent: a finite Galois splitting field $L$ for $T$ is chosen, yielding $T_L$ split and $X_L$ with its pp-divisor $\mathcal{D}_L$; $X/k$ is recovered from $X_L$ plus the descent data—a $\Gamma$-semilinear equivariant action. Equivariant action on $X_L$ corresponds (via Altmann–Hausen theory) to a semilinear action on $\mathcal{D}_L$, subject to detailed cocycle and pullback checks. Conversely, a semilinear object $(\mathcal{D}_L, g_\gamma)$ allows constructing $X_L = \mathrm{Spec}\,A[Y, \mathcal{D}_L]$, equipping it with compatible $\Gamma$-action, and descending it to $X/k$.

6. Representative Examples

Example	Torus/Action	pp-Divisor Structure
$G_m$ on $\mathbb{A}^2$	$T=G_m$, $\lambda \cdot (x,y) = (\lambda x,\lambda^{-1}y)$	$Y = \mathbb{P}^1$, $\omega = \mathbb{Q}_{\geq 0}$; $$\mathcal{D} = [0,\infty) \otimes \{0\} + [-\infty,0] \otimes \{\infty\}$$
Twisted forms of $\mathbb{A}^n$ over $k$	$T$ twisted form of $G_m^n$	$Y = \mathrm{Spec}\,L$, $\omega \otimes \mathrm{Spec}\,L$; semilinear action on $\omega$ corresponds to twisted toric form
Circle action over $\mathbb{R}$	$T = \mathrm{Res}_{\mathbb{C}/\mathbb{R}}(G_m)$	$Y = \mathrm{Spec}\,\mathbb{C}$, $\Delta = 0$; Galois $m \mapsto -m$ twists to unit-circle real form

In the case of $G_m$ acting on $\mathbb{A}^2$, the construction elucidates the passage from combinatorial data (tail-cones/polyhedra) through divisorial formalism to explicit torus actions, confirming both geometric integrality and normality. For forms of affine spaces and arithmetic tori over nonclosed $k$, the semilinear formalism precisely encodes all possible twisted forms, including those arising in the classical theory of toric varieties and arithmetic tori.

7. Broader Implications and Extensions

This framework generalizes the Altmann–Hausen classification from algebraically closed fields to arbitrary $k$ by simultaneously encoding descent of both the torus structure and the polyhedral divisor data via $\Gamma$-semilinear actions. All classical "forms" of toric varieties—such as arithmetic tori, real circle actions, and other twisted affine spaces—are subsumed as special cases. This suggests deeper connections to arithmetic invariants and descent-theoretic schemes, and provides a combinatorial toolkit for the study of equivariant geometry over general fields. A plausible implication is that further categorical equivalences could extend to more general algebraic group actions, provided suitable combinatorial data and descent conditions are formulated (Martinez-Nunez, 28 Jun 2025).