Generalizations of quasielliptic curves

Let $K$ be a ground field of characteristic $p>0$. The goal of this paper is to generalize, in an equivariant way, the rational cuspidal curve

(1.1)

$$X=\operatorname{Spec}K[T^{2},T^{3}]\cup\operatorname{Spec}K[T^{-1}]$$

from the cases $p=2$ and $p=3$, when the automorphism group scheme is non-reduced, to a hierarchy of integral curves $X_{p,n}$ whose automorphism group schemes are likewise non-reduced. Here the index $p>0$ indicates the characteristic, and $p^{n(n+1)/2}$ gives the “size” of non-reducedness.

Our motivation originates from the Enriques classification of algebraic surfaces over ground fields $k=k^{\text{\rm alg}}$: This vast body of theorems on the structure of surfaces $S$ was extended by Bombieri and Mumford to positive characteristics (see [BM77, BM76]). Their main insight was the introduction and analysis of quasielliptic fibrations, which are morphisms $f\colon S\rightarrow B$ whose generic fiber $Y=f^{-1}(\eta)$ is a so-called quasielliptic curve, in other words, a twisted form of (1.1) over the function field $K=k(B)$, with all local rings $\mathscr{O}_{Y,y}$ regular. One knows that such twisted forms exist only over imperfect fields of characteristic $p\leq 3$, and Queen gave explicit equations for them (see [Que71, Que72]), although of rather extrinsic nature.

We discovered the hierarchy $X=X_{p,n}$ somewhat accidentally, while seeking a deeper and more intrinsic understanding of quasielliptic curves. The curves do not reveal themselves in any direct way; one has to understand them through their automorphism group scheme $\operatorname{Aut}_{X/K}$. Its crucial part consists of certain infinitesimal group schemes $U_{n}$ of order $p^{n(n+1)/2}$ acting in a canonical way on the affine line $\mathbb{A}^{1}=\operatorname{Spec}K[T^{-1}]$. The underlying scheme is $\alpha_{p^{n}}\times\alpha_{p^{n-1}}\times\ldots\times\alpha_{p}$, a singleton formed with iterated Frobenius kernels of the additive group, but endowed with a non-commutative group law. Its definition relies on the so-called additive polynomials $\sumop\displaylimits_{i=0}^{n}\lambda_{i}T^{-p^{i}}$, or equivalently the elements of the skew polynomial ring $R[F;\sigma]$, formed over rings with nilpotent elements.

According to Brion’s recent theory of equivariantly normal curves, see [Bri22a], there is a unique compactification $\mathbb{A}^{1}\subset X_{p,n}$ to which the action of $U_{n}$ extends in an optimal way. In general, it is very difficult to unravel the structure of such compactifications, but here we were able to “guess” an explicit description in terms of the numerical semigroups

$${}_{p,n}=\left\langle p^{n},p^{n}-p^{n-1},\ldots,p^{n}-p^{0}\right\rangle\subset\mathbb{N},$$

monoids that each comprise all but finitely many natural numbers. The guesswork was assisted by computer algebra computations with Magma and GAP, performed in a handful of special cases. Our first main result is that the ensuing toric compactification has an intrinsic meaning.

For $3\leq p^{n}\leq 4$, this is precisely the rational cuspidal curve. The second main result unravels the numerical invariants and infinitesimal symmetries of this hierarchy of projective curves.

In this iterated semidirect product, the additive group $\mathbb{G}_{a}$ is normalized by the infinitesimal group scheme $U_{n}$, and both are normalized by the multiplicative group $\mathbb{G}_{m}$. Note that for $3\leq p^{n}\leq 4$, this precisely gives back the computation of Bombieri and Mumford [BM76, Proposition 6], and the above should be seen as a natural generalization.

The computation of the genus relies on general results of Delorme [Del76] on numerical semigroups, applied to our _p,n. The determination of the automorphism group is based on further surprising properties of the projective curves $X=X_{p,n}$: The tangent sheaf ${}_{X/K}=\underline{\operatorname{Hom}}({}^{1}_{X/K},\mathscr{O}_{X})$ turns out to be invertible, actually very ample, giving a canonical inclusion $X\subset\mathbb{P}(\mathfrak{g})=\mathbb{P}^{n+1}$, where $\mathfrak{g}=H^{0}(X,{}_{X/K})$ is the Lie algebra of the automorphism group scheme $G=\operatorname{Aut}_{X/K}$. From the canonical linearization $\mathscr{O}_{X}(1)={}_{X/K}$, we get a matrix representation for $G$, which is crucial to gain control over its structure. Furthermore, $X$ is globally a complete intersection, defined inside $\mathbb{P}^{n+1}$ by the following $n$ homogeneous equations of degree $p$:

$$U_{n-1}^{p}-V^{p-1}Z=0\quad\text{and}\quad U_{j}^{p}-V^{p-1}U_{j+1}=0\quad(0\leq j\leq n-2).$$

Also note that the curves are related by a hierarchy of blowing-ups $X_{p,n-1}=\operatorname{Bl}_{Z}(X_{p,n})$, where the center is the singular point (cf. Lemma 8.3).

Again building on Brion’s theory of equivariantly normal curves, see [Bri22a], we show that our $X=X_{p,n}$ have, over ground fields $K$ with “enough” imperfection, twisted forms $Y$ where all local rings $\mathscr{O}_{Y,y}$ are regular (cf. Theorem 9.4). These have the same structural properties of $X$, except that the singularities get “twisted away.” In turn, the passage from the rational cuspidal curve to quasielliptic curves is generalized to our hierarchy $X=X_{p,n}$.

The above relies on rather general observations, which form the third main result of this paper.

Quasielliptic fibrations play a crucial role in the arithmetic of algebraic surfaces of special type, in particular for K3 surfaces and Enriques surfaces (for an example, see [Sch23b]). We expect that twists over function fields of our $X=X_{p,n}$ play a similar role for surfaces of general type.

By the general theory of non-abelian cohomology and twisted forms, one may view the collection $\operatorname{Twist}(X)$ of isomorphism classes of twisted forms over $S=\operatorname{Spec}(K)$ as non-abelian cohomology $H^{1}(S,\operatorname{Aut}_{X/S})$. For our curves $X=X_{p,n}$, we determined the automorphism group scheme. This raises the question of how to compute non-abelian cohomology for semidirect products in general. We establish effective techniques to do so and are able apply them at least in the case $n\leq 2$. Our fourth main result is as follows.

Perhaps this is the first explicit determination of $\operatorname{Twist}(X)$ via a purely non-abelian cohomological computation of $H^{1}(S,\operatorname{Aut}_{X/S})$ for some relevant hierarchy of schemes $X$. A crucial step in this is the determination of particular twisted forms ${}^{P}\!\mathbb{G}_{a}$ in the form given by Russell [Rus70], a technique likely to be of independent interest.

Let us quote Bombieri and Mumford [BM76, p. 198]: “The study of special low characteristics can be one of two types: amusing or tedious. It all depends on whether the peculiarities encountered are felt to be meaningful variations of the general picture […] or are felt instead to be accidental and random, due for instance to numerological interactions […].” We think that our results amply show that what Bombieri and Mumford have uncovered for $p\leq 3$ is indeed far from accidental, and belong to a structural hierarchy that indeed can be understood from general principles.

The paper is organized as follows: In Section 2 we develop the theory of additive polynomials over rings that contain nilpotents, study the resulting groups of units, and introduce $U_{n}(R)$. The ensuing actions on polynomial rings are discussed in Section 3. Building on these preparations, we give in Section 4 a scheme-theoretic reinterpretation and determine the Lie algebra and the upper and lower central series for the infinitesimal group scheme $U_{n}$. In Section 5 we examine the equivariant compactifications of the affine line $\mathbb{A}^{1}$ and introduce our numerical semigroup _p,n and the ensuing curve $X_{p,n}$, which turns out to be equivariantly normal. We determine the numerical invariants and deduce several crucial geometric consequences in Section 6. In Section 7 we show that our curve can also be seen as a global complete intersection $X_{p,n}\subset\mathbb{P}^{n+1}$. In Section 8 its automorphism group scheme is determined. Section 9 contains general results on the relation between equivariant normality and the existence of twisted forms that are regular, which is then applied to our curves $X_{p,n}$. Section 10 is devoted to twisting and the computation of non-abelian cohomology for semidirect products. We apply this in Section 11 to describe the collection of all twisted forms of $X_{p,n}$ in the cases $n=1$ and $n=2$.

In this section we gather purely algebraic facts that go into the definition of our infinitesimal group scheme $U=U_{n}$ in Section 4. Fix some ring $R$ of characteristic $p>0$, and let $x$ be an indeterminate. Recall that polynomials of the form

$$P(x)=\sumop\displaylimits_{i=0}^{n}\lambda_{i}x^{p^{i}}=\lambda_{0}x+\lambda_{1}x^{p}+\ldots+\lambda_{n}x^{p^{n}}\in R[x]$$

are called additive polynomials. Another widespread designation is $p$-polynomials. Clearly, the set of all such polynomials is stable under addition $P(x)+Q(x)$ and substitution $P(Q(x))$. These two composition laws enjoy the distributive property. In fact, the additive polynomials form an associative ring with respect to these laws, with zero element $P(x)=0$ and unit element $P(x)=x$. Let us call it the ring of additive polynomials.

It can also be seen as the skew polynomial ring $R[F;\sigma]$, where $\sigma\colon R\rightarrow R$ designates the Frobenius map $\lambda\mapsto\lambda^{p}$. Elements are polynomials in the formal symbol $F$, and multiplication is subject to the relations $F\lambda=\lambda^{p}F$. In other words, we have

(2.1)

$$\sumop\displaylimits_{i}\lambda_{i}F^{i}\cdot\sumop\displaylimits_{j}\mu_{j}F^{j}=\sumop\displaylimits_{k}\left(\sumop\displaylimits_{i+j=k}\lambda_{i}\mu_{j}^{p^{i}}\right)F^{k},$$

a modification of the usual Cauchy multiplication. The identification of the skew polynomial ring with the ring of additive polynomials is given by $\lambda\mapsto\lambda x$ and $F\mapsto x^{p}$, so that $\sumop\displaylimits\lambda_{i}F^{i}$ corresponds to $\sumop\displaylimits\lambda_{i}x^{p^{i}}$. For psychological reasons, we strongly prefer to make computations in the skew polynomial ring. In the next section, when it comes to actions on the affine line, we shall turn back to the ring of additive polynomials.

Over ground fields, the ring of additive polynomials was introduced and studied by Ore [Ore33]. A discussion from the perspective of skew polynomial rings was given by Jacobson [Jac43, Chapter 3]. More recent presentations appear in [Gos96, Chapter 1] and [GW04, Chapter 2]. For our purposes, however, it will be crucial to allow nilpotent elements. The following two propositions reveal that nilpotent and invertible elements in $R[F;\sigma]$ are characterized as in usual polynomial rings.

This has the following important consequence.

Given a unit of the form $P=\sumop\displaylimits\lambda_{i}F^{i}$, the inverse $P^{-1}=\sumop\displaylimits\mu_{j}F^{j}$ can be computed as follows: The condition $P\cdot P^{-1}=1$ means $\lambda_{0}\mu_{0}^{p^{0}}=1$ and $\sumop\displaylimits_{i+j=k}\lambda_{i}\mu_{j}^{p^{i}}=0$ for $k\geq 1$, which give the recursion formula

(2.2)

$$\mu_{0}=\lambda_{0}^{-1}\quad\text{and}\quad\mu_{k}=-\frac{1}{\lambda_{0}}\sumop\displaylimits_{i=1}^{k}\lambda_{i}\mu_{k-i}^{p^{i}}\quad(k\geq 1).$$

The skew polynomial ring comes with an infinite-dimensional matrix representation $R[F;\sigma]\rightarrow\operatorname{Mat}_{\infty}(R)$, already determined by the assignments

$$\lambda\longmapsto\begin{pmatrix}\lambda\\ &\lambda^{p}\\ &&\lambda^{p^{2}}\\ &&&\mathinner{\mkern 1.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{$.$}}\mkern 2.0mu\raise 4.0pt\hbox{$.$}\mkern 2.0mu\raise 1.0pt\hbox{$.$}\mkern 1.0mu}\end{pmatrix}\quad\text{and}\quad F\longmapsto\begin{pmatrix}0&1\\ &0&1\\ &&0&1\\ &&&\mathinner{\mkern 1.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{$.$}}\mkern 2.0mu\raise 4.0pt\hbox{$.$}\mkern 2.0mu\raise 1.0pt\hbox{$.$}\mkern 1.0mu}&\mathinner{\mkern 1.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{$.$}}\mkern 2.0mu\raise 4.0pt\hbox{$.$}\mkern 2.0mu\raise 1.0pt\hbox{$.$}\mkern 1.0mu}\end{pmatrix}.$$

More explicitly, this homomorphism is given by

(2.3)

$$\sumop\displaylimits_{i=0}^{n}\lambda_{i}F^{i}\longmapsto\left(\lambda^{p^{r}}_{s-r}\right)_{0\leq r\leq s<\infty}=\begin{pmatrix}\lambda_{0}&\lambda_{1}&\lambda_{2}&\cdots\\ &\lambda_{0}^{p}&\lambda_{1}^{p}&\lambda_{2}^{p}&\cdots\\ &&\lambda_{0}^{p^{2}}&\lambda_{1}^{p^{2}}&\lambda_{2}^{p^{2}}&\cdots\\ &&&\mathinner{\mkern 1.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{$.$}}\mkern 2.0mu\raise 4.0pt\hbox{$.$}\mkern 2.0mu\raise 1.0pt\hbox{$.$}\mkern 1.0mu}&\mathinner{\mkern 1.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{$.$}}\mkern 2.0mu\raise 4.0pt\hbox{$.$}\mkern 2.0mu\raise 1.0pt\hbox{$.$}\mkern 1.0mu}&\mathinner{\mkern 1.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{$.$}}\mkern 2.0mu\raise 4.0pt\hbox{$.$}\mkern 2.0mu\raise 1.0pt\hbox{$.$}\mkern 1.0mu}\end{pmatrix}.$$

Obviously, the map is injective and takes values in the row-finite upper triangular matrices. Note that for each $d\geq 0$, the top left submatrix indexed by $0\leq r,s\leq d-1$ yields a subrepresentation $R[F;\sigma]\rightarrow\operatorname{Mat}_{d}(R)$.

We now examine the unit group $R[F;\sigma]^{\times}$ in more detail, for the time being as an abstract group. It comes with matrix representations $R[F;\sigma]^{\times}\rightarrow\operatorname{GL}_{d}(R)$, $d\geq 0$. Note that this factors over the group of invertible upper triangular matrices $T_{d}(R)\subset\operatorname{GL}_{d}(R)$. Write $\operatorname{Fil}^{d}$ for the kernels. Clearly $\operatorname{Fil}^{0}=R[F;\sigma]^{\times}$, whereas

$$\operatorname{Fil}^{d}=\left\{1+\sumop\displaylimits_{i=d}^{n}\lambda_{i}F^{i}\mid\text{$n\geq d$ and $\lambda_{i}\in\operatorname{Nil}(R)$}\right\}\quad(d\geq 1).$$

These form a descending chain of normal subgroups, in other words, a normal series. Clearly, their intersection contains only the unit element.

The multiplicative character $R[F;\sigma]^{\times}\rightarrow\operatorname{GL}_{1}(R)=R^{\times}$ given by $\sumop\displaylimits\lambda_{i}F^{i}\mapsto\lambda_{0}$ comes with a canonical splitting $\mu\mapsto\mu F^{0}$, so we get a semidirect product $R[F;\sigma]^{\times}=\operatorname{Fil}^{1}\rtimes R^{\times}$. The ensuing conjugacy action of $R^{\times}$ is given by

$$\mu\left(\sumop\displaylimits\lambda_{i}F^{i}\right)\mu^{-1}=\sumop\displaylimits\mu^{1-p^{i}}\lambda_{i}F^{i}.$$

For our applications, it will be important to consider certain smaller subgroups inside the unit group, and the following is crucial throughout.

We keep the set-up as in the previous section. Obviously, the multiplicative monoid of additive polynomials $\sumop\displaylimits\lambda_{i}x^{p^{i}}$ acts on the polynomial ring $R[x]$ via substitution of the indeterminate, in other words by $P(x)\mapsto P(\sumop\displaylimits\lambda_{i}x^{p^{i}})$, and one easily checks that this is an action from the right. In turn, we have a group action $R[x]\times R[F;\sigma]^{\times}\rightarrow R[x]$ from the right, given by

(3.1)

$$Q(x)\ast\sumop\displaylimits\lambda_{i}F^{i}=Q\left(\sumop\displaylimits\lambda_{i}x^{p^{i}}\right).$$

Note that the action of the multiplicative group $\mathbb{G}_{m}(R)=R^{\times}$ via $Q(x)\ast\lambda_{0}=Q(\lambda_{0}x)$ is a special case of this. Furthermore, we have the translation action of the additive group $\mathbb{G}_{a}(R)=R$, defined by

(3.2)

$$Q(x)\ast\alpha=Q(x+\alpha).$$

Obviously, these actions are faithful, and we arrive at inclusions of $\mathbb{G}_{a}(R)$ and $R[F;\sigma]^{\times}$ into the opposite automorphism group of $R[x]$.

In turn, we get an inclusion of $\mathbb{G}_{a}(R)\rtimes R[F;\sigma]^{\times}$ into the opposite automorphism group of $R[x]$. Later, we seek to extend part of this action to certain subrings of $R[x^{-1}]$ in a compatible way. The following observation will be useful: Let $S\subset R[x]$ be the multiplicative system of all monic polynomials. The resulting localization is denoted by $R(x)=S^{-1}R[x]$. Since monic polynomials are regular elements from the polynomial ring, the localization map is injective, and we get an inclusion $R[x]\subset R(x)$.

Fix a ground field $K$ of characteristic $p>0$. In this section we take a more geometric point of view and reinterpret and extend the results of the preceding sections in terms of schemes and group schemes. We now regard

$$U_{n}(R)=U_{n,K}(R)=\left\{1+\sumop\displaylimits_{i=1}^{n}\lambda_{i}F^{i}\in R[T;\sigma]^{\times}\mid\text{ $\lambda_{i}^{p^{n-i+1}}=0$ for $1\leq i\leq n$}\right\}$$

as a group-valued functor $U_{n}$ on the category of $K$-algebras $R$. Clearly, the natural transformation

(4.1)

$$\alpha_{p^{n}}\times\alpha_{p^{n-1}}\times\cdots\times\alpha_{p}\longrightarrow U_{n},\quad(\lambda_{1},\ldots,\lambda_{n})\longmapsto 1+\sumop\displaylimits_{i=1}^{n}\lambda_{i}F^{i}$$

is an isomorphism of set-valued functors, with group laws ignored. In turn, $U_{n}$ is a finite group scheme with coordinate ring $\bigotimesop\displaylimits_{i=1}^{n}K[x_{i}]\,/(x_{i}^{p^{n-i+1}})$ and order $|U_{n}|=h^{0}(\mathscr{O}_{U_{n}})=p^{n(n+1)/2}$. It contains but one point and is thus an infinitesimal group scheme.

One immediately sees that the restriction of (4.1) to $\alpha_{p}^{\oplus n}=\alpha_{p}\times\ldots\times\alpha_{p}$ respects the group laws and gives an inclusion of group schemes $\alpha_{p}^{\oplus n}\subset U_{n}$. Furthermore, for every $m\leq n$, we have canonical inclusions $U_{m}\subset U_{n}$ of group schemes.

Recall that each scheme $X$ over our ground field $K$ comes with a relative Frobenius map $F\colon X\rightarrow X^{(p)}$, given in functorial terms by $X(R)\stackrel{{\scriptstyle F}}{{\rightarrow}}X({}_{F}R)=X^{(p)}(R)$. Here ${}_{F}R$ denotes the abelian group $R$, viewed as an $R$-algebra via the absolute Frobenius map $f\mapsto f^{p}$, and $X^{(p)}=X\otimes_{K}({}_{F}K)$. Note that $R={}_{F}R$ as an $\mathbb{F}_{p}$-algebra. Hence $X(R)=X({}_{F}R)$ and thus $X=X^{(p)}$, provided that $X$ arises as base-change from the prime field $\mathbb{F}_{p}$. For our group scheme $U_{n}$, the relative Frobenius map takes the form

$$U_{n}(R)\longrightarrow U_{n}({}_{F}R)=U_{n}(R),\quad 1+\sumop\displaylimits\lambda_{i}F^{i}\longmapsto 1+\sumop\displaylimits\lambda_{i}^{p}F^{i}.$$

In turn, the relative Frobenius map $F\colon U_{n}\rightarrow U_{n}$ yields an extension

(4.2)

$$0\longrightarrow\alpha_{p}^{\oplus n}\longrightarrow U_{n}\longrightarrow U_{n-1}\longrightarrow 1.$$

By induction on $n\geq 0$, we infer that the finite group scheme $U_{n}$ admits a composition series with quotients isomorphic to $\alpha_{p}$. In particular, $U_{n}$ is unipotent. Since all Lie brackets are trivial, the adjoint representation $\operatorname{ad}\colon\mathfrak{u}_{n}\rightarrow\mathfrak{gl}(\mathfrak{u}_{n})$ of the Lie algebra $\mathfrak{u}_{n}=\operatorname{Lie}(U_{n})$ is trivial, and the adjoint representation $\operatorname{Ad}\colon U_{n}\rightarrow\operatorname{GL}_{\mathfrak{u}_{n}/k}$ of the group scheme factors over the quotient $U_{n-1}$. It is not difficult to determine the latter representation: Since the group $U_{n}(K)$ is trivial, we have

$$\operatorname{Lie}(U_{n})=U_{n}(K[\epsilon])=\left\{1+\epsilon\sumop\displaylimits_{r=1}^{n}\alpha_{r}F^{r}\mid\text{$\alpha_{r}\in K$}\right\},$$

where $\epsilon$ denotes an indeterminate subject to $\epsilon^{2}=0$. The elements $1+\epsilon F^{r}$, $1\leq r\leq n$, form a basis of this $K$-vector space. With $P=\sumop\displaylimits\lambda_{i}F^{i}$, where $\lambda_{0}=1$, and using the relations $\epsilon^{2}=0$ and $F\epsilon=0$, we get

$$P^{-1}\cdot(1+\epsilon F^{s})\cdot P=1+\epsilon F^{s}P=1+\epsilon\sumop\displaylimits_{i=0}^{n-s}\lambda_{i}^{p^{s}}F^{s+i}=\prodop\displaylimits_{i=0}^{n-s}\left(1+\epsilon\lambda_{i}^{p^{s}}F^{s+i}\right).$$

Consequently, $\operatorname{Ad}(P^{-1})$ sends the basis vector $e_{s}=1+\epsilon F^{s}$ to the linear combination $\sumop\displaylimits_{r=s}^{n}\lambda^{p^{s}}_{r-s}e_{r}$. Summing up, in the restricted Lie algebra $\operatorname{Lie}(U_{n})=K^{n}$ all brackets and $p$-powers are zero, and the adjoint representation of the group scheme is given by $(\sumop\displaylimits\lambda_{i}F^{i})^{-1}\mapsto(\lambda^{p^{s}}_{r-s})_{n\geq r\geq s\geq 1}$.

As described in Section 3, the groups $U_{n}(R)$ act from the right on the polynomial ring $R[x]$ via $R$-linear maps. This is obviously functorial in $R$ and thus constitutes an action of the group scheme $U_{n}$ on the affine line $\mathbb{A}^{1}=\operatorname{Spec}K[x]$. Note that this is indeed an action from the left. On $R$-valued points, it is given by

$$(\lambda_{1},\ldots,\lambda_{n})\ast\mu=\sumop\displaylimits_{i=0}^{n}\lambda_{i}F^{i}\ast\mu=\sumop\displaylimits_{i=0}^{n}\lambda_{i}\mu^{p^{i}},$$

where we set $\lambda_{0}=1$ for convenience. Of course, we also have the canonical actions of the multiplicative group $\mathbb{G}_{m}$ and the additive group $\mathbb{G}_{a}$, given via $\lambda_{0}\ast\mu=\lambda_{0}\mu$ and $\alpha\ast\mu=\mu+\alpha$, respectively. The following generalizes a key observation of Bombieri and Mumford [BM76, Proposition 6].

We thus have an iterated semidirect product, for simplicity written as

(4.3)

$$\mathbb{G}_{a}\rtimes U_{n}\rtimes\mathbb{G}_{m}=(\mathbb{G}_{a}\rtimes U_{n})\rtimes\mathbb{G}_{m}=\mathbb{G}_{a}\rtimes(U_{n}\rtimes\mathbb{G}_{m}),$$

acting faithfully on the affine line $\mathbb{A}^{1}=\operatorname{Spec}K[x]$. In turn, we get an inclusion of restricted Lie algebras

$$K\rtimes\operatorname{Lie}(U_{n})\rtimes\mathfrak{gl}_{1}(K)\subset\operatorname{Lie}(\operatorname{Aut}_{\mathbb{A}^{1}/K})=\operatorname{Der}_{K}(K[x]).$$

The elements on the left-hand side can be seen as tuples $(\alpha,\lambda_{1},\ldots,\lambda_{n},\lambda_{0})$ and correspond to the $K$-derivation $\alpha\frac{\partial}{\partial x}+\sumop\displaylimits_{i=1}^{n}\lambda_{i}x^{p^{i}}\frac{\partial}{\partial x}+\lambda_{0}x\frac{\partial}{\partial x}$ of the polynomial ring $K[x]$. For example, the derivation $\lambda_{1}x^{p}\partial/\partial x\in\operatorname{Lie}(U_{n})$ acts via $x\mapsto x+\epsilon\lambda_{1}x^{p}$, which coincides with the action of the group element $1+\epsilon\lambda_{1}F\in U_{n}(k[\epsilon])$.

The spectrum of the function field $K(x)$ comes with a monomorphism

(4.4)

$$\operatorname{Spec}K(x)\longrightarrow\operatorname{Spec}K[x]=\mathbb{A}^{1}.$$

According to Proposition 3.2, there is a unique action on $\operatorname{Spec}K(x)$ that makes the above morphism equivariant.

Let us close this section with some observations on central series. Recall that for a group $G$, the lower central series ${}^{r}={}^{r}G$ and the upper central series $Z_{s}=Z_{s}G$ are inductively defined by

$${}^{0}=G,\quad{}^{r+1}=[G,{}^{r}]\quad\text{and}\quad Z_{0}=\{e\},\quad Z_{s+1}/Z_{s}=Z(G/Z_{s}).$$

The group is nilpotent if ${}^{r}=\{e\}$ for some $r\geq 0$, or equivalently $Z_{s}=G$ for some $s\geq 0$. Then the smallest such integers coincide, and this number $n$ is called the nilpotency class of the group. Note that ${}_{n-r}\subset Z_{r}$, but usually this inclusion is not an equality. We refer to [Hal59, Chapter 10] or [KM79, Chapter 6] for basic facts on nilpotent groups.

For group schemes $G$ of finite type, one has basically the same construction, with sheafification involved. This is straightforward for the higher centers: An $x\in G(R)$ belongs to $Z_{s+1}(R)$ if and only if it commutes with all members of $G(R^{\prime})$ up to elements of $Z_{s}(R^{\prime})$, for all flat extensions $R\subset R^{\prime}$. The situation is more complicated for the ^r because their formation involves schematic images and group scheme closure with respect to the commutator maps $G\times{}^{r}\rightarrow{}^{r+1}$; see [SGA3-1, Exposé $\text{VI}_{B}$, Section 8].

Let us unravel this for our $G=U_{n}$: consider the closed subschemes

(4.5)

$$\{1\}=G_{0}\subset G_{1}\subset\ldots\subset G_{n}=U_{n}$$

defined by $G_{r}(R)=\{1+\sumop\displaylimits_{i=n-r+1}^{n}\lambda_{i}F^{i}\}$.