Algebraic cones

TL;DR

This paper characterizes algebraic cones using actions of the one-dimensional multiplicative algebraic monoid and algebraic group, providing a new perspective on their structure.

Contribution

It introduces a novel characterization of algebraic cones through the actions of specific algebraic monoids and groups.

Findings

Algebraic cones are characterized via monoid and group actions.

Provides a new framework for understanding algebraic cones.

Connects algebraic cones with algebraic monoid and group actions.

Abstract

A characterization of algebraic cones in terms of actions of the one-dimensional multiplicative algebraic monoid ${\bf M}_{\rm m}$ and the algebraic group ${\bf G}_{\rm m}$ are given.

Equations

$$k[X]=\underset{d\in \mathbf{N}}{⨁}k[X{]}_d,k[X{]}_0=k.$$

$${\chi }_d:{\mathbf{G}}_{\mathrm{m}}⟶{\mathbf{G}}_{\mathrm{m}},t\mapsto t^d,$$

$$k[X{]}^{{\mathbf{G}}_{\mathrm{m}}}=k.$$

$$x\in \overline{{\mathbf{G}}_m(y)}.$$

$$k[X]=\underset{d\in \mathbf{Z}}{⨁}k[X{]}_d$$

$$\nu :{\mathbf{P}}^1⟶{\mathbf{P}}^2,(a_0:a_1)\mapsto (p^3:q^{2}t-p^3:q^3-q{p}^2),p=a_1-a_0,q=a_1+a_0.$$

Full text

∗ Supported by grants RFFI 08–01–00095, NSH–1987.2008.1, and the program Contemporary Problems of Theoretical Mathematics of the Russian Academy of Sciences, Branch of Mathematics.

This note answers a question of K. Adjamagbo asked in [A].

Below all algebraic varieties are taken over an algebraically closed field $k$.

An irreducible algebraic variety $X$ is called a cone if $X$ is affine and its coordinate algebra $k[X]$ admits a connected ${\bf N}$-grading:

[TABLE]

Let ${\bf M}_{\rm m}$ be the multiplicative algebraic monoid whose underlying variety is the affine line ${\bf A}^{1}$ and the multiplication $\mu\colon{\bf A}^{1}\times{\bf A}^{1}\to{\bf A}^{1}$ in defined by that in $k$, i.e., $\mu^{*}(T)=T\otimes T$ where $T$ is the standard coordinate function on ${\bf A}^{1}$. The group of units of ${\bf M}_{\rm m}$ is ${\bf M}_{\rm m}\setminus\{0\}={\bf G}_{\rm m}$. We put

[TABLE]

One says that ${\bf M}_{\rm m}$ acts on a variety $Y$ if a morphism $\alpha\colon{\bf M}_{\rm m}\times Y\to Y$ is given, such that $\alpha(g,\alpha(h,y))=\alpha(gh,y)$ and $\alpha(1,y)=y$ for all $g,h,\in{\bf M}_{\rm m}$, $y\in Y$. We write $g(y):=\alpha(g,y)$. The restriction of $\alpha$ to ${\bf G}_{\rm m}\times Y$ is the usual group action of ${\bf G}_{\rm m}$ on $Y$. The set ${\bf M}_{\rm m}(y):=\{g(y)\mid g\in{\bf M}_{\rm m}\}$ is called an ${\bf M}_{m}$-orbit of $y$ (warning: different ${\bf M}_{m}$-orbits may have a nonempty intersection). If ${\bf M}_{\rm m}(y)=y$, then $y$ is called a fixed point of the action.

1. (i)

$X$* is a cone;* 2. (ii)

there is an action of $\;{\bf M}_{\rm m}$ on $X$ with a unique fixed point; 3. (iii)

*there is an action of $\;{\bf G}_{\rm m}$ on $X$ with a fixed point and without other closed orbits. *

Then (i)$\Rightarrow$(ii)$\Rightarrow$(iii) and, if $X$ is normal, (iii)$\Rightarrow$(i).

(i)$\Rightarrow$(ii) Let $X$ be a cone. Consider a grading (1). Then formula $t\cdot f:=t^{d}f$ for $t\in{\bf G}_{m}$, $f\in k[X]_{d}$, defines an action of ${\bf G}_{m}$ on $k[X]$ by $k$-algebra automorphisms. In turn, it defines an (algebraic) action of ${\bf G}_{m}$ on $X$. As grading (1) is connected,

[TABLE]

Since $k[X]^{{\bf G}_{\rm m}}$ separates closed ${\bf G}_{\rm m}$-orbits (see [MF, Cor. A1.3]), from (3) and [Bor, Cor. I.1.8] we deduce that there is a unique such orbit $O$. The coordinate algebra $k[O]$ of $O$ does not contain ${\bf G}_{\rm m}$-stable proper ideals as the zero set of such an ideal would be a proper ${\bf G}_{\rm m}$-stable subset of $O$. Hence, as closedness of $O$ implies surjectivity of the map $k[X]\to k[O]$, $f\mapsto f|_{O}$, the ${\bf G}_{\rm m}$-stable ideal $\bigoplus_{d>0}k[X]_{d}$ vanishes on $O$. Therefore, $O$ is a single point.

By the embedding theorem [PV, Theorem 1.5] we may assume that $X$ is a closed ${\bf G}_{\rm m}$-stable subset of a finite dimensional ${\bf G}_{\rm m}$-module $V$ and $X$ is not contained in its proper submodule. But the set of zeros of all linear functions on $V$ that vanish on $X$ is a ${\bf G}_{\rm m}$-submodule containing $X$. Hence we have the embedding $V^{*}\hookrightarrow k[X]$, $\ell\mapsto\ell|_{X}$, of ${\bf G}_{\rm m}$-modules. As the weights of $V$ are the inverses of that of $V^{*}$, by (3) this yields that the origin $0_{V}$ is the unique ${\bf G}_{\rm m}$-fixed point in $V$; in particular, $O=0_{V}$. By (1) this also yields that, for every ${\bf G}_{\rm m}$-weight $\chi_{d}$ of $V^{*}$, we have $d\in{\bf N}$. Hence by (2) there are $d_{1},\ldots,d_{n}\in{\bf N}$ and a basis $e_{1},\ldots,e_{n}$ of $V$ such that $t(e_{i})=t^{d_{i}}e_{i}$ for every $t\in{\bf G}_{\rm m}$ and $i$. This shows that the action of ${\bf G}_{m}$ on $V$ extends to the action of ${\bf M}_{\rm m}$ on $V$ by putting $0(v)=0_{V}$ for every $v\in V$. As ${\bf M}_{\rm m}$ is the closure of ${\bf G}_{\rm m}$ and $X$ is ${\bf G}_{\rm m}$-stable and closed, $X$ is ${\bf M}_{\rm m}$-stable as well. By the same reason, $0_{V}$ is a fixed point of ${\bf M}_{\rm m}$. There are no other such points because $V^{{\bf G}_{\rm m}}=\{0_{V}\}$.

(ii)$\Rightarrow$(iii) Assume that ${\bf M}_{m}$ acts on an irreducible variety $X$ with a unique fixed point $x$. Restrict this action to ${\bf G}_{m}$. Take a point $y\in X$. If the ${\bf G}_{m}$-orbit ${\bf G}_{m}(y)$ is closed in $X$, then ${\bf G}_{m}(y)={\bf M}_{m}(y)$ since ${\bf G}_{m}$ is dense in ${\bf M}_{m}$.

We claim that then ${\bf G}_{m}(y)$ is a single point, whence ${\bf G}_{m}(y)=x$ because of the uniqueness of $x$. Indeed, assume the contrary, i.e., $\dim{\bf G}_{m}(y)=1$. The orbit map $\varphi_{y}\colon{\bf M}_{\rm m}\to{\bf M}_{m}(y)$,   $g\mapsto g(y)$, is a surjective morphism of one-dimensional smooth varieties such that every fiber is finite and, for every point $z\in{\bf M}_{\rm m}(y)$, $z\neq\varphi_{y}(0)$, the cardinality of $\varphi^{-1}_{y}(z)$ is equal to the order $s$ of the ${\bf G}_{\rm m}$-stabilizer of $y$ while the cardinality of $\varphi^{-1}_{y}(\varphi_{y}(0))$ is $s+1$. By [G, Sect. 2, Cor. 2] this is impossible, a contradiction. Thus, $x$ is a unique closed ${\bf G}_{m}$-orbit.

(iii)$\Rightarrow$(i) Now assume that ${\bf G}_{\rm m}$ acts on $X$ with a unique fixed point $x$ and without other closed orbits. Then, for every point $y\in X$, $y\neq x$, the closure $\overline{{\bf G}_{m}(y)}$ of the orbit ${\bf G}_{m}(y)$ is one-dimensional and

[TABLE]

Assume further that $X$ is normal. Then by Sumihiro’s theorem [S, Cor. 2 of Lemma 8] there is a ${\bf G}_{\rm m}$-stable affine open neighborhood $U$ of $x$. We claim that $X=U$. Indeed, if not, then $X\setminus U$ is a nonempty ${\bf G}_{\rm m}$-stable closed subset in $X$ and (4) is impossible for $y\in X\setminus U$, a contradiction.

Thus, $X=U$, hence $X$ is affine.

As elements of $k[X]^{{\bf G}_{\rm m}}$ are constant on ${\bf G}_{\rm m}$-orbits, (4) implies that $f(y)=f(x)$ for every $f\in k[X]^{{\bf G}_{\rm m}}$ and $y\in X$; whence (3) holds.

Now let $k[X]_{d}$ be the $\chi_{d}$-isotypic component of the ${\bf G}_{\rm m}$-module $k[V]$. Then

[TABLE]

is a ${\bf Z}$-grading of the $k$-algebra $k[X]$. By (3) it is connected, i.e., $k[X]_{0}=k$.

We claim that there are no integers $d_{1}>0$ and $d_{2}<0$ such that $k[X]_{d_{i}}\neq 0$ for $i=1,2$. Indeed, assume the contrary. Then there is a point $y\in X$ such that $k[X]_{d_{i}}$, $i=1,2$, is not in the kernel of $k[X]\to k[\overline{{\bf G}_{m}(y)}]$, $f\mapsto f|_{\overline{{\bf G}_{m}(y)}}$. Hence, for every $i=1,2$, the $\chi_{d_{i}}$-isotypic component of the ${\bf G}_{m}$-module $k[\overline{{\bf G}_{m}(y)}]$ is nonzero. This implies that there is an integer $d\neq 0$ such that the ${\bf G}_{m}$-stable maximal ideal $\{f\in k[\overline{{\bf G}_{m}(y)}]\mid f(x)=0\}$ of $k[\overline{{\bf G}_{m}(y)}]$ has the nonzero $\chi_{d}$- and $\chi^{-1}_{d}$-isotypic components. Let $p$ and $q$ be the nonzero elements of resp. the first and second of them. Then $pq$ is constant on ${\bf G}_{m}(y)$ and hence on $\overline{{\bf G}_{m}(y)}$. As $pq(x)=0$, this means that $pq=0$ contrary to the irreducibility of $\overline{{\bf G}_{m}(y)}$, a contradiction.

Thus, $k[X]_{d}=0$ in (5) either for all negative or for all positive $d$’s. Replacing, if necessary, the action of ${\bf G}_{m}$ on $X$ by $g\cdot y:=g^{-1}(y)$, we may assume that the first possibility is realized, i.e., (1) holds. This completes the proof.  $\square$

[TABLE]

$X$ is the projective plane cubic with an ordinary double point $O=(1:0:0)$ and ${\bf P}^{1}\to X$, $x\mapsto\nu(x)$, is the normalization map. Formula $\alpha(t,(a_{0}:a_{1}))=(a_{0}:ta_{1})$ defines an action of ${\bf G}_{\rm m}$ on ${\bf P}^{1}$ that descends to $X$ by means of $\nu$, see [P]. For this action, $O$ is a unique fixed point and $X\setminus O$ is a ${\bf G}_{\rm m}$-orbit. This action extends to the one of ${\bf M}_{\rm m}$ by putting $0(x)=O$ for every $x\in X$.