ALGEBRAIC CONES

VLADIMIR L. POPOV^{∗}

Abstract. A characterization of algebraic cones in terms of actions of the onedimensional multiplicative algebraic monoid M_{m} and the algebraic group G_{m} is given.

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 N-grading:

k[X] = ⨁_{d∈N} k[X]_{d},  k[X]_{0} = k.  (1)

Let M_{m} be the multiplicative algebraic monoid whose underlying variety is the affine line A^{1} and the multiplication μ: A^{1} × A^{1} → A^{1} is defined by that in k, i.e., μ^{∗}(T) = T ⊗ T where T is the standard coordinate function on A^{1}. The group of units of M_{m} is M_{m} \ {0} = G_{m}. We put

χ_{d}: G_{m} ⟶ G_{m},  t ↦ t^{d},  (2)

One says that M_{m} acts on a variety Y if a morphism α: M_{m} × Y → Y is given, such that α(g, α(h, y)) = α(gh, y) and α(1, y) = y for all g, h, ∈ M_{m}, y ∈ Y. We write g(y) := α(g, y). The restriction of α to G_{m} × Y is the usual group action of G_{m} on Y. The set M_{m}(y) := {g(y) | g ∈ M_{m}} is called an M_{m}-orbit of y (warning: different M_{m}-orbits may have a nonempty intersection). If M_{m}(y) = y, then y is called a fixed point of the action.

Theorem. Let X be an irreducible algebraic variety. Consider the properties: (i) X is a cone; (ii) there is an action of M_{m} on X with a unique fixed point; (iii) there is an action of G_{m} on X with a fixed point and without other closed

orbits. Then (i)⇒(ii)⇒(iii) and, if X is normal, (iii)⇒(i).

Proof. We may assume that dim X > 0.

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

k[X]^{G_{m}} = k.  (3)

Since k[X]^{G_{m}} separates closed G_{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 G_{m}-stable proper ideals as the zero set of such an ideal would be a proper G_{m}-stable subset of O. Hence, as closedness of O implies surjectivity of the map k[X] → k[O], f ↦ f|_{O}, the G_{m}-stable ideal ⨁_{d>0} k[X]_{d} vanishes on O. Therefore, O is a single point.

Date: August 7, 2009.

∗ Supported by grants ���� �����������, ��������������, and the program Contemporary Problems of Theoretical Mathematics of the Russian Academy of Sciences, Branch of Mathematics.

By the embedding theorem [PV, Theorem 1.5] we may assume that X is a closed G_{m}-stable subset of a finite dimensional G_{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 submodule containing X. Hence we have the embedding V^{∗} ↪ k[X], ℓ ↦ ℓ|_{X}, of G_{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 G_{m}-fixed point in V; in particular, O = 0_{V}. By (1) this also yields that, for every G_{m}-weight χ_{d} of V^{∗}, we have d ∈ N. Hence by (2) there are d_{1}, …, d_{n} ∈ N and a basis e_{1}, …, e_{n} of V such that t(e_{i}) = t^{d_{i}}e_{i} for every t ∈ G_{m} and i. This shows that the action of G_{m} on V extends to the action of M_{m} on V by putting 0(v) = 0_{V} for every v ∈ V. As M_{m} is the closure of G_{m} and X is G_{m}-stable and closed, X is M_{m}-stable as well. By the same reason, 0_{V} is a fixed point of M_{m}. There are no other such points because V^{G_{m}} = {0_{V} }.

(ii)⇒(iii) Assume that M_{m} acts on an irreducible variety X with a unique fixed point x. Restrict this action to G_{m}. Take a point y ∈ X. If the G_{m}-orbit G_{m}(y) is closed in X, then G_{m}(y) = M_{m}(y) since G_{m} is dense in M_{m}.

We claim that then G_{m}(y) is a single point, whence G_{m}(y) = x because of the uniqueness of x. Indeed, assume the contrary, i.e., dim G_{m}(y) = 1. The orbit map φ_{y}: M_{m} → M_{m}(y), g ↦ g(y), is a surjective morphism of one-dimensional smooth varieties such that every fiber is finite and, for every point z ∈ M_{m}(y), z ≠ φ_{y}(0), the cardinality of φ_{y}^{−1}(z) is equal to the order s of the G_{m}-stabilizer of y while the cardinality of φ_{y}^{−1}(φ_{y}(0)) is s + 1. By [G, Sect. 2, Cor. 2] this is impossible, a contradiction. Thus, x is a unique closed G_{m}-orbit.

(iii)⇒(i) Now assume that G_{m} acts on X with a unique fixed point x and without other closed orbits. Then, for every point y ∈ X, y ≠ x, the closure \overline{G_{m}(y)} of the orbit G_{m}(y) is one-dimensional and

x ∈ \overline{G_{m}(y)}.  (4)

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

Thus, X = U, hence X is affine.

As elements of k[X]^{G_{m}} are constant on G_{m}-orbits, (4) implies that f(y) = f(x) for every f ∈ k[X]^{G_{m}} and y ∈ X; whence (3) holds.

Now let k[X]_{d} be the χ_{d}-isotypic component of the G_{m}-module k[V]. Then

k[X] = ⨁_{d∈Z} k[X]_{d}  (5)

is a 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}} ≠ 0 for i = 1, 2. Indeed, assume the contrary. Then there is a point y ∈ X such that k[X]_{d_{i}}, i = 1, 2, is not in the kernel of k[X] → k[\overline{G_{m}(y)}], f ↦ f|\overline{G_{m}(y)}. Hence, for every

i = 1, 2, the χ_{d_{i}}-isotypic component of the G_{m}-module k[\overline{G_{m}(y)}] is nonzero. This implies that there is an integer d ≠ 0 such that the G_{m}-stable maximal ideal {f ∈ k[\overline{G_{m}(y)}] | f(x) = 0\} of k[\overline{G_{m}(y)}] has the nonzero χ_{d}- and χ_{d}^{−1}-isotypic components. Let p and q be the nonzero elements of resp. the first and second of them. Then pq is constant on G_{m}(y) and hence on \overline{G_{m}(y)}. As pq(x) = 0, this means that pq = 0 contrary to the irreducibility of \overline{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 G_{m} on X by g · y := g^{−1}(y), we may assume that the first possibility is realized, i.e., (1) holds. This completes the proof. □

Remark. The following example shows that, in general, without the assumption of normality of X the implications (iii)⇒(i) and (ii)⇒(i) do not hold.

Example. Let X be the image of the morphism

ν: P^{1} ⟶ P^{2}, (a_{0} : a_{1}) ↦ (p^{3} : q^{2}t − p^{3} : q^{3} − qp^{2}), p = a_{1} − a_{0}, q = a_{1} + a_{0}.

X is the projective plane cubic with an ordinary double point O = (1 : 0 : 0) and P^{1} → X, x ↦ ν(x), is the normalization map. Formula α(t, (a_{0} : a_{1})) = (a_{0} : ta_{1}) defines an action of G_{m} on P^{1} that descends to X by means of ν, see [P]. For this action, O is a unique fixed point and X \ O is an orbit. This action extends to the one of M_{m} by putting 0(x) = O for every x ∈ X.

References

[A] K. Adjamagbo, Letter to V. L. Popov, July 9, 2009.

[Bor] A. Borel, Linear Algebraic Groups, 2nd enlarged ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, 1991.

[G] A. Grothendieck, Compléments de géométrie algébrique. Espaces de transformations, in: Séminaire C. Chevalley, 1956–1958. Classification de groupes de Lie algébriques, Vol. 1, Exposé no. 5, Secr. math. ENS, Paris, 1958.

[MF] D. Mumford, J. Fogarty, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 34, Springer-Verlag, Berlin, 1982.

[P] V. L. Popov, Algebraic curves with an infinite automorphism group, Math. Notes 23 (1978), 102–108.

[PV] V. L. Popov, E. B. Vinberg, Invariant Theory, in: Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284.

[S] H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.

Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow, 119991, Russia

   E-mail address: popovvl@orc.ru