aa r X i v : . [ m a t h . AG ] F e b LIMIT POINTS AND ADDITIVE GROUP ACTIONS
IVAN ARZHANTSEV
Abstract.
We show that an effective action of the one-dimensional torus G m on a normalaffine algebraic variety X can be extended to an effective action of a semi-direct product G m ⋌ G a with the same general orbit closures if and only if there is a divisor D on X that consists of G m -fixed points. This result is applied to the study of orbits of theautomorphism group Aut( X ) on X . Introduction
Let X be an irreducible affine variety over an algebraically closed field K of characteristiczero. Assume that X is equipped with an effective action G m × X → X of the one-dimensional algebraic torus G m . If the orbit G m · x of some point x ∈ X is non-closed, thecomplement G m · x \ G m · x consists of one point x called the limit point of the orbit G m · x .Limit points play an important role in the theory of algebraic transformation groups.Let G be a reductive algebraic group acting linearly in a vector space V . A vector v ∈ V iscalled nilpotent if the closure of the orbit G · v contains zero. Equivalently, all homogeneousinvariant polynomials of positive degree on V vanish at v . The Hilbert-Mumford Criterionproved by Hilbert (1893) for the action of the unimodular group in a space of forms and byMumford (1965) in the general situation claims that a vector v ∈ V is nilpotent if and onlyif there is a one-dimensional torus G m ⊆ G such that ∈ G m · v ; see e.g. [16, Theorem 5.1].More generally, Richardson proved that if a reductive algebraic group acts regularlyof an affine variety X and x ∈ X is some point, then there is a one-dimensional torus G m ⊆ G such that the closure G m · x intersects the (unique) closed G -orbit in G · x ; see[16, Theorem 6.9].The study of limit points of a G m -action on a smooth projective variety X and the G m -module structure on the tangent space of a limit point leads to the so-called Bialynicki-Birula decomposition of the variety X into locally closed G m -invariant subsets [8]. Thisdecomposition is used in many problems of Algebraic Geometry, Topology and Represen-tation Theory.One may ask whether the limit point x of a non-closed orbit G m · x on an affine variety X is equivalent to the point x in the sense that x can be sent to x by an automorphismof X . In general, this is not the case at least because the point x can be singular while x is smooth. At the same time, the cases when x is equivalent to x are of particular interest,and they are the object of study of this note.One way to guarantee that x and x are equivalent is to find an algebraic group H actingon X such that the orbit H · x contains both the orbit G m · x and the point x . We aregoing to apply this idea in the case when H is the additive group G a = ( K , +) . Mathematics Subject Classification.
Primary 14J50, 14R20; Secondary 13A50, 14L30.
Key words and phrases.
Affine variety, torus action, additive group action, locally nilpotent derivation.The research was supported by Russian Science Foundation, grant 19-11-00056.
Definition 1.
Let X be an irreducible affine variety with an efffective action of the one-dimensional torus G m . We say that a regular action G a × X → X is compatible if thesubgroup G m normalizes the subgroup G a in the automorphsim group Aut( X ) and generic G a -orbits on X coincide with closures of generic G m -orbits.Equivalently, to construct a compatible G a -action on an affine G m -variety X is the sameas to extend the G m -action to an effective action of a semi-direct product G m ⋌ G a suchthat the generic orbit closures of the extended action coincides with generic orbit closuresof the G m -action.Let us recall that any orbit of a unipotent group action on an affine variety is closed [16,Section 1.3]. So generic G a -orbits on X are closed curves isomorphic to the affine line.Now we come to the main result of this note. Theorem 1.
Let X be a normal affine variety with an effective action of the one-dimensional torus G m . Then there exists a compatible G a -action on X if and only if thevariety X contains a prime divisor D that is fixed by G m pointwise. Theorem 1 allows for several applications. We say that an algebraic variety X is almosthomogeneous if there is a regular action G × X → X of an algebraic group G with an openorbit. Corollary 1.
Let X be a normal affine almost homogeneous variety and D be a primedivisor on X . Assume that there is a non-trivial action of a one-dimensional torus G m on X such that D is fixed by G m pointwise. Then the automorphism group Aut( X ) acts on X with an open orbit and this orbit intersects the divisor D . Let us recall that an affine variety X is called rigid , if X admits no non-trivial action ofthe group G a . It is well known that X is rigid if and only if the algebra K [ X ] admits nonon-zero locally nilpotent derivation [12, Section 1.5.1]. Corollary 2.
Let X be a normal affine variety with an effective action of the one-dimensional torus G m . Assume that there is a prime divisor D on X that is fixed by G m pointwise. Then the variety is not rigid. In particular, if dim X > then the automorphismgroup Aut( X ) is not a (finite dimensional) algebraic group. If an affine variety X admits two actions of the torus G m that do not commute, then X admits a non-trivial G a -action; see [11, Section 3] and [6, Proof of Theorem 2.1]. Thisresult plays a key role in the description of the automorphism group of a rigid variety [6].Theorem 1 is also a result of the same type; it guarantees the existence of a non-trivial G a -action on X provided X admits a G m -action of a specific form.We prove Theorem 1 and Corollaries 1-2 in Section 2. In Section 3 we give examplesthat illustrate Theorem 1. In fact, the case of root subgroups on non-degenerate affine toricvarieties [7, Section 2] was the starting point of this work. We discuss this example in detailand present some generalizations. Acknowledgments.
The author is grateful to Alvaro Liendo for a fruitful discussion.2.
Proofs of main results
Let X be an irreducible algebraic variety equipped with a regular action T × X → X ofan algebraic torus T . Consider a regular action G a × X → X such that the image of G a in the automorphism group Aut( X ) is normalized by the image of T . We say that such an IMIT POINTS AND ADDITIVE GROUP ACTIONS 3 action is vertical if the induced action of the group G a on the field K ( X ) T of rational T -invariants is identical, and is horizontal otherwise. Equivalently, a T -normalized G a -actionon X is vertical if a generic G a -orbit on X is contained in the closure of a T -orbit on X .In turn, horizontal G a -actions are characterized by the condition that generic G a -orbitsare transversal to generic T -orbits. In particular, compatible G a -actions defined in theintroduction are precisely vertical actions on affine varieties for T = G m .Let us consider G m -actions on affine varieties more systematically. There is a naturalcorrespondence between effective G m -actions on an affine variety X and Z -gradings on thealgebra A = K [ X ] , i.e. A = M i ∈ Z A i , such that the set of indices i with A i = 0 generates the group Z . Let us recall that ahomogeneous component A i consists of functions f ∈ K [ X ] such that t · f = t i f for all t ∈ G m . Up to automorphism of the torus G m we may assume that there exists a positive i with A i = 0 .A G m -action on an affine variety X is called elliptic if the algebra of invariants K [ X ] G m coincides with the ground field K . Equivalently, the only closed orbit is a G m -fixed point P and all G m -orbits in X contain P in their closures. It terms of gradings, elliptic actionscorrespond to positive gradings A = ⊕ i > A i of the algebra A = K [ X ] with A = K .Oppositely, an effective G m -action is hyperbolic if generic G m -orbits on X are closed.This means that the grading A = ⊕ i ∈ Z A i includes nonzero components of both positiveand negative degrees. Finally, we say that an effective G m -action on an irreducible affinevariety X is parabolic if there is an open subset U ⊆ X such that all G m -orbits in U arenon-closed in X and their closures in X do not intersect pairwise. Lemma 1.
Let X be a normal affine variety equipped with an effective action of thetorus G m . The following conditions are equivalent. a) The action is parabolic. b) There is a prime divisor D in X that is fixed by G m pointwise. c) The grading has the form A = ⊕ i > A i and the transcendence degree of the subalgebra A = K [ X ] G m equals dim X − .Under these conditions, the divisor D coincides with the set of G m -fixed points on X .Proof. Let us consider the quotient morphism π : X → Y = X// G m := Spec K [ X ] G m . Thismorphism separates closed orbits and sends G m -invariant closed subsets on X to closedsubsets on Y [16, Section 4.4]. Since every fiber of π contains a unique closed orbit, all fibersare connected. By [2, Proposition 4] we conclude that generic fibers of π are irreducible.We show that a) implies b). Since the action is parabolic, generic orbit closures on X are separated by the morphism π . This proves that generic fibers of π are one-dimensionaland dim Y = dim X − . Moreover, the subvariety of G m -fixed points on X projects to Y dominantly. We conclude that some irreducible component D of this subvariety is a primedivisor.Now let us check that b) implies c). Since the G m -action on X is effective we have dim Y < dim X . On the other hand, the divisor D projects to a closed subset of Y and theprojection is injective. This means the restriction of π to D is bijective. In particular, allclosed orbits on X are fixed points and dim Y = dim D = dim X − . The first conditionmeans that A i = 0 for all i < and the second one means that the transcendence degreeof A is dim X − . IVAN ARZHANTSEV
Let us show that c) implies a). Condition c) means that dim Y = dim X − and allclosed orbits in X are fixed points. This implies that generic orbits on X are non-closedand their closures coincide with generic fibers of π . We conclude that generic orbit closuresdo not intersect pairwise, i.e., the action is parabolic.Finally, since the restriction of π to D is bijective, every closed orbit on X is a pointin D . (cid:3) Proof of Theorem 1.
Assume first that there is a compatible G a -action on X . Then closuresof generic G m -orbits on X coincide with G a -orbits and so they do not intersect pairwise.This means that the G m -action is parabolic and implication from a) to b) in Lemma 1concludes the proof.Now let us assume that there is a prime divisor D on X that consists of G m -fixed points.Then we have a grading A = ⊕ i > A i as in Lemma 1, c). Let us show that all weightcomponents A s , s > are nonzero. Assume that some A s = 0 . We localize the algebra A by all elements of the subalgebra B := A . The new algebra C = A ( B ) inherits the grading C = ⊕ i > C i from A , and A s = 0 implies C s = 0 . By Lemma 1, a), elements of B separategeneric G m -orbits on X . This implies that the field K := K ( X ) G m of rational invariantscoincides with the field of fractions of the algebra B [16, Lemma 2.1].By construction, any two homogeneous elements in C of the same degree are proportionalover K . This shows that any homogeneous component in C is at most one-dimensionalover K . Since C is a domain, we conclude that C is a semigroup algebra K [Γ] for somesubsemigroup Γ in ( Z > , +) . By assumption, the algebra A is integrally closed in its field offractions. This implies that the algebra C is integrally closed as well. But then the algebra C is isomorphic to the polynomial algebra K [ T ] , where the element T has degree d > .Since the action of G m on X is effective, we have d = 1 and so C s = 0 for all s > .We are going to construct a locally nilpotent derivation δ of the algebra A that is homo-geneous with respect to the grading and such that Ker δ = B . We start with the algebra C = K [ T ] and the derivation δ ′ = ∂∂T . Let a . . . , a m by homogeneous generators of thealgebra A . Then there is an element b ∈ B such that all elements bδ ′ ( a ) , . . . , bδ ′ ( a m ) arein A . Then the derivation δ = bδ ′ is the required LND of the algebra A . Consider the G a -action of X induced by the group { exp ( sδ ) | s ∈ K } of automorphisms of the algebra A .Such a G a -subgroup is normalized by G m because the derivation δ is homogeneous and the G a -action is vertical because by construction δ annihilates all elements in K ( X ) G m . Weconclude that the constructed G a -action is compatible. (cid:3) Remark . The inverse implication in Theorem 1 should also follow from [14, Corollary 2.8].This approach is based on the technique of proper polyhedral divisors of Altmann andHausen [1]. In this note we prefer to use more elementary arguments given above.
Proof of Corollary 1.
Since X is almost homogeneous, there is a regular action G × X → X of an algebraic group G with an open orbit V . Then the union of shifts s · V , where s runsthrough all automorphisms of X , is an open orbit W of the group Aut( X ) on X . ByTheorem 1, there is a compatible G a -action on X . Since the divisor D coincides with theset of G m -fixed points on X , generic G a -orbits intersect both V and D . This shows thatgeneric points on D lie in the same orbit of Aut( X ) as V . So the orbit W intersects thedivisor D . (cid:3) Proof of Corollary 2.
By Theorem 1, there is a compatible G a -action on X . So the variety X is not rigid. It is well known that if an affine variety X of dimension at least admitsa G a -action, then the automorphism group Aut( X ) is not a (finite dimensional) algebraic IMIT POINTS AND ADDITIVE GROUP ACTIONS 5 group. Let us recall the arguments for this. Let δ be a locally nilpotent derivation on thealgebra K [ X ] that corresponds to the compatible G a -action. The kernel Ker δ is a subalgebraof transcendence degree dim X − ; see [12, Principle 11]. In particular, the kernel Ker δ isan infinite dimensional K -vector space. For any finite dimensional subspace V ⊆ Ker δ onecan consider the subgroup { exp ( f δ ) | f ∈ V } of the automorphism group Aut( X ) . This isa commutative unipotent group of dimension dim V . Since dim V is unbounded, all suchsubgroups can not be contained in a finite dimensional algebraic group. (cid:3) Remark . Both Theorem 1 and Corollaries 1-2 do not hold if the variety X is non-normal:one may consider the cuspidal cubic V ( y − x ) in A with the torus action t · ( x, y ) =( t x, t y ) . 3. Examples and applications
Subtorus actions.
Let N be the lattice of one-parameter subgroups of an algebraictorus T . Denote by M the dual lattice of characters of T , and let h· , ·i : N × M → Z be thepairing between N and M . We consider a regular action of T on a normal affine variety X and the corresponding grading A = M u ∈ M A u of the affine algebra A = K [ X ] . Let N Q = N ⊗ Z Q and M Q = M ⊗ Z Q be the rational vectorspaces generated by N and M , respectively. We define the weight monoid M X of the affine T -variety X as the set of weights u ∈ M such that A u = 0 , and the weight cone ω ( X ) asthe cone in M Q generated by the monoid M X . Since the algebra A is finitely generated,the monoid M X is finitely generated and the cone ω ( X ) is polyhedral.We say that a one-dimensional subtorus G m in T is parabolic if the induced G m -actionon X is parabolic. Proposition 1.
Let X be a normal affine T -variety. Then parabolic subtori in T are inbijection with some facets of the cone ω ( X ) . In particular, the number of parabolic subtoriin T is at most finite.Proof. Observe that a one-parameter subgroup G m in T is nothing but an integer linearfunction l on the space M Q . In these terms, the grading on A corresponding to the G m -action on X is A = M i ∈ Z A i , where A i = M l ( u )= i A u . (1)We conclude that such a grading contains no negative component if and only if the inter-section of ω ( X ) with the hyperplane { l = 0 } is a face of ω ( X ) . For any face τ of the cone ω ( X ) we denote by A ( τ ) the subalgebra ⊕ u ∈ τ A u . If a face τ is a facet of a face λ , then A ( τ ) ⊆ A ( λ ) and the transcendence degree of A ( λ ) exceeds the transcendence degree of A ( τ ) by at least one. So for grading (1) without negative components we may have thatthe transcendence degree of the subalgebra A is dim X − only if A has the form A ( λ ) for some facet λ of the cone ω ( X ) . Lemma 1, c) implies that parabolic one-parametersubgroups in T correspond to some facets of ω ( X ) . (cid:3) Now we come to an important particular case of subtorus actions.
IVAN ARZHANTSEV
Affine toric varieties.
Assume that a torus T acts effectively on a normal affinevariety X with an open orbit, i.e., X is an affine toric T -variety. Equivalently, we have dim A u = 1 for any u ∈ M X . Moreover, in this case the algebra A is the semigroup algebraof the semigroup M X = ω ( X ) ∩ M . We refer to [10, 13, 15] for a general theory of toricvarieties.Under these assumptions, the transcendence degree of every subalgebra A ( λ ) , where λ is a facet of the cone ω ( X ) , equals dim X − . So we have a bijection between parabolicsubtori in T and facets of ω ( X ) .Let σ ( X ) := ω ( X ) ∨ be the cone dual to ω ( X ) in the dual space N Q . Then the primitivelattice vectors p , . . . , p m on the rays of σ ( X ) are precisely the parabolic subgroups in T . It is well know that the rays of σ ( X ) are in bijection with T -invariant prime divisors D , . . . , D m on X ; see, e.g., [10, Theorem 3.2.6]. Under this correspondence, the divisor D i is the set of fixed points of the subgroup p i [10, Proposition 3.2.2].We say that a vector e ∈ M is a Demazure root of the cone σ ( X ) if there is s m such that h p s , e i = − and h p i , e i > for all i = s . So, the set R of all Demazure rootsof the cone σ ( X ) is a disjoint union of subsets R , . . . , R m indexed by the correspondingindex s . One can easily check that if dim X > then every set R s is infinite.It is known that Demazure roots of σ ( X ) are in bijection with G a -subgroups in Aut( X ) normalized by T [14, Theorem 1.6]. Moreover, the G a -subgroup H e corresponding to aroot e ∈ R s acts on X compatibly with the G m -subgroup given by the vector p s ∈ N [7,Proposition 2.1]. Summarizing these results, we obtain the following proposition. Proposition 2.
Let X be an affine toric T -varieties. Then parabolic G m -subgroups in T are in bijection with T -invariant prime divisors, and every T -invariant prime divisor on X is the set of G m -fixed points for the corresponding parabolic subtorus. This shows that an open orbit O of the group Aut( X ) on X intersects any T -invariantprime divisor. In fact, it is known that O coincides with the regular locus of X , and if X is non-degenerate and dim X > , then the group Aut( X ) acts on O infinitely transitively[7, Theorem 2.1]. Remark . On a non-normal affine toric variety X , an open Aut( X ) -orbit may not intersectany T -invariant prime divisor on X ; see [9]. Remark . A description of all orbits of the group
Aut( X ) on an affine toric variety X isobtained in [4, Theorem 5.1 and Proposition 5.6].One should note that if we go from the toric case to an arbitrary torus action on a normalaffine variety, the situation may change completely. Let us consider an example of a torusaction with generic orbits of codimension one. Fix positive integers n , n , n with n + 1 = n + n + n . For each i = 0 , , , fix a tuple l i ∈ Z n i > and define a monomial T l i i := T l i i . . . T l ini in i ∈ K [ T ij ; i = 0 , , , j = 1 , . . . , n i ] . By a trinomial we mean a polynomial of the form f = T l + T l + T l . A trinomialhypersurface X is the zero set { f = 0 } in the affine space A n +1 . One can check that X is a normal affine variety of dimension n . The variety X carries a natural effectiveaction of an ( n − -dimensional torus T . By [3, Lemma 2], there is no vertical G a -actionon X . In particular, there is no T -normalized G a -action on X that is compatible with aone-dimensional subtorus in T . IMIT POINTS AND ADDITIVE GROUP ACTIONS 7
Homogeneous fiber spaces.
We are going to apply the results discussed in theprevious subsection to a wider class of almost homogeneous varieties. Let G be a linearalgebraic group and T be a subtorus in G . We take an affine toric T -variety Y and considerthe homogeneous fiber space X = G ∗ T Y , which is the quotient of the direct product G × Y by the T -action t · ( g, y ) = ( gt − , ty ) ; see [16, Section 4.8] for details. The variety G ∗ T Y projects to the homogeneous space G/T , and all fibers of the projection are isomorphic to Y . The G -action on G × Y by left multiplication on the first factor induces the G -actionon X with an open orbit. Moreover, the T -action on G × Y via the action on the secondfactor descents to a T -action on X . Proposition 3.
Let G be a connected linear algebraic group, T be a subtorus in G , and Y be an affine toric T -variety. Consider the homogeneous fiber space X = G ∗ T Y . Then anopen orbit of the group Aut( X ) on X intersects every G -invariant prime divisor on X .Proof. Every G -invariant prime divisor on X has the form G ∗ T D , where D is a T -invariantprime divisor on Y . The action of a parabolic G m -subgroup on Y that fixes D pointwiseextends to X and has the divisor G ∗ T D as the set of fixed points. So the assertion followsfrom Corollary 1. (cid:3) Affine varieties with two orbits.
Finally, let us show that if one adds a homoge-neous divisor to a homogeneous space then the extended space is still homogeneous.
Proposition 4.
Let G be a reductive group and X be a normal affine G -variety consist-ing of two G -orbits O , O with dim O = dim O + 1 . Then the variety X is Aut( X ) -homogeneous.Proof. Since a normal variety is smooth in codimension one, the variety X is smooth. Let K be the stabilizer of a point in the orbit O . By [16, Theorem 6.7], the variety X isisomorphic to G ∗ K V , where V is a K -module. The condition dim O = dim O + 1 impliesthat V is one-dimensional. So K acts on V by some character. The action of G m on G × V by scalar multiplication on the second factor descents to G ∗ K V , and the divisor O consistsof G m -fixed points. So the assertion follows from Corollary 1. (cid:3) Remark . If the group G is semisimple, a more general version of Proposition 4 is givenin [5, Theorem 5.6]. References [1] Klaus Altmann and Juergen Hausen. Polyhedral divisors and algebraic torus actions. Math. Ann. 334(3) (2006), 557–607[2] Ivan Arzhantsev. On actions of reductive groups with one-parameter family of spherical orbits. Sb.Math. 188 (1997), no. 5, 639-655[3] Ivan Arzhantsev. On rigidity of factorial trinomial hypersurfaces. Internat. J. Algebra Comput. 26(2016), no. 5, 1061-1070[4] Ivan Arzhantsev and Ivan Bazhov. On orbits of the automorphism group on an affine toric variety.Cent. Eur. J. Math. 11 (2013), no. 10, 1713–1724[5] Ivan Arzhantsev, Hubert Flenner, Shulim Kaliman, Frank Kutzschebauch, and Mikhail Zaidenberg.Flexible varieties and automorphism groups. Duke Math. J. 162 (2013), no 4, 767–823[6] Ivan Arzhantsev and Sergey Gaifullin. The automorphism group of a rigid affine variety. Math. Nachr.290 (2017), no. 5-6, 662-671[7] Ivan Arzhantsev, Karine Kuyumzhiyan, and Mikhail Zaidenberg. Flag varieties, toric varieties, andsuspensions: three instances of infinite transitivity. Sb. Math. 203 (2012), no. 7, 923–949[8] Andrzej Bialynicki-Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2) 98 (1973),480–497
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