# Infinitely transitive actions on real affine suspensions

###### Abstract.

A group acts infinitely transitively on a set if for every positive integer , its action is -transitive on . Given a real affine algebraic variety of dimension greater than or equal to , we show that, under a mild restriction, if the special automorphism group of (the group generated by one-parameter unipotent subgroups) is infinitely transitive on each connected component of the smooth locus , then for any real affine suspension over , the special automorphism group of is infinitely transitive on each connected component of . This completes the results of [AKZ10] over .

###### Key words and phrases:

Infinite transitive action, real algebraic variety, suspension## Introduction

In this note the algebraic varieties are affine with ground field of characteristic zero. Let be such a variety and let be a non-constant polynomial function on . Recall that the suspension over along is the hypersurface given by the equation .

The paper [AKZ10] shows that in many situations, the infinite homogeneity of an affine variety induces the infinite homogeneity of its iterated suspensions. Namely, if the special automorphism group of an affine variety of dimension at least two acts infinitely transitively on the smooth locus of and if at every smooth point of the tangent space is spanned by the tangent vectors to the orbits of one-parameter additive subgroups, then every suspension over satisfies the same two properties. The proof in such generality is however valid provided that the ground field is algebraically closed. When the ground field is , it is proved that the same result holds under two restrictions: the smooth locus of is connected and the function is surjective. The aim of this note is to settle the real case for any and any .

Note that the notion of affine suspension was introduced in [KZ99] as a particular instance of an affine modification. The latter is an important tool to understand the structure of birational morphisms between affine varieties, see [D05]. Note also that infinite transitivity, sometimes called very transitivity, was recently studied in the context of real algebraic geometry, see [HM09, HM10, BM10].

## 1. Infinitely transitive actions

We recall the notations and state the main results.

###### Definition 1.

A suspension over an affine variety is a hypersurface given by an equation , where is non-constant. In particular, , and the projection on the first factor induces a natural map .

###### Definition 2.

We say that the action of a group on a set is infinitely transitive on each connected component if for every -tuple , it is transitive on -tuples of the form , where are pairwise distinct.

For an algebraic variety , let the special automorphism group be the subgroup of generated by all of its one-parameter subgroups isomorphic to the additive group . Note that the action of does not mix regular and singular points.

Let be an algebraic variety over . We say that a point is flexible if the tangent space is spanned by the tangent vectors to the orbits of one-parameter subgroups , . The variety is called flexible if every smooth point is.

###### Theorem 1 (Infinite transitivity on each connected component).

Let be an affine algebraic variety defined over and . Assume that for each connected component of , the dimension and is non-constant on .

If is flexible and the action of on is infinitely transitive on each connected component, then the suspension is flexible and acts on infinitely transitively on each connected component.

###### Remark 1.

When is connected and , the result is given by [AKZ10, Theorem 3.3]. Notice that under these conditions, is also connected.

###### Remark 2.

If is non-connected, then is not even one-transitive on . Indeed, the action of on fixes each connected component of : every special automorphism admits a decomposition , where each is a one-parameter additive group. For any , the arc then connects to .

###### Remark 3.

The number of connected components can grow on each step even if we started with a variety whose non-singular part is connected. Indeed, if does not attain zero on one of the connected components, say on , then the set splits into and . We may choose one connected component of the suspension and further perform suspensions over this connected component.

As a preliminary part of Theorem 1, we prove the following theorem.

###### Theorem 2 (Infinite transitivity on one connected component).

Let be an affine algebraic variety defined over and . Assume that contains a flexible connected component of dimension at least such that acts infinitely transitively on and is non-constant. Let be a connected component of the smooth locus of the suspension . Then is flexible and acts infinitely transitively on .

Note that the new paper [AFKKZ10] proves that over an algebraically closed field, an affine variety of dimension is flexible if and only if is infinitely transitive on , and even more, if and only if is one-transitive.

## 2. Affine modifications and lifts of automorphisms

In this section we prove the basic results of the theory over the real numbers. The main part is close to the treatment in [AKZ10, §3].

For every geometrically irreducible algebraic variety over the ground field , there is a natural one-to-one correspondence between locally nilpotent derivations (LND’s) on and algebraic actions of one-parameter subgroups . Namely, given a locally nilpotent derivation , the corresponding action is the exponential . Conversely, for every algebraic action of a subgroup , the derivation along the tangent vector field to the orbits of , given by is an LND, see [F06, § 1.5]. The lemma below shows that the same is true for .

Let be a group. Recall that a -module is rational if each belongs to a finite dimensional -invariant linear subspace and the -action on defines a homomorphism of algebraic groups . A -algebra is an algebra with a structure of -module.

###### Lemma 1.

There are one-to-one correspondences between locally nilpotent derivations of , unipotent subgroups , and structures of rational -algebras on .

###### Proof.

For an LND , the corresponding -algebra is defined by the following formula:

For any fixed there exists an with , so this formula gives us a polynomial in . Hence, belongs to a -invariant linear subspace , which shows that is rational as a -algebra.

Conversely, let be a rational -algebra. Let us define

The main point is to prove that for each some power vanishes. Consider a finite dimensional invariant subspace , . Obviously, preserves and the action of on is unipotent. By the Lie-Kolchin theorem, the action of is upper-triangular in some basis of . This means in particular that for some . Note that the actions of on and of on were originally given by the same matrix, hence, this matrix is nilpotent, and the derivation on is in fact locally nilpotent. ∎

Here is the geometric counterpart of the affine suspensions introduced above. Let be a suspension of given by Definition 1. Consider the cylinder over , where . Then is the blow-up of with center along , which is a particular instance of an affine modification, see [KZ99, Example 1.4].

Let be an LND on and let be the associated -action on . Recall the construction of an LND which lifts to (see [AKZ10, Lemma 3.3] or [KZ99]). Let be the lift of on defined by and consider a product by a polynomial such that . Choosing such that the value of on preserves the relation , we get an LND on which satisfies

(1) | |||||

There is some freedom in the choice of . All the derivations obtained in this way annihilate the function , and the corresponding actions preserve the sections . Notice that can also be considered as the blow-up of , and the lifts of LNDs obtained in this way annihilate the function .

We denote by (resp. ) the subgroup of generated by one-parameter subgroup lifted from (resp. ). Recall the following.

###### Lemma 2.

[AKZ10, Lemma 3.2] Let be an affine variety over a field of characteristic and be a suspension of . Then the restriction of the canonical projection satisfies .

We denote by the decomposition of into connected components. If is not surjective, then the suspension over a connected component of is either connected or consists of two components: if does not attain zero, and neither attain zero, but can be either both negative, either both positive.

For every the hyperplane section will be denoted by . We denote by the -coordinate of a point .

Given distinct constants , we let be the subgroup of fixing pointwise the hypersurfaces , . Observe that, as a subgroup of , the group stabilizes all the levels of the function . Likewise, let be the subgroup of maps inducing the identity on the levels of the function .

###### Lemma 3.

If the action of on is infinitely transitive on each connected component, then for every distinct values , the group acts infinitely transitively on each connected component of . The same is true for the action of on .

###### Proof.

Let and be two -tuples of distinct points of . Since restricts to an isomorphism , we get and for . Moreover, two points belong to the same connected component of if and only if their projections belong to the same connected component of . As a consequence, there exists a special automorphism such that . The special automorphism decomposes into exponentials of LND’s. We lift each of these derivations using the polynomial where is determined by . (Compare [AKZ10, Lemma 3.4].) ∎

###### Lemma 4.

Let be a connected component of of dimension at least two. Then for every continuous function and for each , the level set is infinite.

###### Proof.

(See [AKZ10, Lemma 3.6]) We may assume that is non constant. Choose two points such that and . They can be joined by a smooth path in . There exists a tubular neighborhood of diffeomorphic to a cylinder , where and is a ball of dimension . So there is a continuous family of paths joining and within such that any two of them meet only at their ends and . By continuity, on each of these paths there is a point with . In particular, the level set is infinite. ∎

###### Lemma 5.

Let be an affine variety over and be a suspension of . Let be a connected component of of dimension at least two, such that and be the suspension over . If is flexible and the action of is infinitely transitive on , then for every set of distinct points of there exists a special automorphism such that for all .

###### Proof.

We follow the proof of [AKZ10, Lemma 3.5]. We say that the point is hyperbolic if , i.e. . We have to show that the original collection can be moved by means of a special automorphism so that all the points become hyperbolic. Suppose that are already hyperbolic while is not, where . By recursion, it is sufficient to move off while leaving the points hyperbolic. It is enough to consider the following two cases:

Case 1: , , and

Case 2: .

We claim that there exists an automorphism leaving hyperbolic such that in Case 1 the point is hyperbolic as well, and in Case 2 this point satisfies the assumptions of Case 1.

In Case 1 we divide into several disjoint pieces according to different values of so that , where . Assuming that , where for all , we can choose an extra point . Indeed, since , we have . We have , hence .

By Lemma 3 the subgroup acts -transitively on . Therefore we can send the -tuple to fixing the remaining points of . This confirms our claim in Case 1.

In Case 2 we have . It follows from Lemma 2 that belongs to and in the cotangent space . The variety being flexible, there exists an LND such that . Let be a polynomial in and choose a set of generators of the algebra . Then, as in (2), the derivation can be extended to via

Due to our choice, . Hence the action of the associate one-parameter unipotent subgroup pushes the point out of . So the orbit meets the hypersurface in finitely many points. Similarly, for every the orbit meets in finitely many points. Let . For a general value of the image lies outside for all . Since the group preserves , the points are still hyperbolic. Interchanging the roles of and , we achieve that the assumptions of Case 1 are fulfilled for the new collection , as required. ∎

## 3. Infinite transitivity on one connected component

This section is devoted to the proof of Theorem 2. Recall that is an affine variety defined over , is non-constant and . Let be a connected component of , and let be a connected component of .

###### Lemma 6.

Let be a positive integer and let be points in . There exist an automorphism and a nonzero real number such that for each the number is an interior point of .

Moreover, for any finite sets of real numbers disjoint from and disjoint from , the automorphism can be chosen in .

###### Proof.

Acting with , we may assume that the points have pairwise distinct -coordinates. Acting further with , we may assume that these points have also pairwise distinct values of their -coordinates.

The proof depends on the behavior of . If is an interior point of , we let and choose small enough. Then all are close to , and thus are interior points of .

If is non-bounded and , we let and choose great enough. All are then large enough and are thus interior points of . In the case is non-bounded and , the same argument works.

It remains to consider a bounded function , that is . This splits into two cases: either , or .

Case 1. Without loss of generality we may suppose that . Let and . Consider the connected component of the suspension over such that all , and all . Let and be the maximal and minimal values of .

If , we let . Then for any , it is clear that all real numbers are interior points of .

Otherwise, if , we need a non-trivial automorphism . Fix . Note that, acting with the group , any point can be mapped to a point such that is very close to , while all the other points are fixed. Indeed, for a general , the real number . Let in be such that . We endow with two extra coordinates and get . The point can be mapped to by an element of the group . Thus for a general , the automorphism of satisfies for , and .

Choose such that , . As described above, we map to such that . Then, interchanging and , and interchanging and , there is an element of which maps to . Note that . such that

If for the new set , we repeat this procedure. This process is finite since at each step the product reduces by a factor at least . Finally, we get points such that .

Case 2. One of and equals zero. Using Lemma 5, we map the given -tuple in to points . A sufficiently small then fulfills the required condition.

To prove the second part of the lemma, we run the proof once again but we add an extra condition while performing the lift of an automorphism from to . Namely, for an automorphism in , we multiply the polynomial by , and for an automorphism in , we multiply the polynomial by . ∎

###### Proof of Theorem 2.

Fix two -tuples of distinct points and in . By Lemma 6, up to the action of , there exists such that belongs to and for all . We denote by the distinct values of the -coordinates of the given points. We split the set into subsets according to the -coordinate. For each , the set is infinite by Lemma 4. In particular, for each , we get . By Lemma 3, there exists such that and fixes all the points belonging to . Let us denote by the images by . Since the action of on is infinitely transitive by Lemma 3, there exists a special automorphism mapping the -tuple to . ∎

###### Lemma 7.

If acts infinitely transitively on where , then the flexibility of implies the flexibility of any connected component of ,

###### Proof.

Clearly, is flexible if one point is and if the group acts transitively on . Since the function is non-constant, at some point with . Due to our assumption is flexible. Hence there exist locally nilpotent derivations , where , such that the corresponding vector fields span the tangent space , i.e.

It follows that for some index .

Let now be a point such that . Since , the point is hyperbolic. Make a lift as in (2) with . We obtain LNDs

If we interchange and and let , we get another LND

Let us show that the corresponding vector fields span the tangent space at , as required. We can view as LNDs in preserving the ideal , so that the corresponding vector fields are tangent to the hypersurface

The values of these vector fields at the point yield an -matrix

The first rows of are linearly independent, and the last one is independent from the preceding since . Therefore . So these locally nilpotent vector fields indeed span the tangent space at , as claimed. ∎

## 4. Infinite transitivity on each connected component

The purpose of this section is the proof of Theorem 1. As above, is an affine variety defined over , is non-constant, and . Moreover, we assume that acts infinitely transitively on each connected component of . Recall that denotes the -coordinate of the point .

###### Lemma 8.

Assume that is a flexible variety. For every finite set of points in , there exists an automorphism such that all are pairwise distinct, and that all are pairwise distinct.

###### Proof.

We cannot use the starting argument of the proof of Lemma 6, since several points of may have the same projection in .

We denote the set of projections by . Note that all belong to by Lemma 2. Up to a special automorphism of we can assume that all are pairwise distinct. This is possible since is non-constant on each connected component, and the action of on is infinite transitive on each connected component. Consider the images of under the projection . If , the projections and cannot coincide. Otherwise . If , we get also since the points and are distinct.

Thanks to Lemma 5, keeping if , we may assume that and for each . We split the set into several subsets according to the -coordinate. Let be such that . Using Lemma 3, we act by an element of to get points with pairwise distinct -coordinates. Arguing likewise with -actions, we get points with pairwise distinct - and -coordinates. ∎

###### Proof of Theorem 1.

We denote by the number of connected components of and we suppose that the action of on is infinitely transitive on each connected component. Consider a suspension and decompose into connected components. Recall that over each connected component of there is either one or two connected components of . Fix some integers such that and choose two -tuples and in such that for each , the component contains exactly points of and points of . Let .

By Lemma 8 applied to , we may suppose that the values of the -coordinates are pairwise distinct and that the values of the -coordinates are also pairwise distinct.

We want to choose an -tuple of values such that for all the number is an interior point of . To this end, we repeatedly apply Lemma 6 proceeding on one connected component at each step. Notice that we need to preserve the condition that the values of the -coordinates are pairwise distinct and that the values of the -coordinates are also pairwise distinct. At th step, we let and and we use given by Lemma 6.

Such a choice of provides that for . To control the condition that all -values are pairwise distinct and all -values are pairwise distinct for , we require the following. For each one-parameter subgroup acting in the course of the proof of Lemma 6 (recall that it is non-trivial only for ), the conditions on are

This is true for generic . At each step we get an which fits for all . We may choose the pairwise distinct. At the end, we get a collection as required.

We construct an automorphism mapping to as the product of automorphisms, each of them fixing all the points but one. Since all -values are pairwise distinct, to map , we let where satisfies . If , using the lift defined by (see (2)), we map to . Notice that for , the -values are no longer pairwise distinct.

To map onto , we use, for each , the infinite transitivity of the group , multiplying the corresponding LNDs on by the polynomial where is such that . In this way, for each , we fix the points lying off the th connected component. Finally we get an automorphism which maps