Codimension two torus actions on the affine space

Alvaro Liendo; Charlie Petitjean

REVIEW 3 major objections 3 minor 37 references

Codimension two torus actions on the affine space

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A smooth, contractible affine variety with a complexity-two torus action, one fixed point, and a two-dimensional quotient is fixed by its tangent-space representation and two affine lines.

desk verdict Plausible and important classification result, but Step 1 of Theorem 7 has a real gap that needs repair before the paper is publishable. read the letter →

arxiv 2411.14645 v1 pith:XKDIHZBY submitted 2024-11-22 math.AG

Alvaro Liendo , Charlie Petitjean This is my paper · ORCID

classification math.AG MSC 14L3014R0514R1014R20

keywords linearizationconjectureaffinevarietiescontractibletorusactionscomplexitytwopolyhedraldivisorsAltmann-Hausenpresentationexoticspaces

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a classification for the last open class of torus actions relevant to the linearization conjecture: smooth, contractible affine varieties with a faithful complexity-two torus action, exactly one fixed point, and an algebraic quotient isomorphic to the affine plane modulo a finite cyclic group. The claim is that such a variety is completely determined by two pieces of data: the linear action of the torus on the tangent space at the fixed point, and two curves in the quotient plane that are invariant under the cyclic group, pass through the origin, are each isomorphic to the affine line, and meet only with simple normal crossings. This matters because these varieties are the remaining candidates for counterexamples to the linearization conjecture, and the classification reduces the search to a finite combinatorial list while also producing exotic affine spaces.

What carries the argument

The load-bearing mechanism is the Altmann--Hausen presentation of affine $T$-varieties by polyhedral divisors: a variety $X$ is encoded as $X(Y,D)$, where $Y$ is a normal semiprojective quotient and $D$ is a formal sum of rational polyhedra supported on prime divisors of $Y$. The key tools are Lemma 6, which converts fixed points of one-dimensional subtori into intervals of positive length in these polyhedra, and Proposition 5, the authors' earlier smoothness criterion, which forces local étale linearity and hence simple normal crossings. The Abhyankar--Moh--Suzuki theorem supplies the final step identifying the two curves as copies of $\mathbb{A}^1$.

What would settle it

Find a smooth contractible affine variety with a faithful C*-action whose fixed-point locus has at least two connected components under the surrounding contractibility and quotient assumptions; such an example would collapse the contradiction in Step 1. Alternatively, compute the Altmann--Hausen presentation of any variety satisfying the hypotheses and check whether any prime divisor other than the two curves and the exceptional divisors appears in D; such a divisor would refute the claimed normal form.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 7, stated as Theorem 1 in the introduction: for every smooth, contractible affine variety $X$ with a faithful complexity-two torus action, a unique fixed point $x_0$, and algebraic quotient $\mathbb{A}^2/\!/\mu$, there is an equivariant presentation $X = X(Y,D)$ in the Altmann--Hausen sense, with $D = \Delta_1\otimes D_1 + \Delta_2\otimes D_2 + \sum_{i=3}^n \Delta_i\otimes E_i$. The coefficients $\Delta_i$ are computed from the linear $T$-action on the tangent space $T_{x_0}X$; $D_1$ and $D_2$ are the strict transforms of two $\mu$-invariant curves $C_1,C_2 \subset \mathbb{A}^2$ through the origin, each isomorphic to $\mathbb{A}^1$, intersecting only with simple normal crossings. The proof forces this shape by ruling out extra prime divisors, showing the remaining divisor has simple normal crossings, and applying the Abhyankar--Moh--Suzuki theorem to conclude the curves are affine lines. A corollary is that every fully hyperbolic complexity-two action on the affine space is either linearizable or obtained from a linear action by an equivariant bi-cyclic covering, and the paper exhibits an exotic affine fivefold falling into this classification.

Load-bearing premise

In Step 1 of the proof of Theorem 7, when ruling out an extra prime divisor, the argument relies on the unstated fact that the fixed-point locus of a one-dimensional subtorus action on a smooth contractible affine variety must be connected; if this fact fails, the contradiction does not follow.

Editorial extensions

If this is right

If the theorem is correct, the Altmann--Hausen presentation of any such variety is explicitly determined by linear data at the fixed point and by two curves; no other hidden prime divisors can occur.
Corollary 8 follows: every fully hyperbolic complexity-two torus action on affine space is linearizable or an equivariant bi-cyclic cover of a linear action.
The classification produces a smooth, contractible affine fivefold with a unique fixed point that is not isomorphic to affine space (Example 15, with nontrivial Makar-Limanov invariant), so the candidates for linearization counterexamples are genuinely exotic.
Since the exotic fivefold is topologically contractible but not known to be stably affine, the same construction yields potential counterexamples to the Zariski cancellation problem.
The theorem extends the earlier classification of $\mathbb{C}^*$-actions on contractible threefolds to complexity-two torus actions in all dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the two-curve data $C_1,C_2$ should make linearizability algorithmically testable: the action is linear precisely when an origin-preserving automorphism of the plane moves both curves to coordinate axes; this follows from Corollary 8 and Proposition 3 but is not stated as a criterion in the paper.
Editorial inference: the same two-curve data may control finer invariants of the resulting variety; computing the Makar-Limanov invariant as a function of $C_1,C_2$ would tell which members of the family are exotic, a computation the paper only performs on one example.
Editorial inference: the classification highlights an unstated topological fact that deserves explicit proof: the fixed-point locus of a $\mathbb{C}^*$-action on a smooth contractible affine variety is connected. Either this is a known theorem, or the proof of Step 1 needs a replacement argument.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Main take: this is a plausible and potentially important classification result in a genuinely hard area, but the proof as written has a real gap in Step 1 of Theorem 7, plus a few other compressed places. It deserves a serious referee, but it is not ready to be accepted as is.

The genuinely new content is Theorem 7/Theorem 1, extending Koras–Russell and Petitjean's C*-complexity-two results to arbitrary-dimensional torus actions with complexity two. The classification data—the linear action on the tangent space plus two A1 curves in A2/µ with SNC intersections—is natural and the statement is clean. Corollary 8 gives a structural consequence for actions on affine space, and Examples 10–15 illustrate the framework, including an interesting candidate exotic A5 built from the GGP24 preprint. The Altmann–Hausen machinery is appropriate, and the authors' earlier work [LP17] and [P18] is applied as tools, not reverse-engineered. No sign of circularity.

The soft spots are real. Proposition 3's proof is one sentence; identifying Y as a blow-up of A2/µ from the fan of the quotient toric surface needs more detail. The bigger issue is Step 1 of Theorem 7. The authors exclude an extra divisor D0 by choosing a one-dimensional subtorus T′ and invoking Lemma 6 to get T′-fixed points over π(D0), then say the fixed locus is disconnected because D0 avoids an étale neighborhood of the image of x0, contradicting an unstated claim that the fixed locus of a C*-action on a smooth contractible affine variety is connected. That lemma is not stated or cited, and the disconnection claim does not follow from the avoidance premise alone—a connected set can meet a divisor and a point outside a chosen neighborhood. The stress-test note is right. This step needs either a citation of the connectedness result with correct hypotheses or a genuine separation argument, e.g., via Bialynicki-Birula decompositions. Also, Corollary 8's final sentence says more than the proof establishes, and Theorem 1 is stated without a converse, which is fine but should be explicit. Example 15's exoticness rests on a preprint [GGP24], so treat it as conditional.

Bottom line: the main theorem is credible and worth attention, but the write-up is too compressed exactly where it matters. If the authors fill the gap in Step 1 and expand Prop. 3, this becomes a solid paper. I would send it to a knowledgeable referee but expect major revision, not desk rejection.

Referee Report

3 major / 3 minor

Summary. The paper proposes a classification of smooth, contractible affine varieties X of dimension n with a faithful algebraic torus action of complexity two, a unique fixed point, and algebraic quotient isomorphic to A2//mu for a finite cyclic group mu (Theorem 1, stated more precisely as Theorem 7). Using the Altmann–Hausen polyhedral divisor framework, the authors claim that such an X is determined by (a) the linear T-action on the tangent space at the fixed point and (b) two mu-invariant curves C1, C2 in A2, each isomorphic to A1 and meeting only with simple normal crossings, whose strict transforms appear as the divisor components D1, D2 in the polyhedral divisor. The paper also states a corollary that every fully hyperbolic complexity-two torus action on affine space is either linearizable or obtained by an equivariant bi-cyclic cover of a linear action, and it gives several examples, including an explicit candidate exotic space.

Significance. If correct, the classification would be a substantial advance on the linearization conjecture for torus actions, reducing the remaining complexity-two case to a very concrete datum: two A1 curves in A2 with SNC crossings. The use of the Altmann–Hausen machinery is appropriate, and the examples are explicit and useful, including a candidate exotic A4 whose Makar-Limanov invariant is reported to be nontrivial. The paper is not machine-checked and contains no parameter fitting; the main result is a structural theorem. However, the proof of the central theorem has at least one load-bearing gap: the exclusion of extra prime divisors in Step 1 of Theorem 7 depends on an unstated and nontrivial connectedness statement for fixed-point loci of C*-actions. Because that step is essential for the claimed classification, the result cannot be considered established as written.

major comments (3)

[Section 2, Step 1 of the proof of Theorem 7] The exclusion of an additional prime divisor D0 relies on two unstated claims. First, the proof asserts that the fixed-point locus of a C*-action on a smooth contractible affine variety must be connected; neither a proof nor a citation is given at the point of use. This is a nontrivial statement (Smith theory gives acyclicity for finite p-groups, and passing to the full C*-fixed locus needs an additional argument). Second, even if connectedness is granted, the proof has not shown that the T'-fixed points lying over pi(D0) are in a different connected component from the fixed point x0. Lemma 6 only describes where T'-fixed points lie; it says nothing about connected components. The statement 'D0 does not intersect an etale neighborhood of the image of x0' does not imply disconnection: in an affine variety a connected subvariety can meet both a divisor D0 and a neighborhood of a point y0 even when D0 avoids that neighborhood, for example a line through 0 and a point at coordinate 2 in A1. The authors need either a proof or citation of the connectedness lemma and a genuine separation argument, for instance from a Bialynicki-Birula decomposition of the T'-action, or a different way to exclude D0. As written, Step 1 and hence Theorem 7 are unsupported.
[Section 2, Step 2 of the proof of Theorem 7] The proof that the curves D1 and D2 are smooth and form an SNC divisor is too compressed. The sentence 'they must be locally isomorphic to toric curves and all toric curves are smooth' is not a proof: it is not established how Proposition 5 gives local isomorphisms of the divisors themselves to toric curves, and the phrase 'toric curves' is not defined in this context. The intended argument presumably uses the etale local description of smooth T-varieties to show that each prime divisor in the polyhedral divisor is a coordinate hyperplane or an exceptional divisor in a linear model, but this is not spelled out. Since Step 2 is the only place where the SNC condition in Theorem 1(b) is justified, it must be expanded.
[Section 2, Step 3(i) of the proof of Theorem 7] The proof that each Ci is isomorphic to A1 is not sufficiently detailed. The step cites [K93, Theorem A] and asserts that X is obtained by a cyclic cover and is contractible, then claims that the divisors used for the cyclic covering must be Zk-acyclic for almost every k, and that 'Zk-acyclic for almost every k implies that they are, in fact, Zk-acyclic.' Neither the precise statement of [K93, Theorem A] nor the implication 'almost every k' to 'every k' is justified, and the role of contractibility in forcing the branch divisors to have the homology of a point is not explained. Because this is the step that produces the conclusion C1, C2 ≅ A1, the argument needs to be written out with full citations.

minor comments (3)

[Throughout] There are several typos and textual artifacts, including 'dimention' in the Introduction, 'strongly convex' misspelled as 'strongly convex' in Section 1.2, and the corrupted reference 'Bia/suppress lynicki-Birula' in the bibliography.
[Lemma 9] In the proof of Lemma 9, the expression x_i^{a_i q} is used for a rational number q; such an expression is not a well-defined monomial in the coordinate ring. The argument should be formulated using the fact that equality of weight vectors forces the matrix P to have the stated block form without introducing rational powers of coordinates.
[Example 11] The displayed algebra C[v, x1, x2, x3, x4]/(x3x4 - x1x3 - v - v^2) appears to contain a typo: substituting u = x1x2 into the two relations x1x2 = u and x3x4 = u + v + v^2 yields the relation x3x4 - x1x2 - v - v^2, not x3x4 - x1x3 - v - v^2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found; the derivation relies on independent prior theorems, although Theorem 7 Step 1 contains a nontrivial unstated fixed-locus connectedness lemma.

full rationale

The proof of Theorem 7 proceeds from the Altmann–Hausen presentation of affine T-varieties and applies prior results as tools: [LP17, Theorem 7] for étale local smoothness, [P18, Proof of Proposition 6] plus [FZ03, Theorem 4.18] for the one-dimensional torus case in Lemma 6, and external results such as the Abhyankar–Moh–Suzuki theorem and [KPR89, Lemma 5.6] for the structure of the fixed curves. None of these inputs is a restatement of Theorem 1 or of the data (a)–(b) being derived; each has an independent proof and none is fitted or reverse-engineered from the conclusion. The paper therefore does not commit circularity. The only significant issue is in Step 1 of the proof of Theorem 7: the assertion that the T′-fixed locus 'is not connected' and the resulting contradiction rest on an unstated lemma that the fixed locus of a C*-action on a smooth contractible affine variety is connected, and the claimed disconnection is not fully justified. This is a proof gap affecting correctness, but it is not a circularity because it does not reduce the theorem to its hypotheses or to a self-citation chain. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof rests on several substantial theorems from the literature, including the Altmann-Hausen presentation, a smoothness criterion from the authors' prior paper, fixed-point classifications, and quotient classifications. None of these are re-derived here. The only unstated auxiliary assumption identified is the connectedness of fixed-point loci used in Step 1.

assumptions (6)

domain assumption Altmann-Hausen p-divisor presentation: every affine T-variety is Spec of an algebra built from a polyhedral divisor on a semiprojective base.
Section 1.2 invokes [AH06] as the representation theorem on which the entire proof is built.
domain assumption Smoothness criterion from [LP17, Theorem 7]: minimal combinatorial data is locally etale-isomorphic to linear affine-space data.
Proposition 5 uses a prior theorem by the same authors to translate smoothness into local linear models.
domain assumption Fixed-point criterion from [P18, Proposition 6] and [FZ03], extended in Lemma 6 to 1-dimensional subtori.
Lemma 6 relies on this criterion to locate fixed points over divisors.
domain assumption Quotient classification [GKR08]: a 2-dimensional algebraic quotient of affine space by a reductive group is A2//µ with µ cyclic.
The introduction and Prop. 3 use this to fix the shape of the quotient.
domain assumption Abhyankar-Moh-Suzuki theorem [AM75, S74]: smooth A1 curves in A2 are conjugate to coordinate lines.
Step 3 of Theorem 7 uses it to force the curves Ci to be copies of A1.
ad hoc to paper Connectedness of the fixed-point locus of a C*-action on a smooth contractible affine variety.
Theorem 7, Step one uses this to turn non-connected fixed points into a contradiction, but no proof or citation is given.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Codimension two torus actions on the affine space." pith.science (2026). https://pith.science/paper/XKDIHZBY

@misc{pith2026241114645,
  author       = {Pith},
  title        = {Pith review of: Codimension two torus actions on the affine space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XKDIHZBY}},
  note         = {Machine review of arXiv:2411.14645}
}

read the original abstract

In this paper, we classify smooth, contractible affine varieties equipped with faithful torus actions of complexity two, having a unique fixed point and a two-dimensional algebraic quotient isomorphic to a toric blow-up of a toric surface. These varieties are of particular interest as they represent the simplest candidates for potential counterexamples to the linearization conjecture in affine geometry.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 36 canonical work pages

[1]

Abhyankar, Shreeram S.; Moh, Tzuong Tsieng Embeddings of the line in the plane. J. Reine Angew. Math.276(1975), 148–166

work page 1975
[2]

Altmann, Klaus; Hausen, Jürgen Polyhedral divisors and algebraic torus actions. Math. Ann.334(2006), no.3, 557–607

work page 2006
[3]

Linearizing certain reductive group actions

Bass, H.; Haboush, W. Linearizing certain reductive group actions. Trans. Amer. Math. Soc.292(1985), no.2, 463–482

work page 1985
[4]

Remarks on the action of an algebraic torus on k^n

Białynicki-Birula, A. Remarks on the action of an algebraic torus on k^n . II. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.15(1967), 123–125

work page 1967
[5]

Toric varieties

Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties. Grad. Stud. Math., 124 American Mathematical Society, Providence, RI, 2011

work page 2011
[6]

Demazure, Michel Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. (4)3(1970), 507–588

work page 1970
[7]

Flenner, Hubert; Zaidenberg, Mikhail Normal affine surfaces with C ^* -actions. Osaka J. Math.40(2003), no.4, 981–1009

work page 2003
[8]

Second edition Encyclopaedia Math

Freudenburg, Gene Algebraic theory of locally nilpotent derivations. Second edition Encyclopaedia Math. Sci., 136 Invariant Theory Algebr. Transform. Groups, VII Springer-Verlag, Berlin, 2017. xxii+319 pp. ISBN:978-3-662-55348-0 ISBN:978-3-662-55350-3

work page 2017

Show all 37 references

[9]

On embedding of linear hypersurfaces Parnashree Ghosh, Neena Gupta, Ananya Pal https://arxiv.org/abs/2405.07205

arXiv
[10]

The action of an algebraic torus on the affine plane

Gutwirth, A. The action of an algebraic torus on the affine plane. Trans. Amer. Math. Soc.105(1962), 407–414

1962
[11]

V.; Koras, Mariusz; Russell, Peter Two dimensional quotients of C ^ n by a reductive group

Gurjar, R. V.; Koras, Mariusz; Russell, Peter Two dimensional quotients of C ^ n by a reductive group. Electron. Res. Announc. Math. Sci.15(2008), 62–64

2008
[12]

Jelonek, Zbigniew(PL-JAGL) The Jacobian conjecture and the extensions of polynomial embeddings. Math. Ann.294(1992), no.2, 289–293

1992
[13]

Jung, Heinrich W. E. Über ganze birationale Transformationen der Ebene. J. Reine Angew. Math.184(1942), 161–174

1942
[14]

Kaliman, Shulim Smooth contractible hypersurfaces in C ^ n and exotic algebraic structures on C ^ 3 . Math. Z.214(1993), no.3, 499–509

1993
[15]

SKaliman, Shulim; Makar-Limanov, Leonid On the Russell-Koras contractible threefolds. J. Algebraic Geom.6(1997), no.2, 247–268

1997
[16]

C ^* -actions on C ^3 are linearizable.(English summary) Electron

Kaliman, S.; Koras, M.; Makar-Limanov, L.; Russell, P. C ^* -actions on C ^3 are linearizable.(English summary) Electron. Res. Announc. Amer. Math. Soc.3(1997), 63–71

1997
[17]

Affine modifications and affine hypersurfaces with a very transitive automorphism group

Kaliman, Sh.; Zaidenberg, M. Affine modifications and affine hypersurfaces with a very transitive automorphism group. Transform. Groups4(1999), no.1, 53–95

1999
[18]

Automorphism group of a polynomial ring and algebraic group action on an affine space

Kambayashi, T. Automorphism group of a polynomial ring and algebraic group action on an affine space. J. Algebra60(1979), no.2, 439–451

1979
[19]

On linearizing algebraic torus actions

Kambayashi, T.; Russell, P. On linearizing algebraic torus actions. J. Pure Appl. Algebra23(1982), no.3, 243–250

1982
[20]

Knop, Friedrich Nichtlinearisierbare Operationen halbeinfacher Gruppen auf affinen Räumen. Invent. Math.105(1991), no.1, 217–220

1991
[21]

CMS Conf

Koras, Mariusz; Russell, Peter G _ m -actions on A ^ 3 .Proceedings of the 1984 Vancouver conference in algebraic geometry, 269–276. CMS Conf. Proc., 6 Published by the American Mathematical Society, Providence, RI; for the, 1986 ISBN:0-8218-6010-0

1984
[22]

CMS Conf

Koras, Mariusz; Russell, Peter Codimension 2 torus actions on affine n-space.Group actions and invariant theory (Montreal, PQ, 1988), 103–110. CMS Conf. Proc., 10 Published by the American Mathematical Society, Providence, RI; for the, 1989 ISBN:0-8218-6015-1

1988
[23]

CMS Conf

Koras, Mariusz; Russell, Peter On linearizing "good" C ^* -actions on C ^3 .Group actions and invariant theory (Montreal, PQ, 1988), 93–102. CMS Conf. Proc., 10 Published by the American Mathematical Society, Providence, RI; for the, 1989 ISBN:0-8218-6015-1

1988
[24]

Koras, P

M. Koras, P. Russell. Contractible threefolds and C ^ * -actions on C ^ 3 . J. Algebraic Geom. 6 (1997), no. 4, 671- 695

1997
[25]

1994/95 Astérisque(1996), no.237, Exp

Kraft, Hanspeter Challenging problems on affine n-space.(English summary) Séminaire Bourbaki, Vol. 1994/95 Astérisque(1996), no.237, Exp. No. 802, 5, 295–317

1996
[26]

Semisimple group actions on the three-dimensional affine space are linear

Kraft, Hanspeter; Popov, Vladimir L. Semisimple group actions on the three-dimensional affine space are linear. Comment. Math. Helv.60(1985), no.3, 466–479

1985
[27]

Quotient varieties

Kraft, Hanspeter; Petrie, Ted; Randall, John D. Quotient varieties. Adv. Math.74(1989), no.2, 145–162

1989
[28]

Reductive group actions on affine space with one-dimensional quotient.Group actions and invariant theory (Montreal, PQ, 1988), 125–132

Kraft, Hanspeter; Schwarz, Gerald W. Reductive group actions on affine space with one-dimensional quotient.Group actions and invariant theory (Montreal, PQ, 1988), 125–132. CMS Conf. Proc., 10 Published by the American Mathematical Society, Providence, RI; for the, 1989

1988
[29]

Reductive group actions with one-dimensional quotient

Kraft, Hanspeter; Schwarz, Gerald W. Reductive group actions with one-dimensional quotient. Inst. Hautes Études Sci. Publ. Math.(1992), no.76, 1–97

1992
[30]

Algebra490(2017), 204–218

Liendo, Alvaro; Petitjean, Charlie Smooth varieties with torus actions.(English summary) J. Algebra490(2017), 204–218

2017
[31]

On the hypersurface x+x^2y+z^2+t^3=0 in C ^4 or a C ^3 -like threefold which is not C3

Makar-Limanov, L. On the hypersurface x+x^2y+z^2+t^3=0 in C ^4 or a C ^3 -like threefold which is not C3. Israel J. Math.96(1996), 419–429

1996
[32]

Masuda, Mikiya; Moser-Jauslin, Lucy; Petrie, Ted Equivariant algebraic vector bundles over representations of reductive groups: applications. Proc. Nat. Acad. Sci. U.S.A.88(1991), no.20, 9065–9066

1991
[33]

Panyushev, D. I. Semisimple groups of automorphisms of a four-dimensional affine space. Izv. Akad. Nauk SSSR Ser. Mat.47(1983), no.4, 881–894

1983
[34]

Manuscripta Math.156(2018), no.3-4, 399–408

Petitjean, Charlie Smooth contractible threefolds with hyperbolic Gm-actions via polyhedral divisors. Manuscripta Math.156(2018), no.3-4, 399–408

2018
[35]

Petitjean, Charlie Cyclic covers of affine T -varieties. J. Pure Appl. Algebra219(2015), no.9, 4265–4277

2015
[36]

Exotic algebraic group actions

Schwarz, Gerald W. Exotic algebraic group actions. C. R. Acad. Sci. Paris Sér. I Math.309(1989), no.2, 89–94

1989
[37]

Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace C ^2 .(French) J. Math. Soc. Japan26(1974), 241–257

1974

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Abhyankar, Shreeram S.; Moh, Tzuong Tsieng Embeddings of the line in the plane. J. Reine Angew. Math.276(1975), 148–166

work page 1975

[2] [2]

Altmann, Klaus; Hausen, Jürgen Polyhedral divisors and algebraic torus actions. Math. Ann.334(2006), no.3, 557–607

work page 2006

[3] [3]

Linearizing certain reductive group actions

Bass, H.; Haboush, W. Linearizing certain reductive group actions. Trans. Amer. Math. Soc.292(1985), no.2, 463–482

work page 1985

[4] [4]

Remarks on the action of an algebraic torus on k^n

Białynicki-Birula, A. Remarks on the action of an algebraic torus on k^n . II. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.15(1967), 123–125

work page 1967

[5] [5]

Toric varieties

Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties. Grad. Stud. Math., 124 American Mathematical Society, Providence, RI, 2011

work page 2011

[6] [6]

Demazure, Michel Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. (4)3(1970), 507–588

work page 1970

[7] [7]

Flenner, Hubert; Zaidenberg, Mikhail Normal affine surfaces with C ^* -actions. Osaka J. Math.40(2003), no.4, 981–1009

work page 2003

[8] [8]

Second edition Encyclopaedia Math

Freudenburg, Gene Algebraic theory of locally nilpotent derivations. Second edition Encyclopaedia Math. Sci., 136 Invariant Theory Algebr. Transform. Groups, VII Springer-Verlag, Berlin, 2017. xxii+319 pp. ISBN:978-3-662-55348-0 ISBN:978-3-662-55350-3

work page 2017