The hypersurface 𝑥+𝑥²𝑦+𝑧²+𝑡³=0 over a field of arbitrary characteristic

Crachiola, Anthony

doi:10.1090/S0002-9939-05-08171-2

The hypersurface $x + x^2y + z^2 + t^3 = 0$ over a field of arbitrary characteristic
HTML articles powered by AMS MathViewer

by Anthony J. Crachiola PDF

Proc. Amer. Math. Soc. 134 (2006), 1289-1298 Request permission

Abstract:

We develop techniques for computing the AK invariant of domains with arbitrary characteristic. As an example, we show that for any field $\mathbf {k}$ the ring $\mathbf {k}[X,Y,Z,T] / (X + X^2 Y + Z^2 + T^3)$ is not isomorphic to a polynomial ring over $\mathbf {k}$.

References

T. Bandman and L. Makar-Limanov, Affine surfaces with $\textrm {AK}(S)=\Bbb C$, Michigan Math. J. 49 (2001), no. 3, 567–582. MR 1872757, DOI 10.1307/mmj/1012409971
A. Crachiola, On the AK invariant of certain domains, Ph.D. thesis, Wayne State University, 2004.
A. Crachiola and L. Makar-Limanov, On the rigidity of small domains, J. Algebra 284 (2005), no. 1, 1–12. MR 2115001, DOI 10.1016/j.jalgebra.2004.09.015
H. Derksen, More on the hypersurface $x+x^2y+z^2+t^3=0$ in $\mathbf {C}^4$, preprint, 1995, 4 pages.
Harm Derksen, Ofer Hadas, and Leonid Makar-Limanov, Newton polytopes of invariants of additive group actions, J. Pure Appl. Algebra 156 (2001), no. 2-3, 187–197. MR 1808822, DOI 10.1016/S0022-4049(99)00151-6
A. Dubouloz, Generalized Danielewski surfaces, preprint, 2004, 24 pages.
Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. MR 1790619, DOI 10.1007/978-3-0348-8440-2
Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106–110. MR 531454
H. Hasse and F.K. Schmidt, Noch eine Bergründung der Theorie der höheren Differentialquotienten in einem algebraischen Functionenkörper einer Unbestimmten, J. Reine Angew. Math. 177(1937), 215-237.
S. Kaliman, M. Koras, L. Makar-Limanov, and P. Russell, $\mathbf C^*$-actions on $\mathbf C^3$ are linearizable, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 63–71. MR 1464577, DOI 10.1090/S1079-6762-97-00025-5
M. Koras and Peter Russell, Contractible threefolds and $\textbf {C}^*$-actions on $\textbf {C}^3$, J. Algebraic Geom. 6 (1997), no. 4, 671–695. MR 1487230
L. Makar-Limanov, On the hypersurface $x+x^2y+z^2+t^3=0$ in $\textbf {C}^4$ or a $\textbf {C}^3$-like threefold which is not $\textbf {C}^3$. part B, Israel J. Math. 96 (1996), no. part B, 419–429. MR 1433698, DOI 10.1007/BF02937314
Leonid Makar-Limanov, Again $x+x^2y+z^2+t^3=0$, Affine algebraic geometry, Contemp. Math., vol. 369, Amer. Math. Soc., Providence, RI, 2005, pp. 177–182. MR 2126661, DOI 10.1090/conm/369/06811
Masayoshi Miyanishi, $G_{a}$-action of the affine plane, Nagoya Math. J. 41 (1971), 97–100. MR 281719, DOI 10.1017/S0027763000014094
Masayoshi Miyanishi and Tohru Sugie, Affine surfaces containing cylinderlike open sets, J. Math. Kyoto Univ. 20 (1980), no. 1, 11–42. MR 564667, DOI 10.1215/kjm/1250522319
Rudolf Rentschler, Opérations du groupe additif sur le plan affine, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A384–A387 (French). MR 232770
Peter Russell, On affine-ruled rational surfaces, Math. Ann. 255 (1981), no. 3, 287–302. MR 615851, DOI 10.1007/BF01450704