Nonsingular affine 𝑘*-surfaces

Rynes, Jean

doi:10.1090/S0002-9947-1992-1062868-8

Nonsingular affine $k^ *$-surfaces
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Trans. Amer. Math. Soc. 332 (1992), 889-921 Request permission

Abstract:

Nonsingular affine ${k^{\ast } }$-surfaces are classified as certain invariant open subsets of projective ${k^{\ast }}$-surfaces. A graph is defined which is an equivariant isomorphism invariant of an affine ${k^{\ast }}$-surface. Over the complex numbers, it is proved that the only acyclic affine surface which admits an effective action of the group ${{\mathbf {C}}^{\ast } }$ is ${{\mathbf {C}}^2}$ which admits only linear actions of ${{\mathbf {C}}^{\ast }}$.

References

A. Białynicki-Birula, 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 (English, with Russian summary). MR 215831
A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497. MR 366940, DOI 10.2307/1970915
H. Bass and W. Haboush, Linearizing certain reductive group actions, Trans. Amer. Math. Soc. 292 (1985), no. 2, 463–482. MR 808732, DOI 10.1090/S0002-9947-1985-0808732-4

Intersection homology and exceptional orbits of

surfaces

Jacob Eli Goodman, Affine open subsets of algebraic varieties and ample divisors, Ann. of Math. (2) 89 (1969), 160–183. MR 242843, DOI 10.2307/1970814
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977, DOI 10.1007/BFb0067839
Robin Hartshorne, Equivalence relations on algebraic cycles and subvarieties of small codimension, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1973, pp. 129–164. MR 0369359
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
T. Kambayashi and P. Russell, On linearizing algebraic torus actions, J. Pure Appl. Algebra 23 (1982), no. 3, 243–250. MR 644276, DOI 10.1016/0022-4049(82)90100-1
Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984 (German). MR 768181, DOI 10.1007/978-3-322-83813-1
Daniel Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128 (French). MR 254100, DOI 10.24033/bsmf.1675
Domingo Luna, Slices étales, Sur les groupes algébriques, Bull. Soc. Math. France, Paris, Mémoire 33, Soc. Math. France, Paris, 1973, pp. 81–105 (French). MR 0342523, DOI 10.24033/msmf.110
Domingo Luna, Sur les orbites fermées des groupes algébriques réductifs, Invent. Math. 16 (1972), 1–5 (French). MR 294351, DOI 10.1007/BF01391210

Morse theory

John Fogarty, Invariant theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0240104
P. Orlik and P. Wagreich, Algebraic surfaces with $k^*$-action, Acta Math. 138 (1977), no. 1-2, 43–81. MR 460342, DOI 10.1007/BF02392313
Hideyasu Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28. MR 337963, DOI 10.1215/kjm/1250523277