The divisor classes of the hypersurface $z^{p^{m}}=G(x_{1},\cdots ,x_{n})$ in characteristic $p>0$
HTML articles powered by AMS MathViewer
- by Jeffrey Lang PDF
- Trans. Amer. Math. Soc. 278 (1983), 613-634 Request permission
Abstract:
In this article we use P. Samuel’s purely inseparable descent techniques to study the divisor class groups of normal affine hypersurfaces of the form ${z^p} = G({x_1},\ldots ,{x_n})$ and develop an inductive procedure for studying those of the form ${z^{p^m}} = G$. We obtain results concerning the order and type of these groups and apply this theory to some specific examples.References
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- Kiyoshi Baba, On $p$-radical descent of higher exponent, Osaka Math. J. 18 (1981), no. 3, 725–748. MR 635730
- Piotr Blass, Zariski surfaces, C. R. Math. Rep. Acad. Sci. Canada 2 (1980/81), no. 1, 31–33. MR 564489 —, Some geometric applications of a differential equation in characteristic $p > 0$ to the theory of algebraic surfaces (to appear).
- Robert M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York-Heidelberg, 1973. MR 0382254, DOI 10.1007/978-3-642-88405-4
- Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106–110. MR 531454
- R. Ganong, Plane Frobenius sandwiches, Proc. Amer. Math. Soc. 84 (1982), no. 4, 474–478. MR 643732, DOI 10.1090/S0002-9939-1982-0643732-1
- Richard Ganong, On plane curves with one place at infinity, J. Reine Angew. Math. 307(308) (1979), 173–193. MR 534219, DOI 10.1515/crll.1979.307-308.173
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725 A. Grothendieck, E.G. A. II, Inst. Hautes Etudes Sci. Publ. Math. No. 8 (1961).
- Nicole Hallier, Quelques propriétés arithmétiques des dérivations, C. R. Acad. Sci. Paris 258 (1964), 6041–6044 (French). MR 171802
- Nicole Hallier, Étude des dérivations de certains corps, C. R. Acad. Sci. Paris 261 (1965), 3716–3718 (French). MR 188211
- Nicole Hallier, Utilisation des groupes de cohomologie dans la théorie de la descente $p$-radicielle, C. R. Acad. Sci. Paris 261 (1965), 3922–3924 (French). MR 197437 —, Quelques propriétiés d’une dérivation particulière, C. R. Acad. Sci. Paris Ser. A 262 (1966), 553-556.
- Nathan Jacobson, Lectures in abstract algebra. Vol III: Theory of fields and Galois theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, 1964. MR 0172871, DOI 10.1007/978-1-4612-9872-4
- Jeffrey Lang, An example related to the affine theorem of Castelnuovo, Michigan Math. J. 28 (1981), no. 3, 375–380. MR 629369 —, The divisor classes of the surface ${z^{p^n}} = G(x,y)$ over fields of characteristic $p > 0$, Ph.D. Thesis, Purdue, 1981.
- Joseph Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151–207. MR 491722, DOI 10.2307/1971141 —, Rational singularities, Inst. Hautes Etudes Sci. Publ. Math. No. 36.
- Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
- Masayoshi Miyanishi, Regular subrings of a polynomial ring, Osaka Math. J. 17 (1980), no. 2, 329–338. MR 587754
- 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 M. Miyanishi and P. Russell, Purely inseparable coverings of the affine plane of exponent one (to appear). P. Russell, Hamburger-Noether expansion and approximate roots (to appear). —, Affine ruled surfaces, Math. Ann. (to appear).
- P. Samuel, Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, No. 30, Tata Institute of Fundamental Research, Bombay, 1964. Notes by M. Pavman Murthy. MR 0214579
- Pierre Samuel, Classes de diviseurs et dérivées logarithmiques, Topology 3 (1964), no. suppl, suppl. 1, 81–96 (French). MR 166213, DOI 10.1016/0040-9383(64)90006-0
- I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917, DOI 10.1007/978-3-642-96200-4
- Balwant Singh, On a conjecture of Samuel, Math. Z. 105 (1968), 157–159. MR 228478, DOI 10.1007/BF01110441
- Shuen Yuan, On logarithmic derivatives, Bull. Soc. Math. France 96 (1968), 41–52. MR 237482, DOI 10.24033/bsmf.1659 O. Zariski, On Castelnuovo’s criterion of rationality ${p_a} = {p_g} = 0$ of an algebraic surface, Illinois J. Math. 2 (1958).
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 278 (1983), 613-634
- MSC: Primary 14J05; Secondary 13B10, 14C22
- DOI: https://doi.org/10.1090/S0002-9947-1983-0701514-1
- MathSciNet review: 701514