Subadjoint ideals and hyperplane sections

Chiarli, Nadia; Greco, Silvio

doi:10.1090/S0002-9939-96-03126-7

Subadjoint ideals and hyperplane sections
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by Nadia Chiarli and Silvio Greco PDF

Proc. Amer. Math. Soc. 124 (1996), 1035-1041 Request permission

Abstract:

We study the behaviour of the notion of “sub-adjoint ideal to a projective variety" with respect to general hyperplane sections. As an application we show that the two classical definitions of sub-adjoint hypersurface given respectively by Enriques and Zariski are equivalent.

References

E. Arbarello and C. Ciliberto, Adjoint hypersurfaces to curves in $\textbf {P}^{r}$ following Petri, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 1–21. MR 686937
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
Piotr Blass and Joseph Lipman, Remarks on adjoints and arithmetic genera of algebraic varieties, Amer. J. Math. 101 (1979), no. 2, 331–336. MR 527995, DOI 10.2307/2373981
G. Castelnuovo, Sui multipli di una serie lineare di gruppi di punti appartenenti ad una curva algebrica, Rend. Circ. Mat. Palermo 7 (1893), 89-110.
Nadia Chiarli, A sharp Castelnuovo bound for the normalization of certain projective surfaces, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 145–151. MR 1272026
—, Sub-adjoint Ideals to Projective Varieties, Ricerche Mat. (to appear).
Caterina Cumino, Silvio Greco, and Mirella Manaresi, An axiomatic approach to the second theorem of Bertini, J. Algebra 98 (1986), no. 1, 171–182. MR 825140, DOI 10.1016/0021-8693(86)90020-7
F. Enriques:, Introduzione alla geometria sopra le superficie algebriche, Mem. Soc. Ital. Sci. detta dei 40, III serie 10 (1896), 1-81.
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Silvio Greco and Paolo Valabrega, On the theory of adjoints, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 98–123. MR 555694
Silvio Greco and Paolo Valabrega, On the theory of adjoints. II, Rend. Circ. Mat. Palermo (2) 31 (1982), no. 1, 5–15 (English, with Italian summary). MR 664394, DOI 10.1007/BF02849535
Jean-Pierre Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR 725671
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
E. Stagnaro, Canonical and pluricanonical adjoints I, II, III (Preprints 1991,1992), 1991, 1992.
Oscar Zariski, An introduction to the theory of algebraic surfaces, Lecture Notes in Mathematics, No. 83, Springer-Verlag, Berlin-New York, 1969. Notes by James Cohn, Harvard University, 1957–58. MR 0263819
Oscar Zariski, Algebraic surfaces, Second supplemented edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 61, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford. MR 0469915