×

The last coefficient of the Samuel polynomial. (English) Zbl 0625.14001

Let X be a Noetherian scheme, proper over an Artinian ring, and let I be a coherent ideal of \({\mathcal O}_ X\). Let \(\pi: \bar X\to X\) be the blowing up of X along I. Then it is well known that the Samuel function \(S_ I=\chi (X,{\mathcal O}_ X/I^ n)\) is a polynomial in n for \(n\gg 0\) and that every coefficient of this polynomial, except the last one, can be expressed in terms of the exceptional divisor of \(\pi\). Using standard methods from EGA III \([=\) Éléments de géométrie algébrique. III, Publ. Math., Inst. Hautes Étud. Sci. 11 (1962; Zbl 0118.362) and 17 (1963; Zbl 0122.161) by A. Grothendieck] and SGA 6 \([=\) Sém. Géom. algébr. 1966/67, Lect. Notes Math. 225 (1971)] the authors compute the last coefficient of \(S_ I(n)\), for \(n\gg 0\), as the difference \(\chi(X,{\mathcal O}_ X) - \chi(\bar X,{\mathcal O}_{\bar X})\) in the Euler characteristics; see theorem 2.4, 3.2) and also theorem 2.6 where I is supposed to be an \({\mathfrak m}_ x\)-primary ideal in (\({\mathcal O}_{X,x},{\mathfrak m}_ x)\) for a closed point \(x\in X\).
Reviewer: M.Herrmann

MSC:

14A10 Varieties and morphisms
13H15 Multiplicity theory and related topics
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] Berthelot, P. , Grothendieck, A. and Illusie, L. : Séminaire de Géométrie Algébrique du Bois Maire 1966/67. S.G.A. 6 . Lecture Notes in Math. 225. Springer (1971). · Zbl 0218.14001
[2] Grothendieck, A. and Dieudonné, J. : Eléments de Géometrie Algébrique III . Inst. Hautes Etudes Sci. Publ. Math. n\overset o\to - 11. Paris (1961). |
[3] Hartshorne, R. : Algebraic Geometry . Graduate Texts in Math. 52. Springer (1977). · Zbl 0367.14001
[4] Kleiman, S. : Toward a numerical theory of ampleness . Annals of Math. 84 (1966) 293-344. · Zbl 0146.17001 · doi:10.2307/1970447
[5] Nagata, M. : Imbedding of an abstract variety in a complete variety . J. Math. Kyoto Univ. 2 (1962) 1-10. · Zbl 0109.39503 · doi:10.1215/kjm/1250524969
[6] Ramanujam, C.P. : On a geometric interpretation of multiplicity . Inv. Math. 22 (1973) 63-67. · Zbl 0265.14004 · doi:10.1007/BF01425574
[7] Severi, F. : Il teorema di Riemann-Roch per curve, superficie e varieta; questioni collegate . Erg. der Math. und. ihr. Grenz. Springer, Berlin-Göttingen- Heidelberg (1958). · Zbl 0111.17801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.