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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On Euler characteristics associated to exceptional divisors
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by Willem Veys PDF
Trans. Amer. Math. Soc. 347 (1995), 3287-3300 Request permission

Abstract:

Let $k$ be an algebraically closed field and $f \in k[{x_1}, \ldots ,{x_{n + 1}}]$. Fix an embedded resolution $h:X \to {\mathbb {A}^{n + 1}}\quad {\text {of}}\quad {f^{ - 1}}\{ 0\}$ and denote by ${E_i}$, $i \in S$, the irreducible components of ${h^{ - 1}}({f^{ - 1}}\{ 0\} )$ with multiplicity ${N_i}$ in the divisor of $f{\text {o}}h$. Put also $\mathring {E}_i \coloneq {E_i}\backslash { \cup _{j \ne i}}{E_j}$, and denote by $\chi ({E_i})$ its Euler characteristic. Several conjectures concerning Igusa’s local zeta function and the topological zeta function of $f$ motivate the study of Euler characteristics associated to subsets ${ \cup _{i \in T}}{E_i}$ of ${ \cup _{i \in S}}{E_i}$, which are maximal connected with respect to the property that $d|{N_i}$ for all $i \in T$. Here $d \in \mathbb {N},d > 1$. We prove that if $h$ maps ${ \cup _{i \in T}}{E_i}$ to a point, then \[ (-1)^n \sum _{i \in T} \chi (\mathring {E}_i) \geqslant 0 \] This generalizes a well-known result for curves. We also prove some vanishing results concerning the $\chi (\mathring {E}_i)$ for such a maximal connected subset ${ \cup _{i \in T}}{E_i}$ and give an application on the above-mentioned zeta functions, yielding some confirmation of the holomorphy conjecture for those zeta functions.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 3287-3300
  • MSC: Primary 11S40; Secondary 14E15, 32S45
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1308026-3
  • MathSciNet review: 1308026