The number of 1-codimensional cycles on projective varieties
type: Item
instance of: scholarly article
title: The number of 1-codimensional cycles on projective varieties
zbMATH Open document ID: 1197.14005
DOI: 10.1215/0023608X-2009-012
author: Satoshi Takagi
published in: Kyoto Journal of Mathematics
publication date: 2010-07-13T00:00:00Z
review text: The author investigates the converging radius of the ``generalized zeta function''. Let \(X\) be a projective scheme defined over \(\mathbb{F}_q\), and let \(N_d(X)\) denote the number of effective zero-dimensional cycles \(Y\) defined over \(\mathbb{F}_q\) with \(\deg Y=d\). The author observes that the usual zeta function \(\zeta(X;t)\) of \(X\) can be expressed as \(\sum_{d=0}^\infty N_d(X)t^d\), and proposes the definition of a generalized zeta function in the following way. Let \((X,H)\) be a polarized quasi-projective scheme of dimension \(n\) over \(\mathbb{F}_q\) with a fixed compactification \((\overline{X},\overline{H})\). For an integer \(l\) with \(0\leq l\leq n-1\) and any positive integer \(d\), let \(N_d(X,H,l;\mathbb{F}_q)\) be the number of effective \(l\)-dimensional cycles \(Y\) defined over \(\mathbb{F}_q\) with \(\deg_{\overline{H}}Y=d\), and he sets \(Z(X,H,l;t)=\sum_{d=0}^\infty N_d(X,H,l;\mathbb{F}_q)t^{d^{l+1}}\), which is the proposed generalized zeta function. The main theorem of this paper shows that the function converges at the origin \(t=0\) with a natural converging radius.
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 5735949
zbMATH Keywords: algebraic cycles
zbMATH Keywords: zeta function
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5735949