The number of 1-codimensional cycles on projective varieties
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.
scientific article; zbMATH DE number 5735949