Periodicity of the fixed locus of multiples of a divisor on a surface
Let \(D\) be an effective divisor on a nonsingular projective surface \(S\). Let \(B_ n\) be the fixed component of \(| nD |\). It is known from O. Zariski that there exists a divisor \(E\) on \(S\) such that \(B_ n - nE\) is bounded in \(n\) and also that \(B_ n - nE\) is periodic in some cases. In the paper under review the authors prove that \(B_ n - nE\) is periodic for an arbitrary effective divisor \(D\) on \(S\) if the ground field \(k\) is of characteristic 0. That this result does not hold if \(\text{char} (k) = p > 0\) is shown in a previous work by the same authors [``On a problem of Zariski on dimensions of linear systems'', Ann. Math., II. Ser. 137, 531-559 (1993)].
scientific article; zbMATH DE number 559005