Minimal pencils on smooth surfaces in \(\mathbb{P}^{3}\)
type: Item
instance of: scholarly article
title: Minimal pencils on smooth surfaces in \(\mathbb{P}^{3}\)
zbMATH Open document ID: 1155.14030
author: Kazuhiro Konno
published in: Osaka Journal of Mathematics
publication date: 2008-11-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1221656653
review text: In this paper two numerical birational invariants for surfaces are introduced, the minimal genus and the minimal slope; both are a possible generalization to dimension 2 of the notion of gonality of a curve. The author studies in detail the behavior of these invariants in the case of smooth surfaces in \(\mathbb P^3\). Let \(S\) be a complex smooth projective surface and \(\Phi\) a non-constant rational function on it. It is possible to associate in a canonical way to \(\Phi\) a relatively minimal fibration \(f\colon X\longrightarrow B\), where \(X\) is a smooth surface birationally equivalent to \(S\) and \(B\) a smooth curve together with a finite morphism \(\tau\colon B\longrightarrow \mathbb P^1\) such that \(\Phi\) is (birationally) identified with \(f\circ \tau\). Define the genus of \(\Phi\) as the genus of the fibration \(f\), and the slope of \(\Phi\) as the slope \(K_f^2/\chi_f\) (when \(f\) is not a fibre bundle). The minimal genus (resp. slope) of \(S\) is then the minimum \(\mu_g\) (resp. \(\mu_s\)) of the genus (resp. slope) as \(\Phi\) moves in the rational function field of \(S\). These invariants are different -- although likely deeply connected -- from the degree of irrationality [see \textit{T. T. Moh} and \textit{W. Henzer}, J. Algebra 77, 62--73 (1982; Zbl 0533.14010) and \textit{H. Yoshihara}, J. Algebra 167, 634--640 (1994; Zbl 0834.14019)] which is another natural generalization of the gonality to varieties of higher dimension. In the case of surfaces of degree \(d\geq 2\) in \(\mathbb P^3\) the main results of the paper are the following lower bounds (see Theorems 2.1 and 3.1): (1) \(\mu_g(S)\geq (n-2)(n-3)/2\), (2) \(\mu_s(S)\geq 6(n-4)/(n-3)\). Moreover, the author proves that for \(d\geq 7\) equalities in (1) and (2) are satisfied if and only if \(S\) contains a line. If \(S\) does not contain lines the values of \(\mu_g\) and \(\mu_g\) are computed. Note that these results has the same flavor of the one of \textit{M. Namba} [Families of …
reviewed by: Lidia Stoppino
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14D06
zbMATH DE Number: 5363232
zbMATH Keywords: fibration
zbMATH Keywords: surfaces
zbMATH Keywords: slope
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5363232