Minimal pencils on smooth surfaces in \(\mathbb{P}^{3}\)
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 meromorphic functions on compact Riemann surfaces. Lecture Notes in Mathematics. 767. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0417.32008)] on the gonality of plane curves. In the last part of the paper the author develops a study of the pencils with minimal invariants for surfaces of degree 5 and 6. In particular, he can exibit explicit equations for the surfaces containing such pencils and give a classification of the possible pencils.
scientific article; zbMATH DE number 5363232