Special Homogeneous Linear Systems on Hirzebruch Surfaces

The Segre-Gimigliano-Harbourne-Hirschowitz Conjecture can be naturally formulated for Hirzebruch surfaces Fn. We show that this Conjecture holds for imposed base points of equal multiplicity bounded by 8. 1. Linear systems on Hirzebruch surfaces Our goal is to prove Conjecture 4 for linear systems on Hirzebruch surfaces with imposed base points of equal multiplicity bounded by 8. This Conjecture, being a natural reformulation of the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture, has been stated in [Laf 02, Conjecture 2.6]. In the same paper it is shown (Theorem 7.1) that this Conjecture holds for systems with imposed base points of equal multiplicity bounded by 3. We will also give another proof of [Laf 02, Proposition 2.7], where the proof contains a serious mistake (for more details see the proof of Proposition 29). Our method will also work for greater values of multiplicities, but the computational part (realized with the help of computers) becomes very large and timeconsuming. But it is possible to carry our computations further to obtain the proof for m1 = · · · = mr = 9, 10, . . . or to find a counterexample. The author would like to thank Michał Kapustka and Tomasz Szemberg for valuable discussions. By Fn, n ≥ 0, we denote the rational ruled surface (called the n-th Hirzebruch surface) given by Fn = P(OP1 ⊕ OP1(n)) over the field K of characteristic 0. The Picard group Pic(Fn) can be freely generated by the class of a fiber Fn and the class of the section Hn such that F 2 n = 0, H 2 n = n, Fn ·Hn = 1. The irreducible section with self-intersection −n will be denoted by Γn, we have Γn ∈ |Hn − nFn|. The class of Γn in Pic(Fn) will also be denoted by Γn. Let a, b be integers. By Ln(a, b) we will denote the complete linear system associated to the line bundle aFn+ bHn. Lemma 1. If on Fn the class aFn + bHn contains an effective divisor then there exists non-negative integers a, b, q (q > 0 if and only if a < 0) such that the base locus of |aFn+bHn| is qΓn and aFn+bHn is linearly equivalent to qΓn+a Fn+b Hn. Moreover, we have dimLn(a, b) = (b + 1)(2a + 2 + nb) 2 − 1 Proof. For the proof see [Laf 02, Proposition 2.2]. Now we pick r points p1, . . . , pr ∈ Fn in general position, let m1, . . . ,mr be nonnegative integers. By Ln(a, b;m1, . . . ,mr) we denote the linear system of curves in 1991 Mathematics Subject Classification. 14H50; 13P10.