Linear stability of projected canonical curves with applications to the slope of fibred surfaces

Abstract

Let f : S −→ B be a non locally trivial relatively minimal fibred surface. We prove a lower bound for the slope of f depending increasingly from the relative irregularity of f and the Clifford index of the general fibres.

On the slope conjecture of Barja and Stoppino for fibred surfaces

“…Assume the above claim. Then (2)(3)(4)(5) follows immediately by applying Lemma 2.3 to the new fibrationf . It remains to prove the above claim.…”

“…Let τ i : Γ i → Γ i be the normalization, and D i = τ * i O P r i −1 (1) ∈ Pic Γ i be the pullingback of the hyperplane section. Then (3)(4) follows from the facts that…”

“…It is clear if deg(ι i ) ≥ 3. If deg(ι i ) = 2, then by (3)(4) together with the assumption γ i ≥ g/6, one obtains…”

“…Thus, it is natural to investigate the influence of some properties of the fibration on the behaviour of the slope. For instance, according to [14,3], one knows that the Clifford index of the general fiber has some meaning to the lower bound of the slope. We would like to be concerned about the following conjecture of Barja and Stoppino (cf. [3, Conjecture 1.1]) on the influence of the relative irregularity q f := q − b on the lower bound of the slope.…”

“…Pirola constructed in [21] the first example which does not satisfy (1)(2)(3), see also [3, Remark 4.6]. To our knowledge, the only known counterexamples to the bound (1-3) belong to the extremal case q f = g − 1.…”