The slope of fibred surfaces: unitary rank and Clifford index. (English) Zbl 1525.14042
Let \(f: S\to B\) be a relatively minimal fibred surface. \(K_f:=K_S-f^*K_B\) is called the relative canonical divisor and \(\chi_f:=\chi(\mathcal{O}_S)-\chi(\mathcal{O}_F)\chi(\mathcal{O}_B)\) is called the relative Euler characteristic, where \(F\) is a general fiber. Denote by \(g\) the genus of \(F\). The first slope inequality is of the form
\[
K_f^2\ge 4\frac{g-1}{g}\chi_f,
\]
which is proved by Xiao, Cornalba and Harris. There are several improvements of the slope inequality considering the influence of the relative irregularity \(q_f\), the gonality of the fibration and so on.
In this paper, the authors prove new slope inequalities for relatively minimal fibred surfaces. These new slope inequalities depend on relative irregularity \(q_f\), the unitary rank \(u_f\) and the Clifford index \(c_f\). The proof is based on Xiao’s method and a new Clifford-type inequality for subcanonical systems on non-hyperelliptic curves.
In this paper, the authors prove new slope inequalities for relatively minimal fibred surfaces. These new slope inequalities depend on relative irregularity \(q_f\), the unitary rank \(u_f\) and the Clifford index \(c_f\). The proof is based on Xiao’s method and a new Clifford-type inequality for subcanonical systems on non-hyperelliptic curves.
Reviewer: Jingshan Chen (Beijing)
MSC:
| 14J10 | Families, moduli, classification: algebraic theory |
| 14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |
| 14D06 | Fibrations, degenerations in algebraic geometry |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14J29 | Surfaces of general type |
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