19 jan 2021 -- 16:00
Geometry in Como - Online
The talk will take place on Microsoft Teams. The link will be made available here on January 18th, 2021. Please write to giovanni.bazzoni@uninsubria.it if you'd like to be added to our mailing list.
Abstract.
In this talk I will give an overview of the so-called slope inequalities for fibred surfaces: these are in general lower bounds for the slope between the self-intersection of the relative canonical sheaf and the relative Euler characteristic of the structure sheaf. These inequalities were studied first in two seminal papers by Cornalba-Harris and Xiao in the '80's, via two different thechniques. However, there is a key assumption that lies underneath both techniques: the linear stability of the canonical system on the general fibres, which is equivalet to the classical Clifford's theorem. I will then focus in the influence on the slope of the following invariants of the fibred surfaces: the relative irregularity, the unitary rank and the gonality and Clifford index. I will describe results due to Xiao, Barja and myself, and recently by Lu and Zuo (who introduced a third new method). Eventually, I will describe an improved bound obtained very recently in collaboration with my Ph.D. student Enea Riva. This result is obtained via the Xiao's method. The key assumption is a new Clifford-type bound for non complete subacanonical systems which -interestingly enough- is not a linear stability result.