Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
home | mail | papers | authors | news | seminars | events | open positions | login

Slope inequalities for fibred surfaces

Lidia Stoppino

created by bazzoni on 14 Dec 2020
modified on 18 Jan 2021

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 if you'd like to be added to our mailing list.


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.

Credits | Cookie policy | HTML 5 | CSS 2.1