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Slope inequality for families of curves over surfaces

Zhang, T.

Slope inequality for families of curves over surfaces Thumbnail


T. Zhang


In this paper, we investigate the general notion of the slope for families of curves f:X→Yf:X→Y . The main result is an answer to the above question when dimY=2dim⁡Y=2 , and we prove a lower bound for this new slope in this case over fields of any characteristic. Both the notion and the slope inequality are compatible with the theory for dimY=0,1dim⁡Y=0,1 in a very natural way, and this gives a strong evidence that the slope for an n-fold fibration of curves f:X→Yf:X→Y may be KnX/Y/chn−1(f∗ωX/Y)KX/Yn/chn−1(f∗ωX/Y) . Rather than the usual stability methods, the whole proof of the slope inequality here is based on a completely new method using characteristic p>0p>0 geometry. A simpler version of this method yields a new proof of the slope inequality when dimY=1dim⁡Y=1 .

Journal Article Type Article
Acceptance Date Apr 20, 2017
Online Publication Date May 18, 2017
Publication Date Aug 1, 2018
Deposit Date May 17, 2017
Publicly Available Date May 18, 2018
Journal Mathematische Annalen
Print ISSN 0025-5831
Electronic ISSN 1432-1807
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 371
Issue 3-4
Pages 1095-1136
Public URL


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