Slope inequality for families of curves over surfaces

Zhang, T.

Abstract

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 .

Citation

Zhang, T. (2018). Slope inequality for families of curves over surfaces. Mathematische Annalen, 371(3-4), 1095-1136. https://doi.org/10.1007/s00208-017-1551-1

• 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
• DOI: https://doi.org/10.1007/s00208-017-1551-1
• Public URL: https://durham-repository.worktribe.com/output/1358164