Asian Journal of Mathematics

Volume 22 (2018)

Number 4

Special issue in honor of Ngaiming Mok (3 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

Sakai’s theorem for $\mathbb{Q}$-divisors on surfaces and applications

Pages: 761 – 786



Fei Ye (Department of Mathematics & Computer Science, Queensborough Community College, City University of New York, Bayside, N.Y., U.S.A.)

Tong Zhang (Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China)

Zhixian Zhu (Department of Mathematics, University of California at Riverside)


In this paper, we present a characterization of a big $\mathbb{Q}$-divisor $D$ on a smooth projective surface $S$ with $D^2 \gt 0$ and $H^1(\mathcal{O}_S (- \lceil D \rceil)) \neq 0$, which generalizes a result of Sakai for $D$ integral. As applications of this result for $\mathbb{Q}$-divisors, we prove results on base-point-freeness and very-ampleness of the adjoint linear system $\lvert K_S + \lceil D\rceil \rvert$. These results can be viewed as refinements of previous results on smooth surfaces of Ein–Lazarsfeld and Maşek.


$\mathbb{Q}$-divisor, adjoint linear system, vanishing theorem

2010 Mathematics Subject Classification

14C20, 14E25, 14F17, 14J99

Fei Ye was partially supported by PSC-CUNY cycle 47 and 48 Research Awards.

Tong Zhang was supported by the Science and Technology Commission of Shanghai Municipality (STCSM), grant No. 18dz2260400 and a Leverhulme Trust Research Project Grant ECF-2016-269.

Received 24 October 2016

Accepted 13 June 2017

Published 20 September 2018