Algebraic surfaces of general type with K2=3pg−5

Abstract

In this paper, we study minimal algebraic surfaces of general type with $K^2=3p_g-5.$ Since the case $p_g\le 4$ has been studied by Horikawa, Zucconi, and Bauer, we always assume $p_g\ge 5$ here. We prove that for such a surface, the canonical map is generically finite of degree 1 or 2. If the degree of the canonical map is 2, then the canonical image is a ruled surface, moreover, for any fixed $(p_g\ge 14,K^2=3p_g-5),$ there exists such a regular surface. If the degree of the canonical map is 1, then the surface is regular (i.e. q = 0) and its canonical linear system is base point free or has one simple base point. In the case where the canonical map is birational and the canonical linear system is base point free, we study the property of its canonical image and prove that for any fixed $(p_g\ge 5,K^2=3p_g-5),$ there exists such a surface.

Keywords:

• Algebraic surface of general type
• canonical map
• Castelnuovo’s second inequality
• classification

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14J29
• 14J10
• 14J15

Acknowledgments

The author would like to thank Professor Jinxing Cai for his encouragement and some good suggestions. The author is grateful to Professor Kazuhiro Konno and Professor Yi Gu for some useful discussion.

Additional information

Funding

This work was supported by the NSFC (No. 11471020) and “The Fundamental Research Funds of Shandong University” (No. 11140061340345).