A Note on Del Pezzo Fibrations of Degree 1

Abstract

The purpose of this paper is to extend the result in Park (Park, J. (2001). Birational maps of del Pezzo fibrations. J. Reine Angew. Math. 538:213–221) in the case of del Pezzo fibrations of degree 1. To this end we investigate the anticanonical linear systems of del Pezzo surfaces of degree 1. We then classify all possible effective anticanonical divisors on Gorenstein del Pezzo surfaces of degree 1 with canonical singularities.

Key Words:

• Anticanonical divisors
• Birational maps
• Del Pezzo fibrations
• Del Pezzo surfaces
• Log canonical thresholds

Mathematical Subject Classification (2000):

• 14D06
• 14E05
• 14J45

Acknowledgments

The author would like to express gratitude to Professor Vyacheslav V. Shokurov, who gave him many pieces of invaluable advice. This work was partially supported by NSF Grant DMS-0100991.