Hyperplane sections of non-standard graded rings

Abstract

The purpose of this paper is to explain a method on the generalization of the Bertini-type theorem on standard graded rings to the non-standard graded case of certain type.

Keywords:

• Bertini theorem
• non-standard graded ring
• normal ring

2020 Mathematics Subject Classification:

• 13B22
• 13C20
• 14C20

Acknowledgments

The author is grateful to Professor L. Katthän and Professor K. Yanagawa for sharing illuminating discussions. The author is grateful to the referee for some kind comments.

Notes

1 Here, the module-finite condition ensures that $\varphi$ is indeed a morphism. In general, the map of graded rings does not necessarily induce a map on the corresponding projective schemes; rather it is defined over some subset.

Additional information

Funding

This work was supported by JSPS KAKENHI Grant Numbers JP23K03077 and JP23H01066.