Understanding the structure of open algebraic surfaces and normal algebraic surfaces of logarithmic Kodaira dimension one or less

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400042
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Niigata University
• Principal Investigator: Kojima Hideo 新潟大学, 自然科学系, 教授 (90332824)
• Co-Investigator(Renkei-kenkyūsha): KISHIMOTO Takashi 埼玉大学, 理工学研究科, 准教授 (20372576) ; SAITO Natsuo 広島市立大学, 情報科学研究科, 講師 (70382372) ; TAKAHASHI Takeshi 新潟大学, 自然科学系, 准教授 (60390431)
• Research Collaborator: NAGAMINE Takanori
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 代数幾何学 / 開代数曲面 / 正規代数曲面 / 対数的小平次元 / 多項式環 / 高階導分

Outline of Final Research Achievements

I have studied open algebraic surfaces, normal algebraic surfaces and kernels of higher derivations in polynomial rings. I proved that, for an irrational open algebraic surface, its logarithmic Kodaira dimension is non-negative if and only if its logarithmic 12 genus is positive. I studied normal del Pezzo surfaces of Picard rank one with only rational log canonical singularities by using structure theorems on open algebraic surfaces and some results on Q-homology planes and gave partial classification results for those surfaces with one or four singular points. I also gave a sufficient condition for the kernel of a locally finite higher derivations in the polynomial ring in three variables to be a polynomial ring. Furthermore, I applied these results to some problems on affine algebraic varieties.

Free Research Field

代数幾何学