2023 Fiscal Year Final Research Report
On normal canonical surfaces and admissible singularities
| Project/Area Number |
19K03446
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Review Section |
Basic Section 11010:Algebra-related
|
|---|
| Research Institution | Kansai University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2019-04-01 – 2024-03-31
|
| Keywords | 一般型代数曲面 / 標準写像 / 正規特異点 |
|---|
| Outline of Final Research Achievements |
A minimal surface of general type whose canonical map is birational onto the image is called a canonical surface. Such surfaces with geometric genus 4 have a long history since Enriques' book. It has not been recognized the existence of normal ones except in the trivial case of quintic surfaces, before my work on normal canonical surfaces. In this research, I tried to construct new examples of normal canonical sextic surfaces whose volume is either 10 or 11, by deforming double coverings of rational surfaces. However, I failed to show the surfaces thus obtained is actually canonical.
|
| Free Research Field |
代数幾何学
|
| Academic Significance and Societal Importance of the Research Achievements |
期待される最終的な結論は得られなかったものの，一般型２次元正規特異点の数値的な不変量による分類や２重被覆で構成した曲面，とくに超楕円的な曲線束をもつ曲面の変形については新たな知見が得られた．また，幾何種数が大きい場合の正規標準曲面の研究により，種数３の非超楕円的ファイバー芽についての認識がいっそう深くなった．本研究を通じて，幾何種数４の場合には，次数が７以上の正規標準曲面は存在しないという予想に至った．こういった未解決問題の提示は，当該分野の研究進展にひとつの道筋を示すものである．
|
