Research on combinatorial descriptions of dual defects of toric varieties

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K14162
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Okayama University (2021-2023); Nagoya University (2017-2020)
• Principal Investigator: Ito Atsushi 岡山大学, 環境生命自然科学学域, 准教授 (90712240)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: トーリック多様体 / 双対欠損 / アーベル多様体 / セシャドリ定数

Outline of Final Research Achievements

I could not obtain descriptions of dual defects of toric varieties using corresponding polytopes. On the other hand, I obtained the following results during the studies.
I gave a new algebraic geometric proof of the reconstruction theorem in computer vision . I I gave some conditions for satisfying the property (N_p) concerning the syzygies of ample line bundles on abelian varieties, using intersection numbers and other invariants. I defined a generalization of Seshadri constant, an invariant that measures the positivity of line bundles on algebraic varieties, and studied its properties. I constructed a counterexample to a certain conjecture on the K-stability of Fano varieties. I studied birational geometry of some Calabi-Yau 3folds and showed that the movable cone conjecture is satisfied for the 3folds.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

トーリック多様体の双対欠損の記述は得られなかったものの，代数幾何学において非常に重要な対象である直線束やファノ多様体に関し興味深い新たな知見をいくつも得ることができた．とくにアーベル多様体上の直線束の研究には大きな進展があった．またコンピュータビジョンにおける重要な定理に対し代数幾何学的な視点を提供したことも意味がある．