Research on the structure of the Frobenius push-forwards on algebraic varieties and singularities in positive characteristic

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400035
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo University of Agriculture and Technology
• Principal Investigator: Hara Nobuo 東京農工大学, 工学(系)研究科(研究院), 教授 (90298167)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 代数幾何 / 正標数 / フロベニウス直像 / F爆発 / 大域的F有限型 / 大域的F正則 / 特異点 / ベクトル束

Outline of Final Research Achievements

We studied the Frobenius push-forwards on algebraic varieties and their singularities in positive characteristic p, focusing on a few classes of projective varieties and normal surface singularities. Our results are as follows.
1. We classified the structure of the F-blowup sequence of a simple elliptic singularity in terms of the characteristic p, the self-intersection number of the exceptional elliptic curve E on the minimal resolution, and whether E is ordinary or supersingular.
2. We studied the Frobenius push-forward of the structure sheaf of the surface obtained by blowing up the projective plane at n points in general position. In case n=4 (del Pezzo surface of degree 5), we determined all the indecomposable direct summands of the iterated Frobenius push-forwards, and proved that their isomorphism classes are finite (GFFRT). We also proved that these Frobenius summands generate the derived category in case n=4. In case n=10, we constructed a rational surface that is not GFFRT.

Free Research Field

代数学