正標数の代数多様体と特異点のフロベニウス写像に関する研究
メンバー： 原伸生
分野： 数学
所属： 工学研究院数理科学部門
キーワード： 代数幾何学、特異点、正標数、フロベニウス写像
研究概要
正標数pの体、例えば有限体やその代数閉包などの上で定義される代数多様体には、フロベニウス写像とよばれるp乗環準同型から定まる純非分離射が作用しており、これが正標数の代数幾何が、通常よく考えられる標数0（例えば複素数体上）の代数幾何との相違点の起因となっている。
本研究は、正標数の代数多様体のフロベニウス射に関わる概念である、F特異点、F爆発、フロベニウス直像の構造などについて、局所的および大域的な考察を行い、これらが代数多様体あるいは特異点の性質や、とくに正標数の病理的現象と如何に関わっているかを明らかにすることを目的としている。
主要論文・参考事項
1. N. Hara, Looking out for Frobenius summands on a blown-up surface of $P2$, submitted.
2. N. Hara, Structure of the F-blowups of simple elliptic singularities, J. Math. Sci. Univ. Tokyo 22 (Kodaira centennial issue) (2015), 193-218.
3. N. Hara, T. Sawada and T. Yasuda, F-blowups of normal surface singularities, Algebra Number Theory 7:3 (2013), 733--763
4. N. Hara, F-blowups of F-regular surface singularities, Proc. Amer. Math. Soc. 140 (2012), 2215--2226.
5. N. Hara and T. Sawada, Splitting of Frobenius sandwiches, RIMS Kokyuroku Bessatsu B24 (2011), 121-141.
6. N. Hara, F-pure thresholds and F-jumping exponents in dimension two, with an Appendix by Paul Monsky, Math. Research Lett. 13 (2006), 747-760.
7. N. Hara, A characteristic p analog of multiplier ideals and applications, Comm. Algebra 33 (2005), 3375-3388.
8. N. Hara and S. Takagi, On a generalization of test ideals, Nagoya Math. J., 175 (2004), 59-74.
9. N. Hara and K. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), 3143-3174.
10. N. Hara and K.-i. Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geometry 11 (2002), 363-392.
11. N. Hara, K.-i. Watanabe and K. Yoshida, Rees algebras of F-regular type, J. Algebra 247 (2002), 191-218.
12. N. Hara, K.-i. Watanabe and K. Yoshida, F-rationality of Rees algebras, J. Algebra 247 (2002), 153-190.
13. N.~Hara and K. E. Smith, The strong test ideal, Illinois J. Math. 45 (2001), 949-964.
14. N. Hara, Kawachi's invariant for fat points, J. Pure Appl. Algebra 165 (2001), 201-211.
15. N. Hara, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 1885-1906.
16. N. Hara, A characteristic p proof of Wahl's vanishing theorem for rational surface singularities, Arch. Math. (Basel) 73 (1999), 256-261.
17. N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981-996.
18. N. Hara, Classification of two-dimensional F-regular and F-pure singularities, Adv. Math. 133 (1998), 33-53.
19. N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps (announcement of [16]), 数理解析研究所講究録 964 (1996), 138-144.
20. N ~Hara and K.-i. Watanabe, The injectivity of Frobenius acting on cohomology and local cohomology modules, Manuscripta Math. 90 (1996), 301-315.
21. N. Hara, F-injectivity in negative degree and tight closure in graded complete intersection rings, C. R. Math. Rep. Acad. Sci. Canada 17 (6) (1995), 247-252.
22. N. Hara, F-regularity and F-purity of graded rings, J. Algebra 172 (1995), 804--818.
お問い合わせ先
東京農工大学・先端産学連携研究推進センター
urac[at]ml.tuat.ac.jp（[at]を@に変換してください）
Research on Frobenius maps acting on algebraic varieties and singularities in positive characteristic
Research members： Dr. Nobuo Hara
Research fields： Mathematics
Departments： Division of Mathematical Siences, Institute of Engineering
Keywords： algebraic geometry, singularity, positive characteristic, Frobenius map
Summary
In algebraic geometry over a field of positive characteristic, varieties admit the action of the Frobenius morphism, which is a purely inseparable morphism induced from p-th power ring endomorphism, and this causes the crucial difference between positive characteristic geometry and geometry in characteristic 0 (say, over complex numbers).
This research is aiming at better understanding of local and global properties of algebraic varieites and singularities in positive characteristic, in particular their pathologies, with respect to the concepts involoving the Frobenius morphism such as F-singularities, F-blowups and the structural study of Frobenius push-forawards.
Reference articles and patents
1. N. Hara, Looking out for Frobenius summands on a blown-up surface of $P2$, submitted.
2. N. Hara, Structure of the F-blowups of simple elliptic singularities, J. Math. Sci. Univ. Tokyo 22 (Kodaira centennial issue) (2015), 193-218.
3. N. Hara, T. Sawada and T. Yasuda, F-blowups of normal surface singularities, Algebra Number Theory 7:3 (2013), 733--763
4. N. Hara, F-blowups of F-regular surface singularities, Proc. Amer. Math. Soc. 140 (2012), 2215--2226.
5. N. Hara and T. Sawada, Splitting of Frobenius sandwiches, RIMS Kokyuroku Bessatsu B24 (2011), 121-141.
6. N. Hara, F-pure thresholds and F-jumping exponents in dimension two, with an Appendix by Paul Monsky, Math. Research Lett. 13 (2006), 747-760.
7. N. Hara, A characteristic p analog of multiplier ideals and applications, Comm. Algebra 33 (2005), 3375-3388.
8. N. Hara and S. Takagi, On a generalization of test ideals, Nagoya Math. J., 175 (2004), 59-74.
9. N. Hara and K. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), 3143-3174.
10. N. Hara and K.-i. Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geometry 11 (2002), 363-392.
11. N. Hara, K.-i. Watanabe and K. Yoshida, Rees algebras of F-regular type, J. Algebra 247 (2002), 191-218.
12. N. Hara, K.-i. Watanabe and K. Yoshida, F-rationality of Rees algebras, J. Algebra 247 (2002), 153-190.
13. N.~Hara and K. E. Smith, The strong test ideal, Illinois J. Math. 45 (2001), 949-964.
14. N. Hara, Kawachi's invariant for fat points, J. Pure Appl. Algebra 165 (2001), 201-211.
15. N. Hara, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 1885-1906.
16. N. Hara, A characteristic p proof of Wahl's vanishing theorem for rational surface singularities, Arch. Math. (Basel) 73 (1999), 256-261.
17. N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981-996.
18. N. Hara, Classification of two-dimensional F-regular and F-pure singularities, Adv. Math. 133 (1998), 33-53.
19. N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps (announcement of [16]), 数理解析研究所講究録 964 (1996), 138-144.
20. N ~Hara and K.-i. Watanabe, The injectivity of Frobenius acting on cohomology and local cohomology modules, Manuscripta Math. 90 (1996), 301-315.
21. N. Hara, F-injectivity in negative degree and tight closure in graded complete intersection rings, C. R. Math. Rep. Acad. Sci. Canada 17 (6) (1995), 247-252.
22. N. Hara, F-regularity and F-purity of graded rings, J. Algebra 172 (1995), 804--818.
Contact
University Research Administration Center(URAC),
Tokyo University of Agriculture andTechnology
urac[at]ml.tuat.ac.jp
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