Articles

UNDERSTANDING SCHUBERT'S BOOK (I)

  • Banghe LI
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  • KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Banghe LI,E-mail:libh@amss.ac.cn

Received date: 2019-11-12

  Revised date: 2020-03-29

  Online published: 2021-04-06

Supported by

This work was partially supported by National Center for Mathematics and Interdisciplinary Sciences, CAS.

Abstract

Hilbert Problem 15 required an understanding of Schubert’s book [1], both its methods and its results. In this paper, following his idea, we prove that the formulas in §6, §7, §10, about the incidence of points, lines and planes, are all correct. As an application, we prove formulas 8 and 9 in §12, which are frequently used in his book.

Cite this article

Banghe LI . UNDERSTANDING SCHUBERT'S BOOK (I)[J]. Acta mathematica scientia, Series B, 2021 , 41(1) : 97 -113 . DOI: 10.1007/s10473-021-0106-4

References

[1] Schubert H. Kalkül der abzählenden Geometrie. Berlin, New York:Springer-Verlag, 1979
[2] Hilbert D. Mathematical Problems. Translated by Dr. Mary Winston Newson. Bull Amer Math Soc, 1902, 50:437-479
[3] Kleiman S L. Problem 15. Rigorous foundation of Schubert's enumerative calculus//Mathematical Developments Arising from Hilbert Problems, Proceeding of Symposia in Pure Mathematics of the American Mathematical Society, Vol 28. American Mathematical Society, 1976
[4] Zeuthen H G. Almindelige Egenskaber ved Systemer af plane Kurver. Naturw. og. math. Afd. 10, Bd. IV, 1873
[5] Li B H. Hilbert Problem 15 and nonstandard analysis (I). Acta Mathematica Scientia, 2020, 40B(1):1-15
[6] Van der Waerden. Einfuehrung in die Algebraische Geometrie. Berlin, New York:Springer-Verlag, 1973(A Chinese translation has been published by Science Press in China in 2008.)
[7] Weil A. Foundation of Algebraic Geometry. Providence, RI:American Mathematical Society, 1962
[8] Li B H. Hilbert Problem 15 and Ritt-Wu Method (I). Journal of Systems Science and Complexity, 2019, 32(1):47-61
[9] Li B H, Wang D K. Hilbert Problem 15 and Ritt-Wu Method (Ⅱ). Journal of Systems Science and Complexity, 2020, https://doi.org/10.1007/s11424-020-9166-0
[10] Kleiman S L. An introduction to the Reprint Edition//Kalkül der abzählenden Geometrie. Springer-Verlag, 1979
[11] Fulton W. Intersection Theory. 2nd ed. Berlin, New York:Springer-Verlag, 1998
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