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Understanding Schubert’s book (II)


In this paper, we give rigorous justification of the ideas put forward in §20, Chapter 4 of Schubert’s book; a section that deals with the enumeration of conics in space. In that section, Schubert introduced two degenerate conditions about conics, i.e., the double line and the two intersection lines. Using these two degenerate conditions, he obtained all relations regarding the following three conditions: conics whose planes pass through a given point, conics intersecting with a given line, and conics which are tangent to a given plane. We use the language of blow-ups to rigorously treat the two degenerate conditions and prove all formulas about degenerate conditions stemming from Schubert’s idea.

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Correspondence to Banghe Li.

Additional information

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

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Li, B. Understanding Schubert’s book (II). Acta Math Sci 42, 1–48 (2022).

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Key words

  • Hilbert Problem 15
  • enumeration geometry
  • blow-up

2010 MR Subject Classification

  • 14H50
  • 14N15