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.

Automated summary

In this paper, the author rigorously justifies the two degenerate conditions introduced by Schubert regarding conics, and proves all formulas related to these conditions using the language of blow-ups. Schubert's ideas about the enumeration of conics in space are systematically treated and proven in this paper.