The enumerative theory of conics after Halphen
type: Item
instance of: scholarly article
title: The enumerative theory of conics after Halphen
zbMATH Open document ID: 0626.14037
author: Eduardo Casas-Alvero
author: Sebastian Xambó
published in: Lecture Notes in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: This book gives a detailed and self-contained exposition of the enumerative theory of conics. The authors survey the classical approaches to the subject and provide modern proofs (several are new), thus tying up with the most recent research. They give a very good introduction to this ``oldest'' part of enumerative geometry. In particular they explain the classical terminology and concepts, and thus facilitate the reading of older papers. The first part, {\S}{\S} 1-14, reworks Halphen's theory about ``proper solutions'', and leads to what are called Halphen's first and second formula. The first of these gives the proper number of members of conics in a 1-dimensional system satisfying a given condition, in terms of local and global characteristic pairs of the condition and the system, and of the degenerations of the system. The second one gives the number of conics properly satisfying \(5\quad given\quad conditions,\) in terms of characteristic numbers of the conditions. The second part of the book gives a general treatment for conditions of any codimension. In particular, the appropriate intersection ring, \(Hal^{\bullet}(W)\), is constructed - this is the abstract ring of De Concini-Procesi [cf. \textit{C. De Concini} and \textit{C. Procesi}, ``Complete symmetric varieties. II'', in Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya 1983, Adv. Stud. Pure Math. 6, 481-513 (1985; Zbl 0596.14041)] in the case of conics.
reviewed by: Ragni Piene
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 14-01
Mathematics Subject Classification ID: 51-01
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 51N15
zbMATH DE Number: 4017091
zbMATH Keywords: polar line
zbMATH Keywords: enumerative theory of conics
zbMATH Keywords: proper solutions
zbMATH Keywords: characteristic numbers
zbMATH Keywords: intersection ring
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0076501
OpenAlex ID: W1553667660
DOI: 10.1007/BFB0076501
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description: scientific article; zbMATH DE number 4017091