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Enumerative geometry of triangles. I. (English) Zbl 0648.14029

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The aim of this series of (three) papers is to prove rigorously some enumerative formulae of Schubert for triple contacts of plane curves which move in suitable families. Two smooth plane curves have at least triple contact at a point P iff they have the same tangent line at P and the same second-order data at P. Here the problem is to give a conceptual approach for the second-order data. If C is a smooth plane curve and \(x,y,z\in C\) are three points approaching a given point P, consider the family \(\Sigma(x,y,z)\) of all concis through \(x,y,z.\) Consider \(P=P^ 2\times P^ 2\times P^ 2\) and \v{P}\(=\check P^ 2\times \check P^ 2\times \check P^ 2\) and define \(W=\{(x,y,z;L,M,N)\in P\times \check P|\) \(x,y\in L,\) \(x,z\in M\) and \(y,z\in N\},\) Then one has a rational map \(W\to G(2,5)\) defined by \((x,y,z;L,M,N)\to \Sigma(x,y,z)\) (one thinks of G(2,5) as the parameter variety for 2-dimensional linear families of conics). Let \(W^*\subset W\times G(2,5)\) be the closure of the graph of the above rational map. The variety \(W^*\), called the model of Schubert triangles, is the interesting object to be studied for the problem the authors have in mind. To this end one also considers the blow-up \(\bar W\) of W along the closed subvariety \(X=\{(x,x,x;L,L,L)|\) \(x\in P^ 2,L\in \check P^ 2,x\in L\}\) (X is just the singular locus of W and \(\bar W\) is smooth), and the full-diagonal blow-up B of P. Then one has a commutative diagram \[ \begin{tikzcd}\bar {W} \ar[r,"p"]\ar[d,"p" '] & W^\ast\ar[d,"qw"] \\ B \ar[r,"p_W" '] & W \end{tikzcd} \] in which \(\bar W\) is identified with the blow-up of B along \(X_ B=p_ W^{-1}(X)\), and also with the blow-up of \(W^*\) along \(X^*=q_ W^{-1}(X)\). Both blow-ups have the same exceptional locus \(\bar X\) of \(\bar W.\) In this first part of the paper the authors begin the program aiming to determine the rational equivalence ring \(A^{\bullet}(W^*)\), by establishing some basic properties of B and \(W^*\) and by computing \(Pic(W^*)\). The variety \(X^*\) is called the variety of the second-order data of \(P^ 2.\)
[See also the following two reviews.]
Reviewer: L.Bădescu

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14C15 (Equivariant) Chow groups and rings; motives
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