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On the geometry of nodal plane cubics: the condition \(p\). (English) Zbl 0766.14041
Enumerative algebraic geometry, Proc. Zeuthen Symp., Copenhagen/Den. 1989, Contemp. Math. 123, 169-187 (1991).
Summary: [For the entire collection see Zbl 0741.00067.]
We compute the number of nodal plane cubics that satisfy eight conditions, of the following four types: \(\mu\), that the nodal cubic go through a point; \(\nu\), that the nodal cubic be tangent to a given line; \(b\), that the node be on a line; and \(p\), that a nodal tangent go through a point. To do so, we construct and study several compactifications of the variety of non-degenerate nodal plane cubics, providing a geometric understanding of Schubert’s contributions and completing his tables for the conditions \(\mu\), \(\nu\), \(b\) and \(p\). Of the nine degenerations we need, two were not considered by Schubert.

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H10 Families, moduli of curves (algebraic)
14N05 Projective techniques in algebraic geometry