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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.

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