The enumerative geometry of plane cubics. II: Nodal and cuspidal cubics
[For part I see Trans. Am. Math. Soc. 317, No.2, 501-539 (1990; Zbl 0703.14035).] A variety of `complete cubics' can be obtained from the \({\mathbb{P}}^ 9\) parametrizing plane cubics, by a sequence of five blow-ups at smooth centers. The construction has been studied in detail and applied elsewhere by the author to compute the `characteristic numbers' of smooth plane cubics, i.e. the number of smooth cubics containing a given number of points and tangent to a given number of lines in general position. More generally, using the construction to compute characteristic numbers for other families of cubics involves computing five Segre classes: if \(B_ 0\subset V_ 0={\mathbb{P}}^ 9,...,B_ 4\subset V_ 4\) are the centers of the blow-ups, and \(F_ 0,...,F_ 4\) are respectively the closure of the parameter set of the family in \({\mathbb{P}}^ 9\) and its proper transforms through the stack of blow-ups, then one needs to compute the five classes \(s(B_ i\cap F_ i,F_ i)\), or the equivalent information \(B_ i\circ F_ i=c(N_{B_ i}V_ i)\cap s(B_ i\cap F_ i,F_ i).\) This is done in the paper for some families of singular cubics, recovering classic enumerative results. Aside from the enumerative application, the computation prompts a close look at Segre classes and at techniques that are being developed for their calculation.
scientific article; zbMATH DE number 4171087
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