Enumerative geometry of nonsingular plane cubics.

*(English)*Zbl 0753.14045
Algebraic geometry, Proc. Conf., Sundance/UT (USA) 1988, Contemp. Math. 116, 85-113 (1991).

[For the entire collection see Zbl 0725.00006.]

From the text: “This paper completes the determination of the characteristic numbers for all plane cubics, begun with cuspidal cubics, and continued with nodal ones. The family of nonsingular plane cubics displays sufficient complexity to make its study a test case for methods aimed at higher degree curves, so we emphasize the general character of our approach. The 9-parameter family of nonsingular cubics has ten characteristic numbers, denoted by \(N_{9,0}\), \(N_{8,1},\ldots,N_{0,9}\). By definition, each \(N_{\alpha,\beta}\) counts the nonsingular cubics which pass through \(\alpha\) general points, and are tangent to \(\beta\) general lines. (Because the condition to pass through a point is linear, we have \(N_{9,0}=1\). The other characteristic numbers, as we shall see, are not trivial.) These numbers were originally found independently by Maillard and Zeuthen in the 1870’s, by essentially heuristic methods. H. Schubert discussed them in his 1879 book [“Kalkül der abzählenden Geometrie” (Reprint 1979; Zbl 0417.51008)], so their rigorous justification, like the justification of the characteristic numbers of the singular cubics, is part of Hilbert’s 15th problem.

In our view, the characteristic numbers reflect the geometry of the closure, denoted by \(\Gamma\), of the graph of the correspondence between a nonsingular cubic and its dual sextic. Indeed, in the \(\mathbb{P}^ 9\) of all plane cubics, the divisor parametrizing the cubics tangent to a given line contains the locus of all nonreduced cubics, so that, for the first six characteristic numbers, the intersections of the corresponding divisors is proper, and it is even transversal for the first five. On \(\Gamma\), however, the intersection will be proper always for general points and lines, and so we can begin. To continue, we need to normalize to obtain a smooth (but noncomplete) parameter space, so that the intersecting subschemes of codimension 1 will become divisors.

A recent paper by Aluffi takes a different approach to finding the characteristic numbers of the smooth cubics: blow up \(\mathbb{P}^ 9\) successively along smooth loci until the intersection of the 9 strict transforms becomes proper. Aluffi obtains a smooth, complete parameter space, which dominates ours birationally. His computations proceed by determining explicitly the relevant intersection calculus, recursively, at each link of the blow-up chain. Earlier Sterz found some of the same characteristic numbers using a general plan in some ways similar to Aluffi’s. On a different tack, Miret and Xambó have vindicated Schubert’s more elaborate approach to the characteristic numbers of cuspidal cubics and nodal cubics, and they plan to do the same for smooth cubics. In addition, Aluffi and L. J. van Gastel [in Enumerative algebraic geometry, Proc. Zeuthen Symp., Copenhagen/Denmark 1989, Contemp. Math. 123, 259-265 (1991; see the following review)], have independently and somewhat differently, determined the first few nontrivial characteristic numbers of curves of higher degree.

We use a more basic parameter space than Aluffi and Sterz, but we need, as did Maillard, Zeuthen, and Schubert, the characteristic numbers of nodal cubics, which depend, in turn, on those for cuspidal cubics and conics. Along the way, we learn some fascinating things about the various singular cubic curves, and, as a benefit, our calculations, at each stage, are surprisingly simple. — Our results, as do Aluffi’s and Sterz’s, confirm the pioneers’ conclusions. Unlike theirs, our work also vindicates, and powerfully generalizes, the underlying strategy of Maillard and Zeuthen, just as the work of Miret and Xambó does that of Schubert. We do so by extending and strengthening the framework of general results about families of plane curves of any degree, with arbitrary degenerations, begun in our previous papers”.

From the text: “This paper completes the determination of the characteristic numbers for all plane cubics, begun with cuspidal cubics, and continued with nodal ones. The family of nonsingular plane cubics displays sufficient complexity to make its study a test case for methods aimed at higher degree curves, so we emphasize the general character of our approach. The 9-parameter family of nonsingular cubics has ten characteristic numbers, denoted by \(N_{9,0}\), \(N_{8,1},\ldots,N_{0,9}\). By definition, each \(N_{\alpha,\beta}\) counts the nonsingular cubics which pass through \(\alpha\) general points, and are tangent to \(\beta\) general lines. (Because the condition to pass through a point is linear, we have \(N_{9,0}=1\). The other characteristic numbers, as we shall see, are not trivial.) These numbers were originally found independently by Maillard and Zeuthen in the 1870’s, by essentially heuristic methods. H. Schubert discussed them in his 1879 book [“Kalkül der abzählenden Geometrie” (Reprint 1979; Zbl 0417.51008)], so their rigorous justification, like the justification of the characteristic numbers of the singular cubics, is part of Hilbert’s 15th problem.

In our view, the characteristic numbers reflect the geometry of the closure, denoted by \(\Gamma\), of the graph of the correspondence between a nonsingular cubic and its dual sextic. Indeed, in the \(\mathbb{P}^ 9\) of all plane cubics, the divisor parametrizing the cubics tangent to a given line contains the locus of all nonreduced cubics, so that, for the first six characteristic numbers, the intersections of the corresponding divisors is proper, and it is even transversal for the first five. On \(\Gamma\), however, the intersection will be proper always for general points and lines, and so we can begin. To continue, we need to normalize to obtain a smooth (but noncomplete) parameter space, so that the intersecting subschemes of codimension 1 will become divisors.

A recent paper by Aluffi takes a different approach to finding the characteristic numbers of the smooth cubics: blow up \(\mathbb{P}^ 9\) successively along smooth loci until the intersection of the 9 strict transforms becomes proper. Aluffi obtains a smooth, complete parameter space, which dominates ours birationally. His computations proceed by determining explicitly the relevant intersection calculus, recursively, at each link of the blow-up chain. Earlier Sterz found some of the same characteristic numbers using a general plan in some ways similar to Aluffi’s. On a different tack, Miret and Xambó have vindicated Schubert’s more elaborate approach to the characteristic numbers of cuspidal cubics and nodal cubics, and they plan to do the same for smooth cubics. In addition, Aluffi and L. J. van Gastel [in Enumerative algebraic geometry, Proc. Zeuthen Symp., Copenhagen/Denmark 1989, Contemp. Math. 123, 259-265 (1991; see the following review)], have independently and somewhat differently, determined the first few nontrivial characteristic numbers of curves of higher degree.

We use a more basic parameter space than Aluffi and Sterz, but we need, as did Maillard, Zeuthen, and Schubert, the characteristic numbers of nodal cubics, which depend, in turn, on those for cuspidal cubics and conics. Along the way, we learn some fascinating things about the various singular cubic curves, and, as a benefit, our calculations, at each stage, are surprisingly simple. — Our results, as do Aluffi’s and Sterz’s, confirm the pioneers’ conclusions. Unlike theirs, our work also vindicates, and powerfully generalizes, the underlying strategy of Maillard and Zeuthen, just as the work of Miret and Xambó does that of Schubert. We do so by extending and strengthening the framework of general results about families of plane curves of any degree, with arbitrary degenerations, begun in our previous papers”.

Reviewer: E.Stagnaro (Padova)

##### MSC:

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14H45 | Special algebraic curves and curves of low genus |