A rational curve of degree d with 3(d-2) flexes and the maximal number of (d-1)(d-2)/2 real double points

V. Kharlamov and F. Sottile


14 August, 2001

    We construct maximally inflected curves of degree d with the maximum number (d-1)(d-2)/2 nodes by deforming a particular reducible curve in a precise way using Shustin's theory [Sh99] to control the singularities in the deformation. The idea for this comes from the two degree 4 curves pictured below. The curve on the right is a deformation of the reducible curve on the left.

Specifically, the reducible curve is defined by
xy (3(x - y)2 + (3x + 3y - 4)2 - 4)   =   0 ,
and is the union of the 2 coordinate axes and an ellipse. The curve on the right is defined by the polynomial obtained by subtracting
(1/20)(x + y)2(x + y - 1)2
from the equation of the reducible curve. This deforming term defines two parallel lines passing through the singularities of the reducible curve.

    The quartic curve on the right has 2 solitary points at (1,0) and (0,1) and a node at the origin. By the genus formula, it is rational. It also has 6 flexes. Four are indicated by circles, and there is one more along each asymptote, as the curvature of the segment of the curve along the asymptote is different in each of the visible branches. (The tubular neighborhood of an axis is a Möbius band.) Thus any parameterization of this curve is a maximally inflected quartic with 6 flexes, 1 node, and 2 solitary points.

    The more sophisticated ideas needed for the general construction are provided by Shustin's theory. We describe the ingredients of this deformation, verify that it is valid, and lastly show that the curve we constructed has the desired properties. Fix an integer d > 2 throughout and let P0 be the union of a conic and any d-2 distinct lines tangent to the conic. Then P0 is a reducible curve of degree d. Each pair of tangents meet and no three meet in a point as the dual curve to the conic is another conic. Thus P0 has (d-2)(d-3)/2 simple nodes and d-2 other singularities at the points of tangency. We deform those tangency singularities while preserving the nodes.

    In a neighborhood V of each point of tangency, Pd is isomorphic to the reducible curve K0 given by the equation

y(y - x2)   =   0 ,
in some neighborhood U of the origin. For each t in R, let Kt be the deformation of K0 defined in U by
y(y - x2) + tx2   =   0 . (1)
For t positive but sufficiently small, Kt has a solitary point at the origin and 2 flexes along each branch near the parabola y=x2, but within U. Moreover, it lies above the x-axis. We display the reducible curve on the left and the deformation Kt for t=1/15 on the right.

    If a certain cohomology group vanishes, then Shustin's Theorem [Theorem 1 in Sh99] guarantees the existence of a real number e>0 and of a deformation Pt of the curve P0 for t in the interval (0,e) such that (1) Pt has degree d, (2) Pt has a node in a neighbourhood of each node of P0, and (3) in a neighborhood V of each point of tangency of P0, Pt is isomorphic to Kt, in the neighborhood U of the origin. Another result of Shustin [Lemma 3 in Sh99] gives the numerical criterion for the vanishing of this cohomology group

(deg Xz - isdz)   <   4(d-1) ,
the sum over all singular points z of the curve P0, where Xz is the scheme of the singularity z and isdz is a positive integer. When z is a node, Xz has multiplicity 1, and when z is a point of tangency, Xz has multiplicity 2. Thus the above sum is at most d-2, and so the deformation exists.

    For 0 < t < e, the curve Pt has (d-2)(d-3)/2 nodes and d-2 solitary points, by the construction. Since it has degree d, it is rational. Also from the construction, we see that each solitary point contributes 2 flexes, accounting for 2(d-2) flexes. Furthermore, there is an additional flex along each asymptote (the original tangent lines) as the concavity of Pt changes while passing through infinity. Thus any parameterization of the curve Pt gives a maximally inflected curve of degree d with 3(d-2) flexes, (d-2)(d-3)/2 nodes and d-2 solitary points. Here is a picture of the reducible curve and the maximally inflected deformation, when d=5.

Theorem. For each k between 0 and d-2, there is a maximally inflected curve with k cusps, 3(d-2)-2k flexes, d-2-k solitary points, and (d-2)(d-3)/2 nodes.

Proof . In the previous construction, we may replace the local model Kt defined in (1) by the local model Lt defined by

y(y - x2) + tx3   =   0 . (2)
For t > 0, Lt has a cusp at the origin and one flex near the origin. We display Lt for t=1/10 with the singular curve L 0 on the left.
For every tangency point where we use the local perturbation Lt, we will get a cusp, no solitary point, and only one flex; the flex along the tangent line does not appear, as the concavity of Lt does not change along that line.
    The result follows as we may do this for any of the d-2 tangency points.

    For t > 0, the cusp in the curve Lt (2) is on the branch to the left of the origin. Had we instead used the perturbation Mt given by

y(y - x2)  -  tx3   =   0 ,
then the cusp is now on the right branch for t>0. There are 3d-2 different ways to use one of these three local models Kt, Lt, or Mt for the perturbation of the tangency singularities. Not all give topologically distinct curves, as there are some symmetries; for example, each of the 2(d-2) ways to have a single cusp are the same topologically. Also, not all possibilities for ordering of cusps and flexes along the curve Pt is possible; On any such curve, each cusp is either preceded or followed by a flex. Thus for instance, we cannot obtain three consecutive cusps with this construction.

References
[Sh99] E. Shustin, Lower deformations of isolated hypersurface singularities, Algebra i Analiz, 10 (1999), pp. 221--249.

Based upon work supported by the National Science Foundation under Grant No. 0070494.