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## 3.ii. Problems not involving general conditions

We give a brief tour of some enumerative problems that do not involve general conditions, but nonetheless raise some interesting questions regarding real solutions.

Schläfli (see the survey of Coxeter [Co]) showed there are 4 possibilities for the number of real lines on a real cubic surface: 27, 15, 7, or 3. Recent developments in enumerative geometry (mirror symmetry) have led to the solution of a large class of similar enumerative problems involving, among other things, the number of rational curves of a fixed degree on a Calabi-Yau threefold. (See the book of Cox and Katz [CK].) For example, on a general quintic hypersurface in P4 there are 2875 lines [Harr], 609,250 conics [Ka], and 371,206,375 twisted cubics. The number of twisted cubics and higher degree rational curves was computed in the seminal paper of Candelas, de la Ossa, Green, and Parkes [COGP]. How many of the curves can be real in problems of this type? For example, how many real lines can there be on a real quintic hypersurface in P4?

A real homogeneous polynomial f(x) is positive semi-definite (psd) if f(x) is non-negative whenever x is real. Hilbert [Hi1] proved that a psd ternary quartic is a sum of three squares of real quadratic forms. In fact, a general quartic is a sum of 3 squares of complex quadratic forms in 63 different ways [Wa]. Powers and Reznick [PR] studied the question of how many ways one may represent a ternary quartic as a real sum or difference of squares. In every instance, they found that 15 of the 63 ways involved real quadratic forms. Is it true that a general psd quartic is a sum or difference of real squares in exactly 15 different ways?§

A general plane curve C of degree d has 3d(d-2) flexes. These are the points on C where the Hessian determinant of the form defining C vanishes. Since the Hessian determinant has degree 3(d-2), we expect there to be 3d(d-2) such points. This involves intersecting the curve with its Hessian curve, and not with a general curve of degree 3(d-2). A real smooth plane cubic has 3 of its 9 flexes real. Zeuthen [Ze2] found that at most 8 of the 24 flexes of a real plane quartic can be real. An example of a plane quartic with 8 real flexes is provided by the Hilbert quartic [Hi2], which is defined by

(x2 + 2y2 - z2) (2x2 + y2 - z2) + z4   =   0 .

We display this curve in Figure 3, marking the flexes with dots.

 Figure 3: Hilbert's quartic: a plane quartic with 8 flexes (It is instructive to view this quartic together with its Hessian

Klein [Klein] later showed that a general real plane curve has at most 1/3 of its flexes real.

Harnack [Harn] proved that a smooth real algebraic curve of genus g has at most g + 1 topological components, and he constructed real algebraic curves of genus g with g + 1 components. In particular, a plane curve of degree d has genus g = (d - 1)(d - 2)/2 and there are real plane curves of degree d with g + 1 components. (An example is provided by Hilbert's quartic, which has genus 3.) Finer topological questions than enumerating the components leads to (part of) Hilbert's 16th problem [Hi3], which asks for the determination of the topological types of smooth projectively embedded real algebraic varieties.

A variant concerns rational plane curves of degree d. A general rational plane curve of degree d has 3(d - 2) flexes and g = (d - 1)(d - 2)/2 nodes. Theorem 4.2 implies that there exist real rational plane curves of degree d with all 3(d - 2) flexes real, which we call maximally inflected curves. See Remark 5.8 for the connection. Such curves have at most g - d + 2 of their nodes real, and there exist curves with the extreme values of 0 and of g - d + 2 real nodes [KS]. For example, a rational quartic (d=4) has 6 flexes and g=3 nodes. If all 6 flexes are real, then at most one node is real. Figure 4 shows maximally inflected quartics with and 0 and 1 nodes. The flexes are marked by dots.

 Figure 4: Rational quartics with 6 real flexes View these curves in motion

Recently, Huisman asked and answered a new question about real curves. A component X of a real algebraic curve is a psuedoline if its homology class [X] in H1(PnR, Z/2Z) is non-zero, and an oval otherwise.

Question 3.2   Given a smooth (irreducible over C) real algebraic curve C in Pn of genus g and degree c, how many real hypersurfaces of degree d are tangent to at least s components of C with order of tangency at least m ?

Let N be that number, when there are finitely many such hypersurfaces.

Theorem 3.3 (Huisman [Hu], Theorem 3.1) When s=g and gm = cd, and the restriction

H0(Pn, O(d))   ---->   H0(C, O(d))

is an isomorphism, then N is finite. Moreover, N is non-zero if and only if
1. Both m and d are odd and C consists of exactly g psuedolines, or
2. m is even and either d is even or all components of C are ovals.
In case (1), N=mg, and in case (2),

 N   = (g+1)mg     if C has g + 1 components mg     if C has g components.

It is notable that this problem can only be stated over the real numbers.

§The answer to this question is YES, with Powers, Reznick, and Scheiderer, we are addressing this and related questions. For example, exactly 8 of the 15 will be sums of squares, while 15 will involve differences of squares.

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