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
**P**^{4} 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 **P**^{4}?

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
3*d*(*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
3*d*(*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

(*x*^{2} + 2*y*^{2} - *z*^{2})
(2*x*^{2} + *y*^{2} - *z*^{2})
+ *z*^{4} = 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 |

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.

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
*H*^{1}(**P**^{n}_{R},
**Z**/2**Z**) is
non-zero, and an *oval* otherwise.

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

is an isomorphism, then

- Both
*m*and*d*are odd and*C*consists of exactly*g*psuedolines, or *m*is even and either*d*is even or all components of*C*are ovals.

N = |
(g+1)m if
^{g}C has g + 1 components |

m if
^{g}C has g components. |

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