1.5. LOWER BOUNDS 9
associativity in quantum cohomology of
P2
to give the elegant recursion
(1.8) Nd =
a+b=d
NaNb
a2b2
3d 4
3a 2

a3b
3d 4
3a 1
,
which begins with the Euclidean declaration that two points determine a line (N1 =
1). These numbers grow quite fast, for example N5 = 87304.
The number of real rational curves which interpolate a given 3d 1 points in
the real plane
RP2
will depend rather subtly on the configuration of the points. To
say anything about the real rational curves would seem impossible. However this is
exactly what Welschinger [162] did, by finding an invariant which does not depend
upon the choice of points.
A rational curve in the plane is necessarily singular—typically it has
(d−1)
2
ordinary double points. Real curves have three types of ordinary double points.
Only two types are visible in
RP2,
and we are familiar with them from rational
cubics, which typically have an ordinary double point. The curve on the left below
has a node with two real branches, and the curve on the right has a solitary point
‘•’, where two complex conjugate branches meet.
The third type of ordinary double point is a pair of complex conjugate ordinary
double points, which are not visible in
RP2.
Theorem 1.13 (Welschinger [162]). The sum,
(1.9)
(−1)#{solitary points in C}
,
over all real rational curves C of degree d interpolating 3d−1 general points in
RP2
does not depend upon the choice of points.
Set Wd to be the sum (1.9). The absolute value of this Welschinger invariant
is then a lower bound on the number of real rational curves of degree d interpo-
lating 3d−1 points in
RP2.
Since N1 = N2 = 1, we have W1 = W2 = 1. Prior to
Welschinger’s discovery, Kharlamov [33, Proposition 4.7.3] (see also Example 9.3)
showed that W3 = 8. The question remained whether any other Welschinger invari-
ants were nontrivial. This was settled in the affirmative by Itenberg, Kharlamov,
and Shustin [77, 78], who used Mikhalkin’s Tropical Correspondence Theorem [99]
to show
(1) If d 0, then Wd
d!
3
. (Hence Wd is positive.)
(2) lim
d→∞
log Nd
log Wd
= 1. (In fact for d large, log Nd 3d log d log Wd.)
In particular, there are always quite a few real rational curves of degree d interpo-
lating 3d−1 points in
RP2.
Since then, Itenberg, Kharlamov, and Shustin [79] gave
a recursive formula for the Welschinger invariant which is based upon Gathmann
and Markwig’s [60] tropicalization of the Caporaso-Harris [26] formula. This shows
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