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