Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ where $L$ is the homology class of a line and $E_i$ are the exceptional divisors. Also, define $$ \delta_{\beta} := < c_1(TX_k), ~\beta>-1= 3n + m_1 + \ldots m_k-1. $$

Let $N_{\beta}$ be the number of genus zero curves in the class $\beta$ passing through $\delta_{\beta}$ generic points.

$\textbf{Questions:} $ I have two questions. In their paper

http://www.ihes.fr/~maxim/TEXTS/WithManinCohFT.pdf

Kontsevich and Mannin give a recursive formula to compute $N_{\beta}$ (page $29$).

1) Is it known that the numbers $N_{\beta}$ that one gets from their formula are actually the enumerative numbers (i.e. they are actually the honest count of curves through the right number of generic points)? A priori, Gromov Witten Invariants need not be enumerative and I suspect the formula given by Kontsevic and Mannin are for the genus zero GW invariants. In particular, on page $26$ of their paper (second last paragraph), they make the remark

"We expect that $N_{\beta}$ counts the number of rational curves in the homology class $\beta$ passing through $\delta_{\beta}$ points, at least in unobstructed problems. "

This remark seems to suggest that at the time of writing the paper they did not know if the numbers are actually enumerative. Is this presently known (i.e are genus zero GW Invariants on Del-Pezzo surfaces enumerative)? The answer is yes for $\mathbb{P}^2$.

2) My second question is how does one actually compute $N_{\beta}$ using their recursive formula? One needs enough initial conditions for the recursion. On page $29$ (just after they state the formula) they say that $N_{\beta}$ is ``expected'' to be one for all indecomposable $\beta$. This seems to imply that $$N_{3L-E_1-E_2-\ldots- E_8} = 1.$$

But as observed by Mark in this post

What are the indecomposable classes on a del-Pezzo surface?

it seems that this number ought to be the same as the number of rational planar cubics through $8$ generic points, i.e. $12$. So what have I misunderstood here?

freegenus $0$ class $\beta$ on a uniruled variety, there is an associated enumerative Gromov-Witten invariant with one point insertion. This is the basis for the Koll'ar - Ruan theorem on symplectic invariance of uniruledness. Of course for rational connectedness this is much harder, with the best results due to Zhiyu Tian. You might also consult the thesis of Damiano Testa. $\endgroup$ – Jason Starr May 28 '15 at 8:58