The Gromov-Witten number for degree 3 curves of genus zero is 12.
I would like to see an example in which exactly 12 irreducible curves of degree 3 and genus 0 pass through 8 common points in the projective plane
I do understand that the 8 points must be such that each point provides additional restriction.
One idea I had was to set up a system of 8 equations $$Bx_i^{2}y_i+Cx_iy_i^{2}+Dy_i^{3}+Ex_i^{2}+Fx_iy_i+Gy_i^{2}+Hx_i+Iy_i=-J-x_i^3$$ for $i=1,2,3,4,5,6,7,8$
Then I can write all of the other coefficients in terms of $J$
Then perhaps there exists a sufficiently large $a\in (0,\infty)$ such that as I continuously vary $J$ from $-a$ to $a$ there will be exactly 12 $J$ values that produce a genus zero curve.
However, I believe this approach might be ineffective because I believe some of the 8 points must be points at infinity