Gromov-Witten invariants in Fano varieties of dimension 3 and index 2: an example

In this talk, I give a formula for counting complex rational curves passing through certain configuration of points on the 3- dimensional projective space $\mathbb{C}P^1\times\mathbb{C}P^1\times\mathbb{C}P^1$ by using the counting on the 2-dimensional projective space $\mathbb{C}P^1\times\mathbb{C}P^1$ blowing-up at 2 points. That gives rise to the fancy formula for Gromov-Witten (GW) invariants in the Fano threefolds of index 2 with related to the GW invariants in one specific type of surfaces which realise one half of their first Chern class.