I'm studying Algebraic geometry by "Enumerative Geometry And String Theory" [Katz]. In section 8.3, he computed the excess contribution 31 and concluded that the number of smooth conics tangent to five given general lines is 32-31=1.
Then, I want to check the number of smooth conics tangent to five given general "conics" like this way. We know the answer 3264. In this situation, the excess contribution should be 7776-3264=4512.
- Q1. To begin with, can I check the excess contribution 4512 by Chern Class ?
The computation p.122-123 in the book is next. $h^i =0 (i \geq 3). h \in H^2(\mathbb{P}^2), H \in H^2(\mathbb{P}^5). H=2h.$
$ \begin{eqnarray} c(N_{\mathbb{P}^2/\mathbb{P}^5})&=&c(T\mathbb{P}^5)|_{{\mathbb{P}^2}}/c(T{\mathbb{P}^2})\\ &=&((1+H)^6)|_{\mathbb{P}^2}/(1+h)^3 \\ &=&1+9h+30h^2 \end{eqnarray}$
Then
$\begin{eqnarray} c(B)&=&((1+2H)^5|_{\mathbb{P}^2})(1+9h+30h^2)^{-1} \\ &=&(1+4h)^5(1+9h+30h^2)^{-1} \\ &=&1+11h+31h^2 \end{eqnarray}$
$c_2(B)=31h^2$. Therefore, the excess contribution is 31.
- Q2. What's the difference lines and conics in this computation?