Step 2 · Blowup square

Oriented cohomology ring of $\Bl_V\mathbb P^5$

Here $\alpha,\eta$ are the hyperplane classes on $\mathbb P^5$ , $V\cong\mathbb P^2$ , and $\zeta=c_1^A(\mathcal O_E(1))$ . We also write $\alpha,\eta$ for their pullbacks by $\pi,p$ .

$E=\mathbb P(\operatorname{Sym}^2\mathcal T_{\mathbb P^2})$

$A^\bullet(E)=\frac{A^\bullet(\mathrm{pt})[\eta,\zeta]}{\left(\begin{gathered}\eta^3,\\\zeta^3+(9\eta+6a_{11}\eta^2)\zeta^2+30\eta^2\zeta\end{gathered}\right)}.$

An $A^\bullet(\mathrm{pt})$ -basis: $\{\eta^a\zeta^b\}_{0\le a,b\le2}$ .

$j^*\alpha=2\eta+a_{11}\eta^2,\qquad j^*e_{0,0}=\chi_A(\zeta).$

j

$\Bl_V\mathbb P^5$

4 kinds of relations, with exceptional generators

$e_{a,b}=j_*(\eta^a\zeta^b),\qquad0\le a,b\le2.$

p

\pi

$V=\nu_2(\mathbb P^2)$

$A^\bullet(V)=\frac{A^\bullet(\mathrm{pt})[\eta]}{(\eta^3)}.$

$i^*\alpha=[2]_F\eta=2\eta+a_{11}\eta^2,\qquad p^*\eta=\eta.$

i

$\mathbb P^5$

$A^\bullet(\mathbb P^5)=\frac{A^\bullet(\mathrm{pt})[\alpha]}{(\alpha^6)}.$

$\begin{aligned}i_*1&=4\alpha^3+3a_{11}\alpha^4+3a_{21}\alpha^5,\\i_*\eta&=2\alpha^4+a_{11}\alpha^5,\qquad i_*\eta^2=\alpha^5.\end{aligned}$