Consider a blow up $\pi:X\longrightarrow Q$ of a smooth quadric hypersurface along a twisted quartic $C_4\subset Q\subset \mathbb{P}^4$. I can't understand why picard number $\rho(X)=2$. I think $Pic(Q)=\mathbb{Z}[l_1]\oplus\mathbb{Z}[l_2]$ and $Pic(X)=\mathbb{Z}[l_1]\oplus\mathbb{Z}[l_2]\oplus\mathbb{Z}[E]$ but I don't know how to find $\rho$. Also is it right, that $K_X=-2\pi^*H +E$ ?
2026-03-27 02:02:48.1774576968
Picard number of a blown up quadric
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