Let $X$ be the double cover of $\mathbb{P}^2$ branched along a divisor which is union of two lines. Then what will be the $\text{Pic}(X)$ ? Is it torsion free ? If yes, then what is its generator ?
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This double cover is a quadratic cone in $\mathbb{P}^3$, its Picard group is $\mathbb{Z}$, it is generated by the hyperplane class, equivalently by the pullback of a line class from $\mathbb{P}^2$.
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$\begingroup$ Suppose it is branched along $l_1 \cup l_2$ and let $C_1, C_2$ are the inverse image of $l_1, l_2$ respectively. Then how to write $\mathcal{O}_X(C_1)$ in terms of pullback of the hyperplane class ? $\endgroup$ Commented Apr 26, 2020 at 6:28
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$\begingroup$ What do you precisely mean by the inverse images of $l_i$? If the pullbacks, then these are just hyperplane classes. $\endgroup$– SashaCommented Apr 26, 2020 at 6:37
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$\begingroup$ By the inverse image i mean $\pi^{-1} (l_i)$ where $\pi : X \to \mathbb{P}^2$ is the covering map. I did not mean the pullback. $\endgroup$ Commented Apr 26, 2020 at 6:59
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$\begingroup$ Being the inverse image of a closed subset $l_i$, $C_i$ is a closed subset (hence a divisor in $X$) $\endgroup$ Commented Apr 26, 2020 at 7:02
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$\begingroup$ As @Sasha explained, $X$ is a quadratic cone. Your $C_i$'s are just 2 generatrices of this cone. They are Weil divisors, not Cartier. $\endgroup$– abxCommented Apr 26, 2020 at 7:25