I am learning the notion of intersection product of a $\mathbb{Q}$-Cartier divisor with an irreducible complete curve on a normal variety. The definition I learned is that if $D$ is a $\mathbb{Q}$-Cartier divisor on a normal variety $X$ and $C$ is an irreducible complete curve on $X$, such that $lD$ is a Cartier divisor for some positive integer $l$, then we define $D.C$ to be $\frac{1}{l} (lD).C$. I would like to show that this notion is well-defined. What I have tried is to start with some local data $\{U_i,f_i\}$ for the Cartier divisor $lD$, and try to see why there is some factor $l$ coming from the intersection product $(lD).C$. But I am stuck at the beginning. Could someone help me with this problem?Thank you so much in advance.
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2$\begingroup$ Why do you expect a factor $l$? Note that this intersection pairing takes values in $\mathbb{Q}$. $\endgroup$– SashaCommented Jan 16, 2023 at 16:45
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$\begingroup$ Thank you very much for your comments. I was thinking that the $l$ in the denominator for the definition of the intersection product needs to be cancelled by the intersection product $lD.C$ so that the resulting intersection product $D.C$ is well-defined. $\endgroup$– BorisCommented Jan 16, 2023 at 19:50
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$\begingroup$ No, it doesn't need to be canceled. What is indeed crucial is that torsion elements of $\mathrm{Pic}(X)$ all have zero intersection products with curves. $\endgroup$– SashaCommented Jan 17, 2023 at 4:16
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$\begingroup$ Thank you very much for your insightful comments. $\endgroup$– BorisCommented Jan 18, 2023 at 13:47
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