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Let $A= k[x,y,z]/(xy-z^2)$ and set $X = \operatorname{Spec}A$. Let $Y$ be the prime divisor of $X$ corresponding to the ideal generated by $y,z$. The example shows that $\operatorname{Cl}(X)$ is generated by $1 \cdot Y \neq 0$ and $2 \cdot Y = 0$. In what sense is this to be interpreted that $X$ is generated by a ruling of $Y$?

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  • $\begingroup$ I'm not sure what your question is. I looked up the example. It doesn't say that $X$ is generated by a ruling of $Y$. It says that $Y$ is a ruling of the cone $X$, and that $Y$ generates $\operatorname{Cl}(X)$. If I'm misunderstanding something, please let me know? $\endgroup$ – jgon Feb 14 at 6:38
  • $\begingroup$ @jgon: By "X generated by a ruling of $Y$" i mean $X$ is the points swept by some continuous motion of $Y$. I believe that is what ruling means informally. $\endgroup$ – Manos Feb 14 at 12:13

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