# Example II.6.5.2 in Hartshorne (Part II)

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 $$Y$$?

• 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? – jgon Feb 14 at 6:38
• @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. – Manos Feb 14 at 12:13