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Dec 20, 2020 at 12:37 comment added D. Dona It just occurred to me that the same works even in $\mathbb{A}^{3}$, forgetting about the $w$ coordinate. In that case the counterexample is truly "minimal", since $\mathbb{A}^{2}$ doesn't have enough room to house a counterexample.
Dec 15, 2020 at 10:15 vote accept H A Helfgott
Dec 15, 2020 at 9:44 history edited pinaki CC BY-SA 4.0
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Dec 15, 2020 at 3:14 history edited pinaki CC BY-SA 4.0
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Dec 14, 2020 at 23:41 comment added pinaki @HAHelfgott: I missed the requirement for irreducibility - now updated the answer.
Dec 14, 2020 at 23:38 history edited pinaki CC BY-SA 4.0
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Dec 14, 2020 at 21:46 comment added H A Helfgott Moreover, $V$ can only be the origin or the $y$-axis, so it seems this isn't a counterexample at all - not even if (as in D. Dona's answer) the condition that $C$ be irreducible is dropped.
Dec 14, 2020 at 21:43 comment added H A Helfgott In this example, $C$ is reducible.
Dec 14, 2020 at 21:28 history answered pinaki CC BY-SA 4.0