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Apr 21, 2011 at 21:38 comment added Karl Schwede Sue, definitely Bertini theorems is the right way to go for references. One other reference I found is Exercise 10.4 of Eisenbud's "Commutative Algebra".
Apr 21, 2011 at 21:35 history edited Karl Schwede CC BY-SA 3.0
Added edit reflecting Long's example; added 4 characters in body; added 10 characters in body
Apr 21, 2011 at 20:43 comment added Sue Sierra The answer below makes it clear that there's no hope. Are you remembering anything else that might be useful to understand when this does happen?
Apr 21, 2011 at 18:29 history answered Karl Schwede CC BY-SA 3.0