Timeline for Intersection of a smooth projective variety and a plane

6 events

when toggle format	what		by	license	comment
Dec 20, 2011 at 16:07	history	edited	Mike Roth	CC BY-SA 3.0	added 14 characters in body
Dec 20, 2011 at 15:50	comment	added	Mike Roth		Dear Jack - Yes, my previous justification for the irreducibility of the general fibre was completely wrong. I was thinking of the case that $X$ was smooth, and so $\overline{\Gamma}$ could be assumed normal. Then the general fibre is normal and so connectedness implies irreducibility. The general case of (possibly) non-normal $\overline{\Gamma}$ requires the use of the section to get around this, as your example shows.
Dec 19, 2011 at 20:32	comment	added	Jack Huizenga		Thanks for the edit. I originally couldn't justify your statement "Since $\overline{\Gamma}$ is irreducible with connected fibers over an irreducible base, its general fiber is irreducible." Of course you are right that the section is what helps out here. For other people who may have been confused, a counterexample is given as follows: consider the space of conic plane curves singular at the origin. This is naturally parameterized by a $\mathbb{P}^2$, and one can check the total space of the universal family is irreducible.
Dec 19, 2011 at 19:42	history	edited	Mike Roth	CC BY-SA 3.0	added 1309 characters in body
Dec 17, 2011 at 12:14	vote	accept	D.Z. Djokovic
Dec 16, 2011 at 21:27	history	answered	Mike Roth	CC BY-SA 3.0