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A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, should one demand $S$ to be excellent? Will anything change if we want singular points not to be dense in any finite type reduced $S$-scheme?

I would also be gratefull for any ('bad' or 'good') examples.

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Are you talking about schemes over a field of positive characteristic? – Qfwfq Apr 17 '10 at 11:23
Given that he says the word "excellent", I think he's talking about schemes. – Kevin Buzzard Apr 17 '10 at 12:03
Excellence satisfied for f.t. over char-0 Dedekind domain or field or complete local noetherian ring (and more), and for excellent $S$ "every" openness result holds: regular locus, CM locus, normal locus, etc. Although one can write down non-excellent dvrs, anything reduced of finite type over any dvr has dense open regular locus: pass to domain case and then either generic fiber is dense open (so apply theory over frac field) or whole thing is over the residue field. Hochster gave (non-semi-local!) 1-dim. noetherian domains non-reg. at all max. ideals! See 1973 paper "Non-openness loci..." – – BCnrd Apr 17 '10 at 13:42
Thank you very much, Brian! – Mikhail Bondarko Apr 17 '10 at 15:01
up vote 3 down vote accepted

For examples with dense singular locus, see William J. Heinzer and Lawrence S. Levy: Domains of Dimension 1 with Infinitely Many Singular Maximal Ideals, Rocky Mountain J. Math. (2007), 203-214. Their examples are affine and noetherian.

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Oops, I did'nt see the end of BCnrd's comment. So my answer is usefulness. – Qing Liu Jun 1 '10 at 20:34

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