Are irreducible components of a flat family flat? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:32:57Z http://mathoverflow.net/feeds/question/114158 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114158/are-irreducible-components-of-a-flat-family-flat Are irreducible components of a flat family flat? quim 2012-11-22T13:50:29Z 2012-11-22T14:14:27Z <p>Let $f:X\rightarrow Y$ be a flat morphism of schemes of finite type over a field $k$, and assume $Y$ is irreducible. Let $X_1, \dots, X_n$ be the scheme-theoretic irreducible components of $X$ (i.e., including embedded components). </p> <ul> <li>Is it true that each $X_i$ is flat over $Y$?</li> <li>If there are counterexamples to flatness of the $X_i$, is it true at least that each of them has equidimensional fibers?</li> </ul> http://mathoverflow.net/questions/114158/are-irreducible-components-of-a-flat-family-flat/114159#114159 Answer by Tom Goodwillie for Are irreducible components of a flat family flat? Tom Goodwillie 2012-11-22T14:14:27Z 2012-11-22T14:14:27Z <p>No to the first question. Let $Y$ be a nodal cubic curve and let $X$ be its connected two-sheeted covering space. Each of the two components of $X$ is the normalization of $Y$.</p>