Give an example about flatness. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:56:45Z http://mathoverflow.net/feeds/question/73401 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73401/give-an-example-about-flatness Give an example about flatness. google 2011-08-22T12:35:19Z 2011-08-22T15:15:54Z <p>Please give an example of a flat family {X_t} of closed subschemes of PP^n such that the family of projective cones of X_t is not a flat family in PP^{n+1}.</p> <p>I still could not find...</p> http://mathoverflow.net/questions/73401/give-an-example-about-flatness/73413#73413 Answer by David Speyer for Give an example about flatness. David Speyer 2011-08-22T15:15:54Z 2011-08-22T15:15:54Z <p>As an example, let $X_t$ be three points in $\mathbb{P}^2$, at positions $(1:0:0)$, $(0:1:0)$ and $(1:1:t)$. This is certainly a flat family. Notice that the points are colinear when $t=0$ and not for $t \neq 0$.</p> <p>For $t \neq 0$, the homogenous ideal of $X_t$ is <code>$$\langle xz-txy, yz-txy, z^2-t^2 xy \rangle$$</code></p> <p>Let $Y$ be the closure of <code>$\langle xz-txy, yz-txy, z^2-t^2 xy \rangle$</code> in $\mathbb{A}^3 \times \mathbb{A}^1$. If I haven't made any errors, the ideal of $Y$ is <code>$$\langle xz-txy, yz-txy, z^2-t^2 xy, xy(x-y) \rangle.$$</code> Since I defined $Y$ as the closure of a flat family over a smooth one dimensional base, it is flat. For $t \neq 0$, the fiber $Y_t$ is the cone on $X_t$.</p> <p>However, at $t=0$, the fiber $Y_0$ is not the cone on $X_0$. They are the same away from the origin, but $Y_0$ has an extra nonreduced bit. Specifically, since $X_0$ lies on the line $z=0$, the cone on $X_0$ has $z$ in its defining ideal, while $Y_0$ has nothing in degree $1$. You can picture $Y_0$ as three coplanar lines, plus a nonreduced tangent vector pointing out the the $z=0$ plane; the cone on $X_0$ doesn't have that extra nonreduced piece.</p> <p>What may be confusing you is that there isn't some non-flat family here for me to point to. The family $Y_t$, which exists, is flat, but its fibers are not the cones on the $X_t$. There is no closed subfamily of $\mathbb{A}^3 \times \mathbb{A}^1$ whose fibers are the cones on the $X_t$. This is the general picture whenever you have a flat family in $\mathbb{P}^N$ which does not lift to a flat family of projective cones.</p>