Under what hypotheses are schematic fixed points of a flat deformation themselves flat? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:30:47Z http://mathoverflow.net/feeds/question/43438 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43438/under-what-hypotheses-are-schematic-fixed-points-of-a-flat-deformation-themselves Under what hypotheses are schematic fixed points of a flat deformation themselves flat? Ben Webster 2010-10-24T22:06:52Z 2010-10-25T21:15:25Z <p>This is something of a follow-up question to <a href="http://mathoverflow.net/questions/43255/" rel="nofollow">this one</a>; I hope people won't think this is a duplicate. At least in my head, it seems like a distinct enough question to merit a fresh start.</p> <p>All my schemes will be finite type over an algebraically closed field $k$. Let $X\to S$ be a flat family of affine schemes over smooth affine base. Let's say for now that each fiber and the whole family have rational singularities, and thus are Cohen-Macaulay. Assume, furthermore, that $X$ has an action of the group scheme $T=(\mathbb{G}_m)_S$; this is the same data as a grading on $k[X]$ such that $k[S]$ has degree 0.</p> <p>Now, we can take the schematic fixed points $X^T$ of this family, which is a subscheme of $X$ whose points over any ring are invariant points of $X$. Concretely, this is the vanishing set of the ideal generated by all functions of non-zero degree. </p> <blockquote> <p>Must the morphism $X^T\to S$ be flat? If not, are there stricter hypotheses than I gave above which would assure it is?</p> </blockquote> <p>For example, consider the family $$X=\mathrm{Spec}[x,y,z,a_0,\dots, a_{n-1}]/(xy=z^n+a_{n-1}z^{n-1}+\cdots + a_0)$$ where $S=\mathrm{Spec}[a_0,\dots, a_{n-1}]$ with $x$ having degree 1, $y$ degree $-1$ and $z,a_i$ having degree 0. In this case $$X^T=\mathrm{Spec}[z,a_0,\dots, a_{n-1}]/(z^n+a_{n-1}z^{n-1}+\cdots + a_0=0),$$ which is, of course, flat over $S$, even though the number of closed points in a fiber (the number of roots of $z^n+a_{n-1}z^{n-1}+\cdots + a_0$) varies from $n$ to $1$.</p> http://mathoverflow.net/questions/43438/under-what-hypotheses-are-schematic-fixed-points-of-a-flat-deformation-themselves/43523#43523 Answer by Angelo for Under what hypotheses are schematic fixed points of a flat deformation themselves flat? Angelo 2010-10-25T14:25:22Z 2010-10-25T21:15:25Z <p>Here is a counterexample. Let $\mathbb G_{\rm m}$ act on $\mathbb A^2$ by $t\cdot(x,y) = (tx,t^{-1}y)$, and let $f\colon \mathbb A^2 \to \mathbb A^1$ be defined by $f(x,y) = xy$.</p> <p>I am positive that when $X$ is smooth over $Y$, the fixed point scheme is also smooth; but I doubt that one can say much more, in general.</p> <p> Here is a variant. Let $\mathbb G_{\rm m}$ act on $\mathbb A^4$ by $t\cdot(x,y, z, w) = (tx,t^{-1}y,tz,t^{-1}w)$, and let $f\colon \mathbb A^4 \to \mathbb A^1$ be defined by $f(x,y) = xy + zw$.</p>