Suppose $\newcommand{\X}{\mathcal{X}}\X$ is an algebraic stack over a field $k$, $\xi$ is a $k$-point which has another $k$-point $x$ in its closure ($x$ is an

isotrivial degenerationof $\xi$). Must the dimension of $Aut(x)$ be larger than the dimension of $Aut(\xi)$?

If $\X=[X/G]$, where $G$ is an algebraic group over $k$, then the answer is yes. An isotrivial degeneration corresponds to a $G$-orbit with another $G$-orbit in its closure. The closure of a $G$-orbit can only contain smaller dimensional orbits, and the dimension of the orbit is complementary to the dimension of the stabilizer of a point in that orbit, so when one orbit degenerates to another, there is always a jump in the dimension of the stabilizer.

Almost every algebraic stack I can think of is étale-locally of the form $[X/G]$, and étale maps preserve the dimensions of automorphism groups. This suggests that the answer is probably "yes."

## Example

Consider the 1-parameter family of $2\times 2$ matrices $\begin{pmatrix}1&t\\ 0&1\end{pmatrix}$. If we are studying matrices up to conjugation, then away from $t=0$, the family is (isomorphic to) the constant family $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$. However, at $t=0$ you get a different Jordan type. So we say that this family is an *isotrivial degeneration* of Jordan types $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}\rightsquigarrow \begin{pmatrix}1&0\\ 0&1\end{pmatrix}$. When you pass to the more degenerate Jordan type, the automorphism group (i.e. the group of ways in which the matrix is self-conjugate) jumps from something 2-dimensional to something 4-dimensional.

## Motivation

It sometimes happens that you want to determine all degeneration relations among a collection of points in an algebraic stack. For example, you may be trying to determine if some map is weakly proper.^{†} If the answer to this question is "yes," then you can rule out certain degenerations by looking at dimensions of automorphism groups.

^{†}See Weakly proper moduli stacks of curves by Alper, Smyth, and van der Wyck.

downbecause of upper semi-continuity of the fiber dimension of the inertia (see mathoverflow.net/questions/193/…). The question is whether the dimensionmust jump upin the case of an isotrivial degeneration. $\endgroup$ – Anton Geraschenko Mar 12 '11 at 22:36