Higher Essential Dimension: First Steps

Abstract: Let $G$ be a linear algebraic group over a field $k$. Loosely speaking, the essential dimension of $G$ measures the number of independent parameters that are required to define a $G$-torsor over a $k$-field. It measures the complexity of $G$-torsors and equivalent objects. One formal way to define it is to say that the essential dimension of $G$ is $\leq m$ if every $G$-torsor over a finite-type $k$-scheme is, away from some codimension-$1$ closed subscheme, the specialization of a $G$-torsor over a finite-type $k$-scheme of dimension $m$.

Recently, for every integer $d\geq 0$, we defined the $d$-essential dimension of $G$, denoted $\mathrm{ed}^{(d)}(G)$, by replacing ``codimension-$1$'' with ``codimension-$(d+1)$''. After recalling ordinary essential dimension and its usages, I will discuss work in progress about the new sequence of invariants $\{\mathrm{ed}^{(d)}(G)\}_{d\geq 0}$ and its asymptotic behavior as $d\to \infty$. For example, $\mathrm{ed}^{(d)}(\mathbf{G}_m)=d$, $\mathrm{ed}^{(d)}(\mathbf{\mu}_n)=d+1$ and $\mathrm{ed}^{(d)}(\mathbf{G}_m\times\mathbf{G}_m)=2d$ in characteristic $0$. Moreover, there is a dichotomy between unipotent and non-unipotent groups: If $G$ is unipotent, the sequence $\{\mathrm{ed}^{(d)}(G)\}_{d\geq 0}$ is bounded, whereas if $G$ is not unipotent, then $\mathrm{ed}^{(d)}(G)\geq d-C_G$ for some constant $C_G$. There are also some interesting anomalies.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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