The finite-degree profile of essential dimension I: general theory and characteristic prime to $|G|$

Source abstract

Essential dimension counts the parameters needed to define a torsor, but it can drop sharply over a field extension, and it does not say how large an extension is needed. For a $G$-torsor $τ$ over a field $F\supseteq k$ we study the \emph{finite-degree profile} $\edd d_k(τ)=\min\{\ed_k(τ\otimes_FF'):[F':F]\le d\}$, a variant of invariants of Farb--Kisin--Wolfson and Farb--Wolfson, and its jump degrees $d_j(τ)$, the least degree of an extension over which $τ$ needs only $j$ parameters. The profile is geometric over every field: $\edd d_k(τ)\le j$ iff some generically free $G$-variety of dimension $\le j$ has a closed point of degree $\le d$ in the free locus of its twist by $τ$. We prove a Sylow divisibility theorem for jump degrees and a valuation-theoretic rank--index theorem, $\edd d_k(\tgen)\ge\min\{\rk B:B\le A,\ [A:B]\le d\}$ for abelian $A$ of order prime to $\operatorname{char}k$, which determines the profile of every abelian group over a field with enough roots of unity, for instance $n-\lfloor\log_2d\rfloor$ for $(\Z/2)^n$. For $\Z/p$ in characteristic $0$ we show $\lceil m/2\rceil\le d_1\le[k(ζ_p+ζ_p^{-1}):k]$, $m=[k(ζ_p):k]$, with equality for $m$ even. For $S_5$ in characteristic $0$ we prove $d_1=4$, so Klein's reduction of the generic quintic is optimal; the lower bound comes from Abel--Jacobi rigidity, not from cohomological invariants. For $n\ge6$, $6\mid d_1(S_n)$.