The finite-degree profile of essential dimension I: general theory and characteristic prime to |G|
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⊇ k we study the \emph{finite-degree profile} edd dk(τ)=min{edk(τ⊗FF'):[F':F]≤ d}, a variant of invariants of Farb--Kisin--Wolfson and Farb--Wolfson, and its jump degrees dj(τ), the least degree of an extension over which τ needs only j parameters. The profile is geometric over every field: edd dk(τ)≤ j iff some generically free G-variety of dimension ≤ j has a closed point of degree ≤ 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 dk(tgen)≥min{rk B:B≤ A, [A:B]≤ d} for abelian A of order prime to chark, which determines the profile of every abelian group over a field with enough roots of unity, for instance n−⌊log2d⌋ for (Z/2)n. For Z/p in characteristic $0$ we show ⌈ m/2⌉≤ d1≤[k(ζp+ζp−1):k], m=[k(ζp):k], with equality for m even. For S5 in characteristic $0$ we prove d1=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≥6, 6| d1(Sn).
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BibTeX
@misc{shukla2026finitedegree,
title = {{The finite-degree profile of essential dimension I: general theory and characteristic prime to $|G|$}},
author = {Abhishek Shukla},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07508},
eprint = {2610.07508},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07508},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The finite-degree profile of essential dimension I: general theory and characteristic prime to |G|
AU - Abhishek Shukla
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07508
UR - https://arxiv.org/abs/2610.07508
AB - 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⊇ k we study the \emph{finite-degree profile} edd d_k(τ)=min{ed_k(τ⊗_FF'):[F':F]≤ 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)≤ j iff some generically free G-variety of dimension ≤ j has a closed point of degree ≤ 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)≥min{rk B:B≤ A, [A:B]≤ d} for abelian A of order prime to chark, which determines the profile of every abelian group over a field with enough roots of unity, for instance n−⌊log_2d⌋ for (Z/2)^n. For Z/p in characteristic $0$ we show ⌈ m/2⌉≤ d_1≤[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≥6, 6| d_1(S_n).
ER -