HEXAGONbeta

The tangent line at a marked point on M0,n/Sn−1M_{0,n}/\mathfrak S_{n-1}, after a table of Grothendieck

Primarily AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometrymath.CO — Combinatoricsmath.HO — History and Overviewcs.LO — Logic in Computer Science

Contributed by Michel Hua ↗

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

Let LnL_n be the line bundle on M0,nM_{0,n} whose fibre is the tangent line to P1\mathbb P^1 at the last marked point, and LL the tensor product of the tangent lines at all of them. We show that, in characteristic zero, the Picard group of the quotient stack [M0,n/Sn−1][M_{0,n}/\mathfrak S_{n-1}] is cyclic of order (n−1)(n−2)(n-1)(n-2) for every n≥3n \ge 3, generated by LnL_n, and that LL has order n−1n-1 in Pic⁡[M0,n/Sn]\operatorname{Pic}[M_{0,n}/\mathfrak S_n]. For even n≥6n \ge 6 the group was known, through the integral Chow ring computed by Edidin and Hu; the odd case does not seem to be in print. The question comes from a table in Grothendieck’s working notes, which gives these orders for four points and only guesses for five and six. The proof takes a few lines once the stack is written as monic squarefree polynomials modulo the affine group. An elementary version, with explicit trivialisations and fixed configurations, is formalised in Lean.

Provenance statement

Drafted with Claude (Claude Code, Anthropic; Opus 5 and Opus 5.5) from the transcription of folder 146 of Grothendieck’s working notes (University of Montpellier), itself made with language models. The search of the literature, the general formula, its proofs and the Lean 4 formalisation were done with the same models, under the author’s direction. The elementary version (explicit trivialisations and fixed configurations, theorems trivL_iff and trivLi_iff of Grothendieck.Folder146) is checked by Lean with mathlib; the dictionary between equivariant line bundles and these functions is not formalised. The author checked the proofs and the references.

Tools used

Lean
LeanVersion 4
Anthropic
Claude OpusVersion 5.5

References

  1. A. Arsie, A. Vistoli, Stacks of cyclic covers of projective spaces, Compos.\ Math. 140 (2004), no. 3, 647--666, doi:10.1112/S0010437X03000253 (https://doi.org/10.1112/S0010437X03000253).DOI
  2. D. Edidin, personal communication, e-mail of 28 October 2026.
  3. D. Edidin, Z. Hu, The integral Chow rings of the stacks of hyperelliptic Weierstrass points, Michigan Math.\ J. 76 (2026), no. 5, 1091--1113, doi:10.1307/mmj/20236479 (https://doi.org/10.1307/mmj/20236479), arXiv:2208.00556.arXivDOI
  4. W. Fulton, M. Olsson, The Picard group of M1,1\mathcal M_{1,1}, Algebra Number Theory 4 (2010), no. 1, 87--104, doi:10.2140/ant.2010.4.87 (https://doi.org/10.2140/ant.2010.4.87).DOI
  5. S. Gorchinskiy, F. Viviani, Picard group of moduli of hyperelliptic curves, Math.\ Z. 258 (2008), no. 2, 319--331, doi:10.1007/s00209-007-0173-9 (https://doi.org/10.1007/s00209-007-0173-9).DOI
  6. M. Hua, Grothendieck Montpellier. Énoncés des lectures modernisées démontrés, Hexagon preprint hexagon:2611.00125v1 (https://hexagonmath.org/2611.00125v1), 2026.hexagon
  7. D. Mumford, Picard groups of moduli problems, in Arithmetical Algebraic Geometry (Purdue, 1963), Harper & Row, New York, 1965, 33--81.
  8. M. Pernice, The integral Chow ring of the stack of 1-pointed hyperelliptic curves, Int.\ Math.\ Res.\ Not. 2022, no. 15, 11539--11574, doi:10.1093/imrn/rnab072 (https://doi.org/10.1093/imrn/rnab072).DOI

Version history

  1. v1 · 5 pagesSubmitted by Michel HuaInitial depositCurrentNov 07, 2026