math.AG — Algebraic Geometry

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

Contributed by Michel Hua

Let $L_n$ be the line bundle on $M_{0,n}$ whose fibre is the tangent line to $\mathbb P^1$ at the last marked point, and $L$ the tensor product of the tangent lines at all of them. We show that, in characteristic zero, the Picard group of the quotient stack $[M_{0,n}/\mathfrak S_{n-1}]$ is cyclic of order $(n-1)(n-2)$ for every $n \ge 3$, generated by $L_n$, and that $L$ has order $n-1$ in $\operatorname{Pic}[M_{0,n}/\mathfrak S_n]$. For even $n \ge 6$ 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.