Relative completions and separating boundary divisors in moduli of curves

Source abstract

We study relative completions of fundamental groups of moduli stacks of curves of compact type and of partial compactifications of the smooth moduli stack. For $g\geq3$, the compact-type completion has commutative prounipotent radical with Lie algebra $Λ^3_0H\oplus H^{\oplus n}$. Fix $1\leq h\leq g-1$. We show that adjoining the one-node loci of the separating divisors whose genus-$h$ side carries at most two markings gives the same relative completion. The proof uses explicit separating-twist calculations in weight $-2$, pointed Johnson homomorphisms, and the lantern relation. Functorial specialization gives the corresponding results in positive characteristic for odd $\ell$ different from the characteristic. As a consequence, extension to compact type of iterated extensions of algebraic symplectic local systems can be tested on these selected divisors. We also prove that any collection of separating divisors giving the same relative completion contains at least $\binom n2$ divisors.