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The group of basis-conjugating automorphisms of the free group of rank $n$,also known as the McCool group or the welded braid group $P\Sigma_n$, containsa much-studied subgroup, called the upper McCool group $P\Sigma_n^+$. Startingfrom the cohomology ring of $P\Sigma_n^+$, we find, by means of a Gr\"obnerbasis computation, a simple presentation for the infinitesimal Alexanderinvariant of this group, from which we determine the resonance varieties andthe Chen ranks of the upper McCool groups. These computations reveal that,unlike for the pure braid group $P_n$ and the full McCool group $P\Sigma_n$,the Chen ranks conjecture does not hold for $P\Sigma_n^+$, for any $n\ge 4$.Consequently, $P\Sigma_n^+$ is not isomorphic to $P_n$ in that range, thusanswering a question of Cohen, Pakianathan, Vershinin, and Wu. We alsodetermine the scheme structure of the resonance varieties$\mathcal{R}_1(P\Sigma_n^+)$, and show that these schemes are not reduced for$n\geq 4$.

Chen ranks and resonance varieties of the upper McCool groups