  Critical points represent a subset of special points tracing cosmological
structures, carrying remarkable topological properties. They thus offer a
richer high-level description of the multiscale cosmic web, being more robust
to systematic effects. For the first time, we characterize here their
clustering statistics in massive neutrino cosmologies, including
cross-correlations, and quantify their simultaneous imprints on the
corresponding web constituents - i.e., halos, filaments, walls, and voids - for
a series of rarity levels. Our first analysis is centered on a
density-threshold-based approach in configuration space. In particular, we show
that the presence of massive neutrinos does affect the baryon acoustic
oscillation peak amplitudes of all of the critical point correlation functions
above/below the rarity threshold, as well as the positions of their
correspondent inflection points at large scales: departures from analogous
measurements carried out in the baseline massless neutrino scenario can reach
up to ~7% in autocorrelations and ~9% in cross-correlations at z=0 when
M_nu=0.1 eV, and are more pronounced for higher neutrino mass values. In turn,
these combined multiscale effects can be used as a novel technique to set upper
limits on the summed neutrino mass and infer the type of hierarchy. Our study
is particularly relevant for ongoing and future large-volume redshift surveys
such as the Dark Energy Spectroscopic Instrument and the Rubin Observatory
Legacy Survey of Space and Time, which will provide unique datasets suitable
for establishing competitive neutrino mass constraints.
