$\chi^{1}$ Fornacis cluster DANCe. Census of stars, structure, and
  kinematics of the cluster with Gaia-EDR3
  Context: The $\chi^{1}$ Fornacis cluster (Alessi 13) is one of a few open
clusters of its age and distance in the Solar neighbourhood that ought to
benefit from more attention as it can serve as a cornerstone for numerous
future studies related to star and planet formation.
  Aims: We take advantage of the early installment of the third data release of
the Gaia space mission in combination with archival data and our own
observations, to expand the census of cluster members and revisit some
properties of the cluster.
  Methods: We applied a probabilistic method to infer membership probabilities
over a field of more than 1000 deg${^2}$ to select the most likely cluster
members and derive the distances, spatial velocities, and physical properties
of the stars in this sample.
  Results: We identify 164 high-probability cluster members (including 61 new
members) covering the magnitude range from 5.1 to 19.6 mag in the G-band. Our
sample of cluster members is complete down to 0.04 M$_{\odot}$. We derive the
distance of $108.4\pm0.3$ pc from Bayesian inference and confirm that the
cluster is comoving with the Tucana-Horologium, Columba, and Carina young
stellar associations. We investigate the kinematics of the cluster from a
subsample of stars with measured radial velocities and we do not detect any
significant expansion or rotation effects in the cluster. Our results suggest
that the cluster is somewhat younger (about 30 Myr) than previously thought.
Based on spectroscopic observations, we argue that the cluster is
mass-segregated and that the distribution of spectral types shows little
variation compared to other young stellar groups.
  Conclusions: In this study, we deliver the most complete census of cluster
members that can be done with Gaia data alone and we use this new sample to
provide an updated picture on the 6D structure of the cluster.
