$\chi^{1}$ 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.