Background: The influence of halo structure of $^6$He, $^8$B, $^{11}$Be and $^{11}$Li nuclei in several mechanisms such as direct reactions and fusion is already established, although not completely understood. The influence of the $^{10}$C Brunnian structure is less known. Purpose: To investigate the influence of the cluster configuration of $^{10}$C on the elastic scattering at an energy close to the Coulomb barrier. Methods: We present experimental data for the elastic scattering of the $^{10}$C+$^{208}$Pb system at $E_{rm lab}$ = 66 MeV. The data are compared to the three- and the four-body continuum-discretized coupled-channels calculations assuming $^9$B+$p$, $^6$Be+$alpha$ and $^8$Be+$p$+$p$ configurations. Results: The experimental angular distribution of the cross sections shows the suppression of the Fresnel peak that is reasonably well reproduced by the continuum-discretized coupled-channels calculations. However, the calculations underestimate the cross sections at backward angles. Couplings to continuum states represent a small effect. Conclusions: The cluster configurations of $^{10}$C assumed in the present work are able to describe some of the features of the data. In order to explain the data at backward angles, experimental data for the breakup and an extension of theoretical formalism towards a four-body cluster seem to be in need to reproduce the measured angular distribution.
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