Resonances in ${}^{12}$C and ${}^{24}$Mg. What do we learn from a microscopic cluster theory?

We discuss resonance properties in three-body systems, with examples on ${}^{12}$C and ${}^{24}$Mg. We use a microscopic cluster model, where the generator coordinate is defined in the hyperspherical formalism. The ${}^{12}$C nucleus is described by an α+α+α structure, whereas ${}^{24}$Mg is considered as an ${}^{16}$O+α+α system. We essentially pay attention to resonances. We review various techniques which may extend variational methods to resonances. We consider 0${}^{+}$ and 2${}^{+}$ states in ${}^{12}$C and ${}^{24}$Mg. We show that the r.m.s. radius of a resonance is strongly sensitive to the variational basis. This has consequences for the Hoyle state (0${}_2^{+}$ state in ${}^{12}$C) whose radius has been calculated or measured in several works. In ${}^{24}$Mg, we identify two 0+ resonances slightly below the three-body threshold.