The ground states of a one-dimensional system of the Frenkel-Kontorova type, but involving piecewise parabolic potentials, including a nonconvex interatomic interaction, have been studied numerically using the method of effective potentials. Part of the phase diagram is identical to one studied earlier for a convex interaction, and part of it exhibits some new phases, first-order phase transitions, multicritical points, and an accumulation point of multicritical points, all associated with the nonconvex interaction.