We study the half-filled Hubbard chain including next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ The model has three phases: one insulating phase with dominant spin-density-wave correlations at large distances (SDWI), another phase with dominant spin-dimer correlations or dimerized insulator (DI), and a third one in which long-distance correlations indicate singlet superconductivity (SS). The boundaries of the SDWI are accurately determined numerically through a crossing of excited energy levels equivalent to the jump in the spin Berry phase. The DI-SS boundary is studied using several indicators such as correlation exponent ${K}_{\ensuremath{\rho}},$ Drude weight ${D}_{c},$ localization parameter ${z}_{L},$ and charge gap ${\ensuremath{\Delta}}_{c}.$