We use molecular-dynamics simulations in two dimensions to investigate the possibility that a core-softened potential can reproduce static and dynamic anomalies found experimentally in liquid water: (i) the increase in specific volume upon cooling, (ii) the increase in isothermal compressibility upon cooling, and (iii) the increase in the diffusion coefficient with pressure. We relate these anomalies to the shape of the potential. We obtain the phase diagram of the system and identify two solid phases: a square crystal (high-density phase) and a triangular crystal (low-density phase). We also discuss the relation between the anomalies observed and the polymorphism of the solid. Finally, we compare the phase diagram of our model system with experimental data, noting especially the line of temperatures of maximum density, the line of pressures of maximum diffusion constant, and the line of temperatures of minimum isothermal compressibility.