  We discuss the evolution of high-order Van Hove singularities (hoVHSs) that
carry faster-than logarithmic divergences over a wide range of parameters in
cuprate-like electronic band dispersions. Numerical analysis gives insight into
the quantization of the VHS power-law-exponent pV and into transitions between
hoVHSs with different values of pV. The cuprates are found to lie in the
parameter regime where the amplitude of the hoVHS is not too large. Our study
indicates that the occurrence of high-temperature superconductivity requires
simultaneous tuning of two different competing orders (antiferromagnetism and
the density wave associated with the hoVHS in cuprates), which is why it is so
rare.
