It is believed that the magnetic fluctuations in cuprate superconductors re
flect the proximity to a quantum phase transition. It will be argued that t
his notion acquires further credibility if combined with the idea that the
superconducting state is in a tight competition with the stripe phase over
a large range of hole concentrations. On basis of existing data and some si
mple considerations, a zero-temperature phase diagram will be proposed with
an unusual topology which is unique to the competition stripe phase superc
onductivity. It is argued that the existence of a state which is at the sam
e time stripe ordered and superconducting (antiferromagnetic supersolid) is
a prerequisite for quantum critical behavior in the magnetic sector. Vario
us predictions follow which can be tested experimentally. (C) 1999 Elsevier
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