  A quantum phase transition may occur in a system at zero temperature when a
controlling parameter is tuned towards a critical point. An important question
is whether such a critical point exists in a particular system and how stable
it is. Here, we identify the critical point of a quantum phase transition as a
singular point in the affine algebraic variety of the characteristic equation
for the Hamiltonian describing the system, with an unstable critical point
being associated with an isolated singular point which has a finite Tjurina
number. The theory is illustrated by studying a model system of
zero-dimensional (finite) Heisenberg spin chain with an impurity, which
exhibits a nontrivial first-order quantum phase transition. Both analytical and
numerical calculations show that the quantum phase transition always exists
when the impurity has a $Z_2$ symmetry but only remains in systems with an even
number of spin sites when the $Z_2$ symmetry is broken.
