  In this paper we construct a parametrization-free embedding technique for
numerically evolving reaction-diffusion PDEs defined on algebraic curves that
possess an isolated singularity. In our approach, we first desingularize the
curve by appealing to techniques from algebraic geometry. We create a family of
smooth curves in higher dimensional space that correspond to the original curve
by projection. Following this, we pose the analogous reaction-diffusion PDE on
each member of this family and show that the solutions (their projection onto
the original domain) approximate the solution of the original problem. Finally,
we compute these approximants numerically by applying the Closest Point Method
which is an embedding technique for solving PDEs on smooth surfaces of
arbitrary dimension or codimension, and is thus suitable for our situation. In
addition, we discuss the potential to generalize the techniques presented for
higher-dimensional surfaces with multiple singularities.
