Minimal generators of the defining ideal of the Rees Algebra associated to monoid parameterizations

We describe a minimal set of generators of the defining ideal of the Rees Algebra associated to a proper parametrization of any monoid hypersurface. In the case of plane curves, we recover a known description for rational parameterizations having a syzygy of minimal degree (μ=1). We also show that our approach can be applied to parameterizations of rational surfaces having a Hilbert–Burch resolution with μ1=μ2=1.