 Monomial curve

Over a field $k$, an affine curve parametrized by positive integer powers of one parameter. Its coordinate ring is the subalgebra generated by those powers, and its defining prime ideal is the kernel of $k[x_1,\dots ,x_n]\to k[t]$ sending $x_i$ to $t^{a_i}$. Equal exponent sums yield binomial relations. Studying those relations connects an additive semigroup of exponents to the geometry of the curve.