 When "X" is an algebraic curve with field of definition the complex numbers, and if "X" has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of "X" (its manifold of complex points).For example, the definition of elliptic curve from algebraic geometry is "connected non-singular projective curve of genus 1 with a given rational point on it".By the Riemann-Roch theorem, an irreducible plane curve of degree formula_1 given by the vanishing locus of a section formula_2 has geometric genus

where "s" is the number of singularities when properly counted.