Characterization of curves in C⁽²⁾

In this paper we characterize the irreducible curves lying in C⁽²⁾. We prove that a curve B has a degree one morphism to C⁽²⁾ with image a curve of degree d, with irreducible preimage in C× C, if and only if there exists an irreducible smooth curve D and morphisms from D to C and B, of degrees d and 2 respectively, forming a diagram which does not reduce.