Bielliptiske

Bielliptiske (bielliptic) refers to objects in algebraic geometry that are related to elliptic curves through a degree-two map. The most common use is for algebraic curves, but the term also appears for certain complex surfaces.

Bielliptiske kurver (bielliptic curves) are smooth projective curves C over an algebraically closed field that admit a morphism of degree 2 to an elliptic curve E; equivalently, there exists an involution i on C such that the quotient C/⟨i⟩ is an elliptic curve. Such a bielliptic involution is a nontrivial automorphism of order two, and by the Riemann–Hurwitz formula the covering C → E is ramified (for genus g(C) ≥ 2). Not all curves are bielliptic; the existence of a bielliptic map imposes constraints on the genus and on the automorphism group of C. Bielliptic curves form a distinct class within the moduli of curves of genus g ≥ 2 and are studied in connection with automorphisms and moduli spaces.

Bielliptiske overflater (bielliptic surfaces) appear in the classification of complex surfaces. A bielliptic surface is a smooth projective surface of Kodaira dimension 0 that is a quotient of a product of two elliptic curves by a finite group acting freely. Such surfaces have irregularity q = 1 and geometric genus p_g = 0, and they admit a universal cover isomorphic to the product of two elliptic curves. They occupy one of the seven families in the Enriques–Kodaira classification of surfaces with Kodaira dimension 0.

In summary, bielliptiske describes objects characterized by a close relationship to elliptic curves via a two-to-one map, most often in the setting of curves or certain elliptic-curve–based surfaces.