Enriques surface
Algebraic surface with special triviality properties
In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex numbers) and are elliptic surfaces of genus 0.
Over fields of characteristic not 2 they are quotients of K3 surfaces by a group of order 2 acting without fixed points and their theory is similar to that of algebraic K3 surfaces. Enriques surfaces were first studied in detail by Enriques (1896) as an answer to a question discussed by Castelnuovo (1895) about whether a surface with q = pg = 0 is necessarily rational, though some of the Reye congruences introduced earlier by Reye (1882) are also examples of Enriques surfaces.
Enriques surfaces can also be defined over other fields.
Over fields of characteristic other than 2, Artin (1960) showed that the theory is similar to that over the complex numbers. Over fields of characteristic 2 the definition is modified, and there are two new families, called singular and supersingular Enriques surfaces, described by Bombieri & Mumford (1976). These two extra families are related to the two non-discrete algebraic group schemes of order 2 in characteristic 2.
Invariants of complex Enriques surfaces
The plurigenera Pn are 1 if n is even and 0 if n is odd. The fundamental group has order 2.
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