Irregular elliptic surfaces of degree 12 in the projective fourspace

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So far only a few families of smooth irregular surfaces are known to exist in P⁴ up to pullbacks by suitable finite morphisms from P⁴ onto P⁴ itself. In this paper we present two different constructions of irregular smooth minimal elliptic surfaces of degree 12 in P⁴. The first is a monad construction while the other uses liaison. The family constructed via liaison includes the surfaces of the first family as a special case.