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We give necessary and sufficient criteria for a smooth Enriques surface S inP^r to be scheme-theoretically an intersection of quadrics. Moreover we provein many cases that, when S contains plane cubic curves, the intersection of thequadrics containing S is the union of S and the 2-planes spanned by the planecubic curves. We also give a new (very quick) proof of the projective normalityof S if the degree of S is at least 12.

Projective normality and the generation of the ideal of an Enriques surface