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Why do we study projective normality of a projective variety ? Does it have anything to do with non-singularity ? Any other purpose to study this ?

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Dear Jack, Projective normality gives an algebraic interpretation of a geometric property of linear systems on a variety: it says that the intersection of the variety with a complete linear system of hypersurfaces in projective space is again a complete linear system on the projectively normal variety. (At least if the variety is non-singular.) This is discussed in the wikipedia entry on this topic. Regards, Matthew –  Matt E Jun 7 '12 at 14:23
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