Question on projective variety

Dear Prof Kim.

In the Algebraic Geometry notes, we defined a Projective Algebraic Variety to be a subset of the projective space defined by the zeros of a homogeneous polynomial equation. Do we require the Algebraic set to be irreducible as well? and if so, why do we require this extra condition?

Thank you very much.



Because you know too much, you’re looking ahead already. We haven’t yet formally discussed the definition of a projective variety. Rather we’re just discussing examples. We’ll get to the issue of irreducibility and so on within a few weeks, and the question of the precisely correct definition.

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