What is the Definition of a Conic in Complex Projective 2 Space?

[Diophantus]

I was just wondering what the more fundamental definition of a conic in complex projective 2 space is. Is it better to say that it is a curve such that the dehomogenisation of its defining equation is a represents a conic in R^2; OR simply a curve defined by a homogeneous degree two polynomial.

Or is it better to define it in a more geometric way?

I suppose this is all a matter of opinion really.

 

[Chris Hillman]

The second (homogeneous quadratic polynomial), I should think. Have you read the textbook by Joe Harris, Algebraic Geometry: a First Course,
Springer, 1992? This has lots of great material on projective spaces and Grassmannians, with an appropriate emphasis on group actions.

 

[Diophantus]

I haven't seen that one but I'll look it up in the library if you recommend it. The two I have been using are Miles Reid's 'Undergraduate Algebraic Geometry', and Cox, Little and O'Shea's 'Ideals, Varieties and Algorithms'. The former isn't that great but is usefull in that it follows my course quite well, the latter is superb though.