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Oct 21, 2010 at 11:43 comment added Sheikraisinrollbank As might be obvious from my answer, I'm not a professional algebraic geometer, so I didn't answer question 2. But I have to admit that despite my preference for varieties, even I occasionally think of them as a special class of schemes. Still, it's important to keep in mind that additional powerful tools are available when one is working over the complex numbers, and thinking of varieties as schemes is not always the best route...
Oct 21, 2010 at 11:30 history answered Sheikraisinrollbank CC BY-SA 2.5