What should affine and projective schemes over $\mathbf{F}_1$ be?

Your question sounds rather vague, but in any case, you might be interested in the paper "On the notion of geometry over $\mathbb{F}_1$" by Alain Connes and Caterina Consani, available on the arXiv (or on Connes' homepage http://www.alainconnes.org). 

