Blowing up points and embedding flat stable planes in the nonorientable compact surface of genus one

We show that the point set of every flat stable plane embeds in the point set of the real projective plane. Connectedness of lines or of the point space is not assumed. We give two largely independent proofs; the first one is more conceptual, while the second one is more direct, and shorter. The first proof uses a new construction called blowing up a point, i.e., replacing it with its line pencil; this amounts to adding a cross cap. This construction seems to be of interest in its own right.