By a drawing of the Fano plane I mean a system of seven simple curves and seven points in the real plane such that

- every point lies on exactly three curves, and every curve contains exactly three points;
- there is a unique curve through every pair of points, and every two curves intersect in exactly one point;
- the curves do not intersect except in the seven points under consideration.

The familiar picture

does not count as a drawing, since the last requirement is not satisfied: there are two "illegal" intersections. In fact, this is easy to fix:

However, this drawing is degenerate in the sense that two of the curves just "touch" each other, without crossing, at some point. And here, eventually, my question goes:

Is every drawing of the Fano plane degenerate?

(Although I can give a topological definition of degeneracy, it is a little technical and, may be, not the smartest possible one, so I prefer to suppress it here.)

iffthat product is $-1$. Since $1 \neq -1$ we're done. – Noam D. Elkies May 15 '13 at 2:59iffwe're in characteristic 2. – Noam D. Elkies May 17 '13 at 16:16