Talk:Smooth projective plane

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Definition

I guess, the operations of joining and intersecting are (defined and) smooth outside of the diagonals; really? Boris Tsirelson (talk) 19:36, 21 September 2017 (UTC)[reply]

"The geometric operations of smooth planes are continuous; hence, each smooth plane is a compact topological plane." — This "hence" is unclear to me. An immediate consequence? Or application of a nontrivial result well-known to experts? Boris Tsirelson (talk) 20:25, 21 September 2017 (UTC)[reply]

A possible source: SMOOTH PROJECTIVE PLANES by BENJAMIN MCKAY. There, the definition requires also that the incidence relation is a smooth embedded manifold. Is this requirement missing here? Also, there compactness is just required by definition (for being called "topological", the more so, "smooth"); and what happens here? Boris Tsirelson (talk) 20:33, 21 September 2017 (UTC)[reply]

I am pretty sure that the author(s) of this page are being a little sloppy here (and elsewhere) and that the geometric operations are restricted to distinct elements (semi mentioned in the lead, but dropped in the formal definition). I don't know what to say about "hence". I am not familiar with this material and am just trying to put this page into an acceptable WP format. --Bill Cherowitzo (talk) 20:43, 21 September 2017 (UTC)[reply]

I am not familiar, too; this is why I just ask questions. I continue:

"Smooth planes exist only with point spaces of dimension 2^m where $1\leq m\leq 4$ " — Not a part of the definition, rather, a theorem. McKay attributes this theorem to Freidenthal; or is it a conjecture of Freidenthal? Boris Tsirelson (talk) 20:46, 21 September 2017 (UTC)[reply]

"The point manifold of a smooth projective plane is homeomorphic to its classical counterpart, and so is the line manifold." — Also not a part of the definition. And, according to McKay, they are (moreover) diffeomorphic. Boris Tsirelson (talk) 20:53, 21 September 2017 (UTC)[reply]