FOM: PS on "arcane"

Lou van den Dries vddries at
Wed Oct 22 16:25:18 EDT 1997

PS Of course, the fact that "genus" is an "arcane" or " technical"
notion from the FOM viewpoint ("it's only of interest to pure
mathematicians" according to that viewpoint) does not take away for
me that it is a fundamental notion, but its fundamental nature
has been *discovered*, it didn't stare one in the face. This seems
a possibility that FOM prefers to overlook. From the FOM viewpoint
it would be the degree of a polynomial perhaps that would be considered
as more fundamental or basic. Now, degree is an invariant under the group
of linear transformations, so it certainly has geometric significance,
but genus is invariant under a much bigger group, the group of birational
transformations. Since "having infinitely many rational solutions"
is also invariant under the group of birational transformations, it
is reasonable that "genus" is more fundamental for the issue of
having infinitely many rational solutions than the superficially
more basic notion of degree. (Sorry, replace "reasonable" by "plausible".)

It seems to me that FOM as championed by Harvey and Steve raises this
kind of superficiality to the level of a foundational principle. 

Lou van den Dries

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