FOM: arcane, genus, algorithm

[John Baldwin]

My understanding of Marker's point in re Faltings is, `There are
easy to states results concerning the shape (to use Dave's word)
of the solution set and number of rational solutions.  This connection
of geometry and arithmetic is foundational.'

Various objections have been raised because precisely stating `shape'
requires the `arcane' notion of genus.  I only note that we rightly view
one of the outstanding achievements of 20th century logic to be
the definition
of algorithm.  Without the Turing analysis, Matijasevic' result is as
vague as `holes in a surface' without an explication of genus.