fair number of algebraic geometers know the story of an example,
published by the young Theodor Vahlen in 1891, which showed that an
algebraic curve in 3-space could not in general be described by less
than 4 equations. It held its own for a full fifty years until Oskar
Perron destroyed it in 1941 with perfectly elementary and transparent
A few algebraic geometers know that, due to the differing attitudes of
Vahlen and Perron to the Nazi regime, Perron's paper triggered a
politically charged debate which even Francesco Severi took part in.
Only after the war this gave way to a discussion of "set- theoretic"
versus "ideal-theoretic (complete) intersections" in Algebraic
Almost no algebraic geometer has looked at Vahlen's original paper,
We will recall the story and take it one step further.
This will bring up the question how mathematics contrives to be a
cumulative science (if it actually is cumulative).
Note. No knowledge of Algebraic Geometry is required to follow most of