The classical Hasse principle states that a quadratic form over the rational numbers
represents zero rationally if and only if it represents zero over every completion
of the rational numbers.
In this talk we address the following problem: does there exist a cubic surface over
the field of rational numbers which contains no rational line, yet contains a line
over every completion of the rational numbers? i.e., does a Hasse type principle hold
for the collection of lines on a cubic surface?
In the talk we shall explain, amongst other things, how one can translate this into a
problem in graph theory which can be tackled using a computer.
This is joint work with Jörg Jahnel.