  We propose a geometric correspondence between (a) linearly degenerate systems
of conservation laws with rectilinear rarefaction curves and (b) congruences of
lines in projective space whose developable surfaces are planar pencils of
lines. We prove that in projective 4-space such congruences are necessarily
linear. Based on the results of Castelnuovo, the classification of
three-component systems is obtained, revealing a close relationship of the
problem with projective geometry of the Veronese variety and the theory of
associativity equations of two-dimensional topological field theory.
