Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P4

Abstract:

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 P ⁴ 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 Veronesé variety V ²⊂P ⁵ and the theory of associativity equations of two-dimensional topological field theory.