Systems of conservation laws of Temple class, equations of associativity and linear congruences in projective space

TL;DR

This paper establishes a geometric link between certain conservation law systems and line congruences in projective space, classifying three-component cases and connecting them to algebraic geometry and topological field theory.

Contribution

It introduces a geometric correspondence between degenerate conservation laws and line congruences, classifies three-component systems in projective 4-space, and relates the problem to Veronese varieties and associativity equations.

Findings

In projective 4-space, congruences are necessarily linear.

Classification of three-component systems is achieved.

Connections to Veronese variety and topological field theory are established.

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 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.