2000/06/04 by Ilya Zakharevich, Zakharevich, Ilya · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematics and Applications #math-ph #math.AP #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.math-ph/0006001
43 pages, preliminary version
arxiv created 2000/06/04 · openalex publication_date 2000/06/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3 they reduce to a special flavor of a non-linear wave equation. The symmetries embedded in the definition of a Veronese web reveal themselves as Bäcklund--Darboux transformations between these non-linear wave equations. On the other hand, the twistor transform identifies Veronese webs with moduli spaces of rational curves on certain complex surfaces. These moduli spaces can be described in terms of the non-linear Riemann problem. This reduces solutions of these non-linear wave equations to the non-linear Riemann problem. We examine these relationships in the particular case of 3-dimensional Veronese webs, simultaneously investigating how these notions relate to general notions of geometry of webs.