Scattering diagrams, sheaves, and curves
Abstract
We review the recent proof of the N.Takahashi's conjecture on genus $0$ GromovWitten invariants of $(\mathbb{P}^2, E)$, where $E$ is a smooth cubic curve in the complex projective plane $\mathbb{P}^2$. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of GromovWitten invariants of $(\mathbb{P}^2, E)$ and the world of moduli spaces of coherent sheaves on $\mathbb{P}^2$. Using this bridge, the N.Takahashi's conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on $\mathbb{P}^2$. This survey is based on a three hours lecture series given as part of the BeijingZurich moduli workshop in Beijing, 912 September 2019.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.08741
 Bibcode:
 2020arXiv200208741B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Symplectic Geometry
 EPrint:
 Expository paper. 18 pages, 1 figure