Séminaires : Séminaire d'Algèbre

Many important algebraic varieties, such as open positroid strata in Grassmannians, Richardson varieties, or augmentation varieties of certain Legendrian links, are known to carry cluster structures. In particular, each such variety is covered, up to codimension 2, by a collection of overlapping open tori. In this talk, I will discuss the ``deep locus'' of a cluster variety, that is, the complement to the union of all cluster toric charts. I will explain a conjectural relation between the deep locus and the natural torus action compatible with the cluster structure. For many positroid strata in Gr(2,n) and Gr(3,n), and for cluster varieties of types ADE, this relation is made precise: we show that the deep locus consists precisely of the points with non-trivial stabilizer for this action. If time permits, I will explain how these results can be applied in the context of homological mirror symmetry and say a few words on the geometry of deep loci. The talk is based on joint work with Marco Castronovo, José Simental, and David Speyer (arXiv:2402.16970).

This talk will be on Zoom only.