Seminar

Syzygies capture intricate geometric properties of a subvariety in projective space, as shown in the work of Hilbert more than a century ago. However, when the ambient space is another smooth projective toric variety X, this relationship fails to hold as tightly. With this in mind, we consider another homological object over the Cox ring of X, called a virtual resolution, which encodes several geometric invariants in close parallel to the case of free resolutions and projective space. We will briefly review the classical case and then discuss joint work with D. Erman and G. Smith on virtual resolutions, as well as other recent results in the area.