Loper, Michael2020-08-252020-08-252020-05https://hdl.handle.net/11299/215209University of Minnesota Ph.D. dissertation. 2020. Major: Mathematics. Advisor: Christine Berkesch. 1 computer file (PDF); vi, 99 pages.Minimal free resolutions enable researchers to use homological algebra to study geometric invariants of subvarieties of projective space. Virtual resolutions for smooth projective toric varieties are a generalization of minimal free resolutions for projective space. They enable researchers to use homological algebra to study a wider range of geometric objects. This thesis investigates what light can be shed by virtual resolutions on toric varieties and also explores properties of virtual resolutions themselves. For example, two algebraic conditions are identified that completely determine when a graded chain complex is a virtual resolution. The invariance of saturated Fitting ideals is also proved and a connection between saturated Fitting ideals and locally free sheaves is shown. After studying virtual resolutions in general, sets of points in the product of two projective lines are explored. In particular, both necessary and sufficient combinatorial conditions are proved for when a given set of points is exactly the solution set to two bihomogeneous polynomials. These sets of points are called virtual complete intersections. Finally, the VirtualResolutions software package for Macaulay2 is introduced. This package contains tools for computing examples and testing conjectures.ensyzygiestoric varietiesvirtual resolutionsThesis or Dissertation