Browsing by Subject "toric varieties"

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    Diagonalization, Direct Summands, and Resolutions of the Diagonal
    (2024-05) Sayrafi, Mahrud
    This thesis concerns the interplay of algebraic invariants like multigraded syzygies and the geometry of toric varieties. An active program in commutative algebra seeks to construct virtual resolutions of ideals and module over multigraded polynomial rings known as Cox rings in order to study algebraic geometry on toric varieties. We solve several problems in different aspects of this program: uniqueness of virtual resolutions on products of projective spaces; existence of short virtual resolutions and Orlov's conjecture in Picard rank 2; Horrocks' splitting criterion for vector bundles on smooth projective toric varieties; Castelnuovo--Mumford regularity and truncations of multigraded modules; bounds on Castelnuovo--Mumford regularity of modules and powers of ideals; and computing direct summand decompositions of multigraded modules and sheaves.
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    Properties Of Virtual Resolutions Over Toric Varieties
    (2020-05) Loper, Michael
    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.