We explore the relationship between multigraded Castelnuovo-Mumford regularity,
truncations, Betti numbers, and virtual resolutions. We prove that on a product of projective …

We study the asymptotic non-vanishing of syzygies for products of projective spaces.
Generalizing the monomial methods of Ein, Erman, and Lazarsfeld we give an explicit range …

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We provide a number of new conjectures and questions concerning the syzygies of P 1× P
1. The conjectures are based on computing the graded Betti tables and related data for large …

In this thesis, we explore three aspects of singularities and syzygies in algebraic geometry.
First, we study singularities in positive characteristic. In particular, we prove an inversion of …

Hirzebruch surfaces a are 1-bundles over a smooth rational curve. They are the minimal
models of rational surfaces; every smooth rational surface can be obtained from a …

Motivated by toric geometry, we lift machinery for understanding the syzygies of curves in
projective space to the setting of products of projective spaces. Using this machinery, we …

We provide a number of new conjectures and questions concerning the syzygies of
$\mathbb {P}^ 1\times\mathbb {P}^ 1$. The conjectures are based on computing the graded …