Multigraded syzygies of toric embeddings

A smooth normal toric variety X is determined by a multigraded polynomial ring S and a monomial ideal encoded by the fan of X. When a normal toric variety Y is embedded into X, recent results show that the multigraded free S-resolution of the ideal for Y has an abundance of rich combinatorial structure. We will explain the important geometric implications provided by this large new suite of explicit cellular resolutions over multigraded rings, which includes a generalization of Beilinson's resolution of the diagonal for projective space. This is ongoing joint work with Lauren Cranton Heller, Mahrud Sayrafi, Greg Smith, and Jay Yang.

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