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In this joint work with Aldo Conca and Rekha Thomas, we concentrate on defining ideals of Segre products, Veronese embeddings, determinantal varieties, and other toric varieties. These classical objects have very nice properties, like being Cohen-Macaulay. In many cases there are characterizations when further nice properties hold (for instance, Gorenstein-ness). We ask the following fundamental question: when are such nice properties inherited by an initial ideal? As an example, take the case of Gorenstein Segre products or Veronese. embeddings. In both cases we construct initial complexes that are also Gorenstein. We also describe shellings of these complexes.