Department of Mathematics
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A variety being Gorenstein can be a useful property to have when considering questions in birational geometry. Although Schubert varieties are Cohen-Macaulay, they are not Gorenstein in general. I will describe a convenient way to find a "Gorensteinization" for a Schubert variety by considering only one blow-up along its boundary divisor. We start by reducing to the local question, one involving Kazhdan-Lusztig varieties. These affine varieties can be degenerated to a toric scheme defined using the Stanley-Reisner ideal of a subword complex. The blow-up of this variety along its boundary is now Gorenstein. Carefully choosing a degeneration of the blow-up allows us to extend this result to Schubert varieties.

In the first talk, I will provide the necessary background to motivate this problem, including definitions and classical results involving Schubert varieties, Kazhdan-Lusztig varieties and Bott-Samelson resolutions. In the second talk I will give an overview of the Gorensteinization construction and discuss applications and future research.