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   https://doi.org/10.1017/nmj.2022.29  

   HASSETT, BRENDAN; TSCHINKEL, YURI (October 2022, Nagoya Mathematical Journal)

   Abstract We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups. 

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2. Cremona transformations and derived equivalences of K3 surfaces

   https://doi.org/10.1112/s0010437x18007145  

   Hassett, Brendan; Lai, Kuan-Wen (July 2018, Compositio Mathematica)

   We exhibit a Cremona transformation of $$\mathbb{P}^{4}$$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties. 

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3. On the relative minimal model program for fourfolds in positive and mixed characteristic

   https://doi.org/10.1017/fmp.2023.6  

   Hacon, Christopher; Witaszek, Jakub (January 2023, Forum of Mathematics, Pi)

   Abstract We show the validity of two special cases of the four-dimensional minimal model program (MMP) in characteristic $p>5$ : for contractions to $${\mathbb {Q}}$$ -factorial fourfolds and in families over curves (‘semistable MMP’). We also provide their mixed characteristic analogues. As a corollary, we show that liftability of positive characteristic threefolds is stable under the MMP and that liftability of three-dimensional Calabi–Yau varieties is a birational invariant. Our results are partially contingent upon the existence of log resolutions. 

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4. Symbols and equivariant birational geometry in small dimensions

   Hassett, Brendan; Kresch, Andrew; Tschinkel, Yuri (January 2021, 2019 Schiermonnikoog conference `Rationality of algebraic varieties')

   Farkas, Gavril; Shen, Mingmin; Taelman, Lenny; van der Geer, Gerard (Ed.)

   We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions. 

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5. Nilpotent varieties in symmetric spaces and twisted affine Schubert varieties

   https://doi.org/10.1090/ert/613  

   Hong, Jiuzu; Korkeathikhun, Korkeat (June 2022, Representation Theory of the American Mathematical Society)

   We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar–Henderson in the twisted setting. We also get some applications to the geometry of the order 2 nilpotent varieties in certain classical symmetric spaces. 

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