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1. -SIGNATURE UNDER BIRATIONAL MORPHISMS

   https://doi.org/10.1017/fms.2019.6  

   MA, LINQUAN; POLSTRA, THOMAS; SCHWEDE, KARL; TUCKER, KEVIN (January 2019, Forum of Mathematics, Sigma)

   We study $$F$$ -signature under proper birational morphisms $$\unicode[STIX]{x1D70B}:Y\rightarrow X$$ , showing that $$F$$ -signature strictly increases for small morphisms or if $$K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$$ . In certain cases, we can even show that the $$F$$ -signature of $$Y$$ is at least twice as that of  $$X$$ . We also provide examples of $$F$$ -signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses. 

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2. Higher Dimensional Algebraic Geometry: A Volume in Honor of V. V. Shokurov

   https://doi.org/10.1017/9781009396233  

   Cheltsov, Ivan; Sarikyan, Arman; Zhuang, Ziquan (December 2024, Cambridge University Press)

   Hacon, Christopher; Xu, Chenyang (Ed.)

   Arising from the 2022 Japan-US Mathematics Institute, this book covers a range of topics in modern algebraic geometry, including birational geometry, classification of varieties in positive and zero characteristic, K-stability, Fano varieties, foliations, the minimal model program and mathematical physics. The volume includes survey articles providing an accessible introduction to current areas of interest for younger researchers. Research papers, written by leading experts in the field, disseminate recent breakthroughs in areas related to the research of V.V. Shokurov, who has been a source of inspiration for birational geometry over the last forty years. 

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3. Birational geometry of Beauville–Mukai systems III: Asymptotic behavior

   https://doi.org/10.1112/blms.13158  

   Qin, Xuqiang; Sawon, Justin (September 2024, Bulletin of the London Mathematical Society)

   Abstract Suppose that a Hilbert scheme of points on a K3 surface of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville–Mukai system on a Fourier–Mukai partner of . We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme. 

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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. Birational maps with transcendental dynamical degree

   https://doi.org/10.1112/plms.12573  

   Bell, Jason P.; Diller, Jeffrey; Jonsson, Mattias; Krieger, Holly (December 2023, Proceedings of the London Mathematical Society)

   Abstract We give examples of birational selfmaps of , whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation. 

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