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1. K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

   https://doi.org/10.1017/S1474748021000384  

   ASCHER, KENNETH; DEVLEMING, KRISTIN; LIU, YUCHEN (September 2021, Journal of the Institute of Mathematics of Jussieu)

   Abstract We show that the K-moduli spaces of log Fano pairs $$\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $$\mathbb {P}^3$$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $$\mathbb {P}^1\times \mathbb {P}^1$$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces. 

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2. Wall crossing for K‐moduli spaces of plane curves

   https://doi.org/10.1112/plms.12615  

   Ascher, Kenneth; DeVleming, Kristin; Liu, Yuchen (June 2024, Proceedings of the London Mathematical Society)

   Abstract We construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces. 

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3. Compactifications of Moduli of Points and Lines in the Projective Plane

   https://doi.org/10.1093/imrn/rnab200  

   Schaffler, Luca; Tevelev, Jenia (August 2021, International Mathematics Research Notices)

   Abstract Projective duality identifies the moduli spaces $$\textbf{B}_n$$ and $$\textbf{X}(3,n)$$ parametrizing linearly general configurations of $$n$$ points in $$\mathbb{P}^2$$ and $$n$$ lines in the dual $$\mathbb{P}^2$$, respectively. The space $$\textbf{X}(3,n)$$ admits Kapranov’s Chow quotient compactification $$\overline{\textbf{X}}(3,n)$$, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ “broken lines”. Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003. 

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4. Lifts of Twisted K3 Surfaces to Characteristic 0

   https://doi.org/10.1093/imrn/rnab334  

   Bragg, Daniel (January 2022, International Mathematics Research Notices)

   Abstract Deligne [9] showed that every K3 surface over an algebraically closed field of positive characteristic admits a lift to characteristic 0. We show the same is true for a twisted K3 surface. To do this, we study the versal deformation spaces of twisted K3 surfaces, which are particularly interesting when the characteristic divides the order of the Brauer class. We also give an algebraic construction of certain moduli spaces of twisted K3 surfaces over $${\operatorname {Spec}}\ \textbf {Z}$$ and apply our deformation theory to study their geometry. As an application of our results, we show that every derived equivalence between twisted K3 surfaces in positive characteristic is orientation preserving. 

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5. Reductivity of the automorphism group of K-polystable Fano varieties.

   Alper, Jarod; Blum, Harold; Halpern-Leistner, Daniel; Xu, Chenyang (July 2020, Inventiones mathematicae)

   null (Ed.)

   We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and Θ-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space. 

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