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1. 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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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 arithmetic of a family of degree - two K3 surfaces

   https://doi.org/10.1017/S0305004118000087  

   BOUYER, FLORIAN; COSTA, EDGAR; FESTI, DINO; NICHOLLS, CHRISTOPHER; WEST, MCKENZIE (May 2019, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x , y , z and w ; let $$\mathcal{X}$$ be the generic element of the family of surfaces in ℙ given by \begin{equation*}X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2.\end{equation*} The surface $$\mathcal{X}$$ is a K3 surface over the function field ℚ( t ). In this paper, we explicitly compute the geometric Picard lattice of $$\mathcal{X}$$ , together with its Galois module structure, as well as derive more results on the arithmetic of $$\mathcal{X}$$ and other elements of the family X . 

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4. Counting special Lagrangian classes and semistable mukai vectors for K3 surfaces

   https://doi.org/10.1007/s10711-023-00823-w  

   Athreya, Jayadev S; Fan, Yu-Wei; Lee, Heather (October 2023, Geometriae Dedicata)

   Motivated by the study of the growth rate of the number of geodesics in flat surfaces with bounded lengths, we study generalizations of such problems for K3 surfaces. In one gener- alization, we give a result regarding the upper bound on the asymptotics of the number of classes of irreducible special Lagrangians in K3 surfaces with bounded period integrals. In another generalization, we give the exact leading term in the asymptotics of the number of Mukai vectors of semistable coherent sheaves on algebraic K3 surfaces with bounded central charges, with respect to generic Bridgeland stability conditions. 

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5. Perfect points on curves of genus 1 and consequences for supersingular K3 surfaces

   https://doi.org/10.1112/S0010437X22007382  

   Bragg, Daniel; Lieblich, Max (May 2022, Compositio Mathematica)

   We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus 1 curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of genus 1 fibrations on supersingular K3 surfaces without purely inseparable multisections. 

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