scholarly journals Group Actions, Cyclic Coverings and Families of K3-Surfaces

2006 ◽  
Vol 49 (4) ◽  
pp. 592-608 ◽  
Author(s):  
Alessandra Sarti
Keyword(s):  
Lattice Theory ◽  
Group Actions ◽  
K3 Surfaces ◽  
Rational Curves ◽  
Picard Number ◽  
Special Fibers ◽  
Cyclic Coverings

AbstractIn this paper we describe six pencils of K3-surfaces which have large Picard number (ρ = 19, 20) and each contains precisely five special fibers: four have A-D-E singularities and one is non-reduced. In particular, we characterize these surfaces as cyclic coverings of some K3-surfaces described in a recent paper by Barth and the author. In many cases, using 3-divisible sets, resp., 2-divisible sets, of rational curves and lattice theory, we describe explicitly the Picard lattices.

scholarly journals K3 surfaces with Picard number three and canonical vector heights

2007 ◽  
Vol 76 (259) ◽  
pp. 1493-1499 ◽  
Author(s):  
Arthur Baragar ◽  
Ronald van Luijk
Keyword(s):  
K3 Surfaces ◽  
Picard Number ◽  
Canonical Vector

scholarly journals Cox rings of K3 surfaces of Picard number three

2021 ◽  
Vol 565 ◽  
pp. 598-626
Author(s):  
Michela Artebani ◽  
Claudia Correa Deisler ◽  
Antonio Laface
Keyword(s):  
K3 Surfaces ◽  
Picard Number ◽  
Cox Rings

scholarly journals On Some Generalized Rapoport–Zink Spaces

2019 ◽  
Vol 72 (5) ◽  
pp. 1111-1187
Author(s):  
Xu Shen
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Connected Components ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Mixed Characteristic ◽  
Special Fibers ◽  
Type Spaces ◽  
Local Invariants

AbstractWe enlarge the class of Rapoport–Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. Such obtained Rapoport–Zink spaces are said to be of abelian type. The class of Rapoport–Zink spaces of abelian type is strictly larger than the class of Rapoport–Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport–Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze.We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport–Zink spaces of abelian type. We construct and study the Ekedahl–Oort stratifications for the special fibers of Rapoport–Zink spaces of abelian type. As an application, we deduce a Rapoport–Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

scholarly journals Density of rational curves on $$K3$$ surfaces

2012 ◽  
Vol 356 (1) ◽  
pp. 331-354 ◽  
Author(s):  
Xi Chen ◽  
James D. Lewis
Keyword(s):  
K3 Surfaces ◽  
Rational Curves

scholarly journals On the Supersingular Reduction of K3 Surfaces with Complex Multiplication

2018 ◽  
Vol 2020 (20) ◽  
pp. 7306-7346
Author(s):  
Kazuhiro Ito
Keyword(s):  
Comparison Theorem ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Explicit Description ◽  
Brauer Group ◽  
K3 Surface ◽  
Good Reduction ◽  
Picard Number ◽  
Reduction Modulo ◽  
Modulo P

Abstract We study the good reduction modulo $p$ of $K3$ surfaces with complex multiplication. If a $K3$ surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions. Our results generalize some results of Shimada for $K3$ surfaces with Picard number $20$. Our methods rely on the main theorem of complex multiplication for $K3$ surfaces by Rizov, an explicit description of the Breuil–Kisin modules associated with Lubin–Tate characters due to Andreatta, Goren, Howard, and Madapusi Pera, and the integral comparison theorem recently established by Bhatt, Morrow, and Scholze.

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