scholarly journals On Cox rings of K3 surfaces

2010 ◽  
Vol 146 (4) ◽  
pp. 964-998 ◽  
Author(s):  
Michela Artebani ◽  
Jürgen Hausen ◽  
Antonio Laface
Keyword(s):  
K3 Surfaces ◽  
Del Pezzo Surfaces ◽  
K3 Surface ◽  
Picard Number ◽  
Finitely Generated ◽  
Generators And Relations ◽  
Cox Rings ◽  
Double Covers ◽  
Del Pezzo ◽  
Effective Cone

AbstractWe study Cox rings of K3 surfaces. A first result is that a K3 surface has a finitely generated Cox ring if and only if its effective cone is rational polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3 surfaces of Picard number two, and explicitly compute the Cox rings of generic K3 surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.

scholarly journals Cox rings of degree one del Pezzo surfaces

2009 ◽  
Vol 3 (7) ◽  
pp. 729-761 ◽  
Author(s):  
Damiano Testa ◽  
Anthony Várilly-Alvarado ◽  
Mauricio Velasco
Keyword(s):  
Del Pezzo Surfaces ◽  
Cox Rings ◽  
Del Pezzo

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 Quotients of del Pezzo surfaces

2019 ◽  
Vol 30 (12) ◽  
pp. 1950068
Author(s):  
Andrey Trepalin
Keyword(s):  
Complete Classification ◽  
Finite Subgroup ◽  
Del Pezzo Surfaces ◽  
Picard Number ◽  
Del Pezzo Surface ◽  
Cyclic Groups ◽  
Trivial Group ◽  
Quotient Surface ◽  
Del Pezzo

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

scholarly journals Hilbert functions of Cox rings of del Pezzo surfaces

2017 ◽  
Vol 480 ◽  
pp. 309-331
Author(s):  
Jinhyung Park ◽  
Joonyeong Won
Keyword(s):  
Hilbert Functions ◽  
Del Pezzo Surfaces ◽  
Cox Rings ◽  
Del Pezzo

scholarly journals Gröbner bases, monomial group actions, and the Cox rings of Del Pezzo surfaces

2007 ◽  
Vol 316 (2) ◽  
pp. 777-801 ◽  
Author(s):  
Mike Stillman ◽  
Damiano Testa ◽  
Mauricio Velasco
Keyword(s):  
Gröbner Bases ◽  
Group Actions ◽  
Grobner Bases ◽  
Del Pezzo Surfaces ◽  
Cox Rings ◽  
Monomial Group ◽  
Del Pezzo

scholarly journals On pliability of del Pezzo fibrations and Cox rings

2017 ◽  
Vol 2017 (723) ◽  
Author(s):  
Hamid Ahmadinezhad
Keyword(s):  
Moduli Space ◽  
Toric Varieties ◽  
Projective Spaces ◽  
Low Rank ◽  
Del Pezzo Surfaces ◽  
Coordinate Ring ◽  
Birational Transformations ◽  
Cox Rings ◽  
Del Pezzo

AbstractWe develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne–Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over

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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