scholarly journals Elliptic Fibrations on K3 Surfaces

2013 ◽  
Vol 57 (1) ◽  
pp. 253-267 ◽  
Author(s):  
Viacheslav V. Nikulin
Keyword(s):  
Finite Number ◽  
Automorphism Groups ◽  
K3 Surfaces ◽  
Rational Curves ◽  
Closed Field ◽  
K3 Surface ◽  
Elliptic Fibrations ◽  
Algebraically Closed Field

AbstractThis paper consists mainly of a review and applications of our old results relating to the title. We discuss how many elliptic fibrations and elliptic fibrations with infinite automorphism groups (or Mordell–Weil groups) an algebraic K3 surface over an algebraically closed field can have. As examples of applications of the same ideas, we also consider K3 surfaces with exotic structures: with a finite number of non-singular rational curves, with a finite number of Enriques involutions, and with naturally arithmetic automorphism groups.

scholarly journals On nilpotent automorphism groups of function fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nurdagül Anbar ◽  
Burçin Güneş
Keyword(s):  
Automorphism Group ◽  
Function Field ◽  
Automorphism Groups ◽  
Function Fields ◽  
Infinite Family ◽  
Nilpotent Subgroup ◽  
Closed Field ◽  
Algebraically Closed Field

Abstract We study the automorphisms of a function field of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. More precisely, we show that the order of a nilpotent subgroup G of its automorphism group is bounded by 16 (g – 1) when G is not a p-group. We show that if |G| = 16(g – 1), then g – 1 is a power of 2. Furthermore, we provide an infinite family of function fields attaining the bound.

scholarly journals Every Curve of Genus not Greater Than Eight Lies on a K3 Surface

2008 ◽  
Vol 190 ◽  
pp. 183-197 ◽  
Author(s):  
Manabu Ide
Keyword(s):  
Smooth Curve ◽  
Ample Divisor ◽  
Closed Field ◽  
K3 Surface ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Smooth Locus ◽  
Double Coverings

Let C be a smooth irreducible complete curve of genus g ≥ 2 over an algebraically closed field of characteristic 0. An ample K3 extension of C is a K3 surface with at worst rational double points which contains C in the smooth locus as an ample divisor.In this paper, we prove that all smooth curve of genera. 2 ≤ g ≤ 8 have ample K3 extensions. We use Bertini type lemmas and double coverings to construct ample K3 extensions.

HOMOGENEOUS SPACE FIBRATIONS OVER SURFACES

2017 ◽  
Vol 18 (2) ◽  
pp. 293-327 ◽  
Author(s):  
Yi Zhu
Keyword(s):  
Homogeneous Space ◽  
Function Field ◽  
Algebraic Surface ◽  
Rational Point ◽  
Homogeneous Spaces ◽  
Rational Curves ◽  
Closed Field ◽  
Generic Fiber ◽  
Algebraically Closed Field ◽  
Simple Connectedness

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface over an algebraically closed field, a variety whose geometric generic fiber is a projective homogeneous space admits a rational point if and only if the elementary obstruction vanishes.

scholarly journals On Commuting Varieties of Nilradicals of Borel Subalgebras of Reductive Lie Algebras

2014 ◽  
Vol 58 (1) ◽  
pp. 169-181 ◽  
Author(s):  
Simon M. Goodwin ◽  
Gerhard Röhrle
Keyword(s):  
Finite Number ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subgroup ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Commuting Variety ◽  
Irreducible Components

AbstractLet G be a connected reductive algebraic group defined over an algebraically closed field of characteristic 0. We consider the commuting variety of the nilradical of the Lie algebra of a Borel subgroup B of G. In case B acts on with only a finite number of orbits, we verify that is equidimensional and that the irreducible components are in correspondence with the distinguishedB-orbits in . We observe that in general is not equidimensional, and determine the irreducible components of in the minimal cases where there are infinitely many B-orbits in .

scholarly journals Some Groups of Type E7

2006 ◽  
Vol 182 ◽  
pp. 259-284 ◽  
Author(s):  
T. A. Springer
Keyword(s):  
Automorphism Group ◽  
Irreducible Representation ◽  
Vector Space ◽  
Algebraic Group ◽  
Automorphism Groups ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Identity Component ◽  
Field Of Definition ◽  
Quartic Form

AbstractAn algebraic group of type E7 over an algebraically closed field has an irreducible representation in a vector space of dimension 56 and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is ≠ 2, 3, taking into account a field of definition F. Certain F-forms of E7 appear in the automorphism groups of quartic forms over F, as well as forms of E6. Many of the results of the paper are known, but are perhaps not easily accessible in the literature.

scholarly journals On K3 Surface Quotients of K3 or Abelian Surfaces

2017 ◽  
Vol 69 (02) ◽  
pp. 338-372
Author(s):  
Alice Garbagnati
Keyword(s):  
Abelian Group ◽  
Moduli Space ◽  
Minimal Model ◽  
K3 Surfaces ◽  
Rational Curves ◽  
The Other ◽  
Abelian Surface ◽  
K3 Surface ◽  
Abelian Surfaces ◽  
Special Cases

Abstract The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group G (respectively of a K3 surface by an Abelian group G) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the K3 surfaces that are (rationally) G-covered by Abelian or K3 surfaces (in the latter case G is an Abelian group). When G has order 2 or G is cyclic and acts on an Abelian surface, this result is already known; we extend it to the other cases. Moreover, we prove that a K3 surface XG is the minimal model of the quotient of an Abelian surface by a group G if and only if a certain configuration of rational curves is present on XG . Again, this result was known only in some special cases, in particular, if G has order 2 or 3.

scholarly journals ORBIT PARAMETRIZATIONS FOR K3 SURFACES

2016 ◽  
Vol 4 ◽  
Author(s):  
MANJUL BHARGAVA ◽  
WEI HO ◽  
ABHINAV KUMAR
Keyword(s):  
Fixed Point ◽  
Recent Work ◽  
Moduli Spaces ◽  
Natural Extension ◽  
Algebraic Groups ◽  
K3 Surfaces ◽  
Closed Field ◽  
Positive Entropy ◽  
Algebraically Closed Field

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose Néron–Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.

scholarly journals Automorphism Groups of Certain Enriques Surfaces

2021 ◽  
Author(s):  
Simon Brandhorst ◽  
Ichiro Shimada
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Rational Curves ◽  
Elliptic Fibrations ◽  
Enriques Surfaces

AbstractWe calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general n-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations.

scholarly journals Elliptic Fibrations and Kodaira Dimensions of Schreieder’s Varieties

2021 ◽  
Author(s):  
Alice Garbagnati
Keyword(s):  
K3 Surfaces ◽  
Base Change ◽  
Kodaira Dimension ◽  
Elliptic Surface ◽  
Minimal Resolution ◽  
Elliptic Fibrations ◽  
Smooth Model ◽  
Birational Model ◽  
Hodge Numbers ◽  
Rational Elliptic Surface

Abstract We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_{(n)}^{(l)}$, which is the quotient of the product of a certain curve $C_{(n)}$ by itself $l$ times by a group $G\simeq \left ({\mathbb{Z}}/n{\mathbb{Z}}\right )^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved that the Kodaira dimension of this smooth model is 1, if $c>1$; if $l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on $n$ and $l$ we prove that $Y_{(n)}^{(l)}$ admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if $l=2$, its minimal resolution is a modular elliptic surface, obtained by a base change of order $n$ on a specific extremal rational elliptic surface; if $l\geq 3$ it has a birational model that admits a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of Kodaira dimension at most 0.

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