scholarly journals F-theory models on K3 surfaces with various Mordell-Weil ranks — constructions that use quadratic base change of rational elliptic surfaces

2018 ◽  
Vol 2018 (5) ◽  
Author(s):  
Yusuke Kimura
Keyword(s):  
K3 Surfaces ◽  
Base Change ◽  
Elliptic Surfaces ◽  
Rational Elliptic Surfaces

scholarly journals ELLIPTIC K3 SURFACES ASSOCIATED WITH THE PRODUCT OF TWO ELLIPTIC CURVES: MORDELL–WEIL LATTICES AND THEIR FIELDS OF DEFINITION

2016 ◽  
Vol 228 ◽  
pp. 124-185 ◽  
Author(s):  
ABHINAV KUMAR ◽  
MASATO KUWATA
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Double Cover ◽  
Base Change ◽  
Elliptic Surfaces ◽  
Low Degree ◽  
Weil Group ◽  
Finite Index Subgroup ◽  
Rational Elliptic Surfaces

To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Masato Kuwata. We give an explicit description of the geometric Mordell–Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the nongeneric case, we describe a method to calculate explicitly a finite index subgroup of the Mordell–Weil group, which may be saturated to give the full group. Our methods rely on several interesting group actions, the use of rational elliptic surfaces, as well as connections to the geometry of low degree curves on cubic and quartic surfaces. We apply our techniques to compute the full Mordell–Weil group in several examples of arithmetic interest, arising from isogenous elliptic curves with complex multiplication, for which these K3 surfaces are singular.

scholarly journals F-theory models with 3 to 8 U(1) factors on K3 surfaces

2021 ◽  
pp. 2150125
Author(s):  
Yusuke Kimura
Keyword(s):  
K3 Surfaces ◽  
Base Change ◽  
Complex Structures ◽  
Elliptic Surfaces ◽  
Specific Complex ◽  
Kummer Surfaces ◽  
Rational Elliptic Surfaces ◽  
Weierstrass Equations

In this study, we construct four-dimensional F-theory models with 3 to 8 U(1) factors on products of K3 surfaces. We provide explicit Weierstrass equations of elliptic K3 surfaces with Mordell–Weil ranks of 3 to 8. We utilize the method of quadratic base change to glue pairs of rational elliptic surfaces together to yield the aforementioned types of K3 surfaces. The moduli of elliptic K3 surfaces constructed in the study include Kummer surfaces of specific complex structures. We show that the tadpole cancels in F-theory compactifications with flux when these Kummer surfaces are paired with appropriately selected attractive K3 surfaces. We determine the matter spectra on F-theory on the pairs.

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.

Extremal rational elliptic surfaces in characteristicp

1991 ◽  
Vol 207 (1) ◽  
pp. 429-437 ◽  
Author(s):  
William E. Lang
Keyword(s):  
Elliptic Surfaces ◽  
Rational Elliptic Surfaces

Arithmetic and geometry of rational elliptic surfaces

2016 ◽  
Vol 46 (6) ◽  
pp. 2061-2076
Author(s):  
Cec[! \' i!]lia Salgado
Keyword(s):  
Elliptic Surfaces ◽  
Rational Elliptic Surfaces

Rational Elliptic Surfaces

2019 ◽  
pp. 145-159
Author(s):  
Matthias Schütt ◽  
Tetsuji Shioda
Keyword(s):  
Elliptic Surfaces ◽  
Rational Elliptic Surfaces

scholarly journals Multiple fibers on rational elliptic surfaces

1988 ◽  
Vol 307 (1) ◽  
pp. 205-205 ◽  
Author(s):  
Brian Harbourne ◽  
William E. Lang
Keyword(s):  
Elliptic Surfaces ◽  
Multiple Fibers ◽  
Rational Elliptic Surfaces
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