Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces

Yusuke Kimura

doi:10.1007/jhep03(2018)045

Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces

Journal of High Energy Physics ◽

10.1007/jhep03(2018)045 ◽

2018 ◽

Vol 2018 (3) ◽

Cited By ~ 4

Author(s):

Yusuke Kimura

Keyword(s):

K3 Surfaces ◽

Elliptic Surfaces ◽

Rational Elliptic Surfaces

Download Full-text

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

Journal of High Energy Physics ◽

10.1007/jhep05(2018)048 ◽

2018 ◽

Vol 2018 (5) ◽

Cited By ~ 4

Author(s):

Yusuke Kimura

Keyword(s):

K3 Surfaces ◽

Base Change ◽

Elliptic Surfaces ◽

Rational Elliptic Surfaces

Download Full-text

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2018.03.010 ◽

2019 ◽

Vol 223 (1) ◽

pp. 277-300 ◽

Cited By ~ 1

Author(s):

Alice Garbagnati ◽

Cecília Salgado

Keyword(s):

Linear Systems ◽

K3 Surfaces ◽

Elliptic Surfaces ◽

Elliptic Fibrations ◽

Rational Elliptic Surfaces

Download Full-text

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

Nagoya Mathematical Journal ◽

10.1017/nmj.2016.56 ◽

2016 ◽

Vol 228 ◽

pp. 124-185 ◽

Cited By ~ 2

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.

Download Full-text

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

International Journal of Modern Physics A ◽

10.1142/s0217751x21501256 ◽

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.

Download Full-text