scholarly journals Threefolds fibred by mirror sextic double planes

2020 ◽  
pp. 1-42
Author(s):  
Remkes Kooistra ◽  
Alan Thompson
Keyword(s):  
Systematic Study ◽  
Betti Numbers ◽  
K3 Surfaces ◽  
Canonical Divisor ◽  
Geometric Properties ◽  
Elliptic Surfaces ◽  
Birational Maps ◽  
Minimal Form ◽  
Birational Model ◽  
Double Planes

Abstract We present a systematic study of threefolds fibred by K3 surfaces that are mirror to sextic double planes. There are many parallels between this theory and the theory of elliptic surfaces. We show that the geometry of such threefolds is controlled by a pair of invariants, called the generalized functional and generalized homological invariants, and we derive an explicit birational model for them, which we call the Weierstrass form. We then describe how to resolve the singularities of the Weierstrass form to obtain the “minimal form”, which has mild singularities and is unique up to birational maps in codimension 2. Finally, we describe some of the geometric properties of threefolds in minimal form, including their singular fibres, canonical divisor, and Betti numbers.

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.

scholarly journals K3 surfaces with non-symplectic involution and compact irreducible G2-manifolds

2011 ◽  
Vol 151 (2) ◽  
pp. 193-218 ◽  
Author(s):  
ALEXEI KOVALEV ◽  
NAM-HOON LEE
Keyword(s):  
Betti Numbers ◽  
Matching Condition ◽  
K3 Surfaces ◽  
Connected Sum ◽  
Fano Threefolds ◽  
New Class ◽  
Blowing Up ◽  
G2 Manifolds

AbstractWe consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G2 developed by the first named author. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors and the latter K3 surfaces should satisfy a certain ‘matching condition’ intertwining on their periods and Kähler classes. Suitable examples of threefolds were previously obtained by blowing up curves in Fano threefolds.In this paper, we give a large new class of suitable algebraic threefolds using theory of K3 surfaces with non-symplectic involution due to Nikulin. These threefolds are not obtainable from Fano threefolds as above, and admit matching pairs leading to topologically new examples of compact irreducible G2-manifolds. ‘Geography’ of the values of Betti numbers b2, b3 for the new (and previously known) examples of irreducible G2 manifolds is also discussed.

scholarly journals Birational Maps of Moduli Spaces of Vector Bundles on $K3$ Surfaces

2011 ◽  
Vol 34 (2) ◽  
pp. 473-491 ◽  
Author(s):  
Masanori KIMURA ◽  
Kōta YOSHIOKA
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
K3 Surfaces ◽  
Birational Maps

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.

Quantitative parallel EELS spectrum imaging developed as a new tool in AEM

1992 ◽  
Vol 50 (2) ◽  
pp. 1202-1203
Author(s):  
Gianluigi Botton ◽  
Gilles L'espérance
Keyword(s):  
Statistical Analysis ◽  
Spatial Resolution ◽  
Personal Computer ◽  
Systematic Study ◽  
Present Author ◽  
Chemical Elements ◽  
Detection Limits ◽  
Research Groups ◽  
Quantitative Performance ◽  
Spectrum Imaging

As interest for parallel EELS spectrum imaging grows in laboratories equipped with commercial spectrometers, different approaches were used in recent years by a few research groups in the development of the technique of spectrum imaging as reported in the literature. Either by controlling, with a personal computer both the microsope and the spectrometer or using more powerful workstations interfaced to conventional multichannel analysers with commercially available programs to control the microscope and the spectrometer, spectrum images can now be obtained. Work on the limits of the technique, in terms of the quantitative performance was reported, however, by the present author where a systematic study of artifacts detection limits, statistical errors as a function of desired spatial resolution and range of chemical elements to be studied in a map was carried out The aim of the present paper is to show an application of quantitative parallel EELS spectrum imaging where statistical analysis is performed at each pixel and interpretation is carried out using criteria established from the statistical analysis and variations in composition are analyzed with the help of information retreived from t/γ maps so that artifacts are avoided.

Sign in / Sign up

Export Citation Format

Share Document