Families of Group Actions, Generic Isotriviality, and Linearization

We study families of reductive group actions on A2 parametrized by curves and show that every faithful action of a non-finite reductive group on A3 is linearizable, i.e. G-isomorphic to a representation of G. The difficulties arise for non-connected groups G. We prove a Generic Equivalence Theorem which says that two affine mor- phisms p: S → Y and q: T → Y of varieties with isomorphic (closed) fibers become isomorphic under a dominant ́etale base change φ : U → Y . A special case is the following result. Call a morphism φ: X → Y a fibration with fiber F if φ is flat and all fibers are (reduced and) isomorphic to F. Then an affine fibration with fiber F admits an ́etale dominant morphism μ: U → Y such that the pull-back is a trivial fiber bundle: U ×Y X ≃ U × F . As an application we give short proofs of the following two (known) results: (a) Every affine A1-fibration over a normal variety is locally trivial in the Zariski-topology; (b) Every affine A2-fibration over a smooth curve is locally trivial in the Zariski-topology

Heritability in the genome-wide association era

Heritability, the fraction of phenotypic variation explained by genetic variation, has been estimated for many phenotypes in a range of populations, organisms, and time points. The recent development of efficient genotyping and sequencing technology has led researchers to attempt to identify the genetic variants responsible for the genetic component of phenotype directly via GWAS. The gap between the phenotypic variance explained by GWAS results and those estimated from classical heritability methods has been termed the “missing heritability problem”. In this work, we examine modern methods for estimating heritability, which use the genotype and sequence data directly. We discuss them in the context of classical heritability methods, the missing heritability problem, and describe their implications for understanding the genetic architecture of complex phenotypes.National Institutes of Health (U.S.) (fellowship 5T32ES007142-27)National Institutes of Health (U.S.) (grant R21 DK084529