A threefold violating a local-to-global principle for rationality

In this note we construct a conditional example of a smooth projective threefold that is irrational over $\mathbb Q$ but is rational at all places. Our example uses a genus $2$ curve $C$ of Bruin--Stoll and is conditional on the Birch--Swinnerton-Dyer conjecture for $C$.Comment: 6 page

Comprehensive methylome map of lineage commitment from haematopoietic progenitors.

Epigenetic modifications must underlie lineage-specific differentiation as terminally differentiated cells express tissue-specific genes, but their DNA sequence is unchanged. Haematopoiesis provides a well-defined model to study epigenetic modifications during cell-fate decisions, as multipotent progenitors (MPPs) differentiate into progressively restricted myeloid or lymphoid progenitors. Although DNA methylation is critical for myeloid versus lymphoid differentiation, as demonstrated by the myeloerythroid bias in Dnmt1 hypomorphs, a comprehensive DNA methylation map of haematopoietic progenitors, or of any multipotent/oligopotent lineage, does not exist. Here we examined 4.6 million CpG sites throughout the genome for MPPs, common lymphoid progenitors (CLPs), common myeloid progenitors (CMPs), granulocyte/macrophage progenitors (GMPs), and thymocyte progenitors (DN1, DN2, DN3). Marked epigenetic plasticity accompanied both lymphoid and myeloid restriction. Myeloid commitment involved less global DNA methylation than lymphoid commitment, supported functionally by myeloid skewing of progenitors following treatment with a DNA methyltransferase inhibitor. Differential DNA methylation correlated with gene expression more strongly at CpG island shores than CpG islands. Many examples of genes and pathways not previously known to be involved in choice between lymphoid/myeloid differentiation have been identified, such as Arl4c and Jdp2. Several transcription factors, including Meis1, were methylated and silenced during differentiation, indicating a role in maintaining an undifferentiated state. Additionally, epigenetic modification of modifiers of the epigenome seems to be important in haematopoietic differentiation. Our results directly demonstrate that modulation of DNA methylation occurs during lineage-specific differentiation and defines a comprehensive map of the methylation and transcriptional changes that accompany myeloid versus lymphoid fate decisions

Symmetries of Fano varieties

We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi--Yau varieties, and log terminal singularities. We show the existence of a constant $m(n)$, so that every symmetric group $S_k$ acting on an $n$-dimensional Fano variety satisfies $k \leq m(n)$. We prove that $m(n)> n+\sqrt{2n}$ for every $n$. On the other hand, we show that $\lim_{n\to \infty} m(n)/(n+1)^2 \leq 1$. However, this asymptotic upper bound is not expected to be sharp. We obtain sharp bounds for certain classes of varieties. For toric varieties, we show that $m(n)=n+2$ for $n\geq 4$. For Fano quasismooth weighted complete intersections, we prove the asymptotic equality $\lim_{n\to \infty} m(n)/(n+1)=1$. Among the Fano weighted complete intersections, we study the maximally symmetric ones and show that they are closely related to the Fano--Fermat varieties, i.e., Fano complete intersections in $\mathbb P^N$ cut out by Fermat hypersurfaces. Finally, we draw a connection between maximally symmetric Fano varieties and boundedness of Fano varieties. For instance, we show that the class of $S_8$-equivariant Fano $4$-folds forms a bounded family. In contrast, the $S_7$-equivariant Fano $4$-folds are unbounded.Comment: 32 pages, 3 table

Conic bundle threefolds differing by a constant Brauer class and connections to rationality

A double cover $Y$ of $\mathbb{P}^1 \times \mathbb{P}^2$ ramified over a general $(2,2)$-divisor will have the structure of a geometrically standard conic bundle ramified over a smooth plane quartic $\Delta \subset \mathbb{P}^2$ via the second projection. These threefolds are rational over algebraically closed fields, but over nonclosed fields, including over $\mathbb{R}$, their rationality is an open problem. In this paper, we characterize rationality over $\mathbb{R}$ when $\Delta(\mathbb{R})$ has at least two connected components (extending work of M. Ji and the second author) and over local fields when all odd degree fibers of the first projection have nonsquare discriminant. We obtain these applications by proving general results comparing the conic bundle structure on $Y$ with the conic bundle structure on a well-chosen intersection of two quadrics. The difference between these two conic bundles is encoded by a constant Brauer class, and we prove that this class measures a certain failure of Galois descent for the codimension 2 Chow group of $Y$.Comment: 18 page

Curve classes on conic bundle threefolds and applications to rationality

We undertake a study of conic bundle threefolds $\pi\colon X\to W$ over geometrically rational surfaces whose associated discriminant covers $\tilde{\Delta}\to\Delta\subset W$ are smooth and geometrically irreducible. First, we determine the structure of the group $\mathrm{CH}^2 X_{\overline{k}}$ of rational equivalence classes of curves. Precisely, we construct a Galois-equivariant group homomorphism from $\mathrm{CH}^2X_{\overline{k}}$ to a group scheme associated to the discriminant cover $\tilde{\Delta}\to \Delta$ of $X$. The target group scheme is a generalization of the Prym variety of $\tilde{\Delta}\to\Delta$ and so our result can be viewed as a generalization of Beauville's result that the algebraically trivial curve classes on $X_{\overline{k}}$ are parametrized by the Prym variety. Next, we use our structural result on curve classes to study rationality obstructions, in particular the refined intermediate Jacobian torsor (IJT) obstruction recently introduced by Hassett--Tschinkel and Benoist--Wittenberg. We show that for conic bundle threefolds there is no strongest (known) rationality obstruction. Precisely, we construct a geometrically rational irrational conic bundle threefold where the IJT obstruction cannot witness irrationality (irrationality is detected through the real topology) and a geometrically rational irrational conic bundle threefold where all classical rationality obstructions vanish and the IJT obstruction is needed to prove irrationality.Comment: 28 pages. Comments welcome! v2: Updated introductio

The K-moduli space of a family of conic bundle threefolds

We describe the 6-dimensional compact K-moduli space of Fano threefolds in deformation family No 2.18. These Fano threefolds are double covers of $\mathbb P^1\times\mathbb P^2$ branched along smooth $(2,2)$-surfaces, and Cheltsov--Fujita--Kishimoto--Park proved that any smooth Fano threefold in this family is K-stable. A member of family No 2.18 admits the structures of a conic bundle and a quadric surface bundle. We prove that K-polystable limits of these Fano threefolds admit conic bundle structures, but not necessarily del Pezzo fibration structures. We study this K-moduli space via the moduli space of log Fano pairs $(\mathbb P^1\times\mathbb P^2, c R)$ for $c=1/2$ and $R$ a $(2,2)$-divisor, which we construct using wall-crossings. In the case where the divisor is proportional to the anti-canonical divisor, the first author, together with Ascher and Liu, developed a framework for wall crossings in K-moduli and proved that there are only finitely many walls, which occur at rational values of the coefficient $c$. This paper constructs the first example of wall-crossing in K-moduli in the non-proportional setting, and we find a wall at an irrational value of $c$. In particular, we obtain explicit descriptions of the GIT and K-moduli spaces (for $c \leq 1/2$) of these $(2,2)$-divisors. Furthermore, using the conic bundle structure, we study the relationship with the GIT moduli space of plane quartic curves.Comment: 128 pages, 11 figures. Comments are welcom

Defining the Molecular Basis of Tumor Metabolism: a Continuing Challenge Since Warburg's Discovery

Cancer cells are the product of genetic disorders that alter crucial intracellular signaling pathways associated with the regulation of cell survival, proliferation, differentiation and death mechanisms. the role of oncogene activation and tumor suppressor inhibition in the onset of cancer is well established. Traditional antitumor therapies target specific molecules, the action/expression of which is altered in cancer cells. However, since the physiology of normal cells involves the same signaling pathways that are disturbed in cancer cells, targeted therapies have to deal with side effects and multidrug resistance, the main causes of therapy failure. Since the pioneering work of Otto Warburg, over 80 years ago, the subversion of normal metabolism displayed by cancer cells has been highlighted by many studies. Recently, the study of tumor metabolism has received much attention because metabolic transformation is a crucial cancer hallmark and a direct consequence of disturbances in the activities of oncogenes and tumor suppressors. in this review we discuss tumor metabolism from the molecular perspective of oncogenes, tumor suppressors and protein signaling pathways relevant to metabolic transformation and tumorigenesis. We also identify the principal unanswered questions surrounding this issue and the attempts to relate these to their potential for future cancer treatment. As will be made clear, tumor metabolism is still only partly understood and the metabolic aspects of transformation constitute a major challenge for science. Nevertheless, cancer metabolism can be exploited to devise novel avenues for the rational treatment of this disease. Copyright (C) 2011 S. Karger AG, BaselFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Univ Fed ABC UFABC, CCNH, Santo Andre, BrazilUniversidade Federal de São Paulo UNIFESP, Dept Ciencias Biol, São Paulo, BrazilUniversidade Federal de São Paulo UNIFESP, Dept Bioquim, São Paulo, BrazilUniv Fed Sao Carlos UFSCar, DFQM, Sorocaba, BrazilUniversidade Federal de São Paulo UNIFESP, Dept Ciencias Biol, São Paulo, BrazilUniversidade Federal de São Paulo UNIFESP, Dept Bioquim, São Paulo, BrazilFAPESP: 10/16050-9FAPESP: 10/11475-1FAPESP: 08/51116-0Web of Scienc