A note on the stratification by automorphisms of smooth plane curves of genus 6

In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$, where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p > 13$. We start with a classification already obtained by the first author and we use standard techniques. Interestingly, in the way to get these families for the different strata, we find two remarkable phenomenons that did not appear before. One is the existence of a non $0$-dimensional final stratum of plane curves. At a first sight it may sound odd, but we will see that this is a normal situation for higher degrees and we will give a explanation for it. We explicitly describe representative families for all strata, except for the stratum with automorphism group $\mathbb{Z}/5\mathbb{Z}$. Here we find the second difference with the lower genus cases where the previous techniques do not fully work. Fortunately, we are still able to prove the existence of such family by applying a version of Luroth's theorem in dimension $2$

The Picard Group of Brauer-Severi Varieties

In this note we provide explicit generators of the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular, for all Brauer-Severi surfaces. To produce these generators we use the Twisting Theory for smooth plane curves

Connecting blazars with ultra high energy cosmic rays and astrophysical neutrinos

We present a strong hint of a connection between high energy $\gamma$-ray emitting blazars, very high energy neutrinos, and ultra high energy cosmic rays. We first identify potential hadronic sources by filtering $\gamma$-ray emitters %from existing catalogs that are in spatial coincidence with the high energy neutrinos detected by IceCube. The neutrino filtered $\gamma$-ray emitters are then correlated with the ultra high energy cosmic rays from the Pierre Auger Observatory and the Telescope Array by scanning in $\gamma$-ray flux ($F_{\gamma}$) and angular separation ($\theta$) between sources and cosmic rays. A maximal excess of 80 cosmic rays (42.5 expected) is found at $\theta\leq10^{\circ}$ from the neutrino filtered $\gamma$-ray emitters selected from the second hard {\it Fermi}-LAT catalogue (2FHL) and for $F_\gamma\left(>50\:\mathrm{GeV}\right)\geq1.8\times10^{-11}\:\mathrm{ph}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. The probability for this to happen is $2.4 \times 10^{-5}$, which translates to $\sim 2.4 \times 10^{-3}$ after compensation for all the considered trials. No excess of cosmic rays is instead observed for the complement sample of $\gamma$-ray emitters (i.e. not in spatial connection with IceCube neutrinos). A likelihood ratio test comparing the connection between the neutrino filtered and the complement source samples with the cosmic rays favours a connection between neutrino filtered emitters and cosmic rays with a probability of $\sim1.8\times10^{-3}$ ($2.9\sigma)$ after compensation for all the considered trials. The neutrino filtered $\gamma$-ray sources that make up the cosmic rays excess are blazars of the high synchrotron peak type. More statistics is needed to further investigate these sources as candidate cosmic ray and neutrino emitters.Comment: Accepted for publication in MNRAS, added one figure (redshift distribution), new IceCube data, and penalty factor for subsets within single catalogue

EZH2 Single Nucleotide Variants (SNVs): Diagnostic and Prognostic Role in 10 Solid Tumor Types

The enhancer of zeste homolog 2 (EZH2) gene encodes a histone methyltransferase that is a catalytic subunit of the Polycomb repressive complex 2 (PRC2) group of proteins that act to repress gene expression. The EZH2 locus is rarely mutated in solid tumors and there is no comprehensive study of EZH2 single nucleotide variants (SNVs) associated with cancer susceptibility, prognosis and response to therapy. Here, for the first time, we review the functional roles of EZH2 DNA variants and propose a putative etiological role in 10 various solid tumors including: esophageal, hepatocellular, oral, urothelial, colorectal, lung and gastric cancers. In particular, we found that the C allele of the EZH2 variant rs3757441 is associated with increased EZH2 RNA expression and poorer prognosis (advanced stage) in at least two malignancies such as colorectal and hepatocellular carcinoma. This suggests that the C allele may be a functional risk variant in multiple malignant tumors. We therefore propose that the rs3757441 single nucleotide variant (SNV) be genotyped and real-time PCR assays be performed in large cohort studies in order to confirm this preliminary finding that could be useful for clinical practice