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OBJECTIVES: The Hepatology Committee of the European Society for Pediatric Gastroenterology, Hepatology and Nutrition (ESPGHAN) aims to educate pediatric gastroenterologists, members of ESPGHAN and professionals from other specialties promoting an exchange of clinical expertise in the field of pediatric hepatology.METHODS: The 2020 single topic ESPGHAN monothematic 3-day conference on pediatric liver disease, was organized in Athens, Greece and was entitled " Acute Liver Failure" (ALF). ALF is a devastating disease with high mortality and in a considerable fraction of patients, the cause remains unresolved. As knowledge in diagnosis and treatment of ALF in infants and children has increased in the past decades, the objective was to update physicians in the field with developments in medical therapy and indications for liver transplantation (LT) and to identify areas for future research in clinical and neurocognitive outcomes in ALF.RESULTS: We recently reported the epidemiology, diagnosis, and initial intensive care management issues in separate manuscript. Herewith we report on the medical treatment, clinical lessons arising from pediatric studies, nutritional and renal replacement therapy (RRT), indications and contraindications for LT, neurocognitive outcomes, new techniques used as bridging to LT, and areas for future research. Oral presentations by experts in various fields are summarized highlighting key learning points.CONCLUSIONS: The current report summarizes the current insights in medical treatment of pediatric ALF and the directions for future research.

OBJECTIVES: The Hepatology Committee of the European Society for Pediatric Gastroenterology, Hepatology and Nutrition (ESPGHAN) aims to educate pediatric gastroenterologists, members of ESPGHAN and professionals from other specialties promoting an exchange of clinical expertise in the field of pediatric hepatology. Herewith we have concentrated on detailing the recent advances in acute liver failure in infants and children.METHODS: The 2020 ESPGHAN monothematic three-day conference on pediatric hepatology disease, entitled "acute liver failure" (ALF), was organized in Athens, Greece. ALF is a devastating disease with high mortality and most cases remain undiagnosed. As knowledge in diagnosis and treatment of ALF in infants and children has increased in the past decades, the objective was to update physicians in the field with the latest research and developments in early recognition, curative therapies and intensive care management, imaging techniques and treatment paradigms in these age groups.RESULTS: In the first session, the definition, epidemiology, various causes of ALF, in neonates and older children and recurrent ALF (RALF) were discussed. The second session was dedicated to new aspects of ALF management including hepatic encephalopathy (HE), coagulopathy, intensive care interventions, acute on chronic liver failure, and the role of imaging in treatment and prognosis. Oral presentations by experts in various fields are summarized highlighting key learning points.CONCLUSIONS: The current report summarizes the major learning points from this meeting. It also identifies areas where there is gap of knowledge, thereby identifying the research agenda for the near future.

We describe a construction of plane quartics with prescribed Galois operation on the 28 bitangents, in the particular case of a Galois-invariant Steiner hexad. As an application, we solve the inverse Galois problem for degree two del Pezzo surfaces in the corresponding particular case.

Objectives: We aimed to investigate national allocation policies for pediatric liver transplantation (LT).Method: A survey was prepared by the European Society for Paediatric Gastroenterology Hepatology and Nutrition Hepatology Committee in collaboration with the North American Studies of Pediatric Liver Transplantation consortium. The survey was sent to pediatric hepatologists and transplant surgeons worldwide. National data were obtained from centrally based registries.Results: Replies were obtained from 15 countries from 5 of the world continents. Overall donation rate varied between 9 and 35 per million inhabitants. The number of pediatric LTs was 4 to 9 per million inhabitants younger than 18 years for 13 of the 15 respondents. In children younger than 2 years mortality on the waiting list (WL) varied between 0 and 20%. In the same age group, there were large differences in the ratio of living donor LT to deceased donor LT and in the ratio of split liver segments to whole liver. These differences were associated with possible discrepancies in WL mortality.Conclusions: Similarities but also differences between countries were detected. The described data may be of importance when trying to reduce WL mortality in the youngest children.

Bile acids (BA) have been found to promote coagulation by increasing tissue factor (TF) activity. The contribution of elevated BA levels and cholestasis to TF decryption within the liver parenchyma and the role of farnesoid X receptor (FXR) in this process remain unclear. We investigated the effects of BA on TF activity and thrombin generation in hepatocytes and correlated these effects with activation of FXR-dependent signaling and apoptosis. HepG2 cells and primary hepatocytes were incubated with chenodeoxycholic acid (CDCA), glycochenodeoxycholic acid (GCDCA), ursodeoxycholic acid (UCDA), or the synthetic FXR agonist GW4064 for 24 h. MTT tests demonstrated cell viability throughout experiments. TF activity was tested via factor Xa generation and thrombin generation was measured by calibrated automated thrombography. Increased TF activity alongside enhanced thrombin generation was observed with CDCA and GW4064 but not with GCDCA and UDCA. TF activity was substantially reduced when FXR activation was blocked with the antagonist DY 268. Quantitative polymerase chain reaction revealed upregulation of FXR target genes only by CDCA and GW4064. Western blot analysis and fluorescence microscopy showed no TF overexpression arguing for TF decryption. Caspase 3 activity measurements and flow cytometric analysis of Annexin V binding showed no signs of apoptosis. Long-term exposure of hepatocytes to nontoxic BA may cause intracellular FXR overstimulation, triggering TF decryption irrespective of the amphiphilic properties of BA. The effect of BA on TF activation correlates with the molecule’s ability to enter the cells and activate FXR. TF decryption occurs independently of apoptotic mechanisms. The potential of various bile acids to induce procoagulant tissue factor (TF) activity in viable HepG2 cells and primary human hepatocytes was investigated. Increased TF activity correlated with the molecules’ ability to enter the cells and activate the farnesoid X receptor (FXR) suggesting a crucial role of FXR in bile acid-mediated TF activity within the liver parenchyma.

ObjectivesAssessing influence of anti-adalimumab (ADA) antibodies (AAA) on serum trough ADA levels and uveitis activity in long-term ADA treatment of juvenile idiopathic arthritis (JIA)-associated uveitis.Patients and interventionsThis prospective observational study included 20 patients from a single centre treated with ADA for active uveitis refractory to conventional disease-modifying antirheumatic drugs. AAA, serum ADA trough levels and uveitis activity were evaluated at regular intervals up to 6 years.ResultsAAA were detected in nine patients (45%). Permanent AAA in seven were associated with undetectable ADA trough levels and loss of response (LOR). Transient AAA were detected in four with measurable ADA trough levels and response of uveitis to treatment, followed in two by permanent AAA associated with LOR. Use of concomitant immunosuppression was significantly higher in patients without AAA (pConclusionsAAA-associated LOR frequently occurs in long-term treatment with ADA for JIA-associated uveitis. Concomitant immunosuppressive therapy significantly reduces the risk of LOR due to AAA.

We investigated ‘rare’ bile acids (BA) as potential markers in septic neonates. ‘Rare’ (C-6 hydroxylated BA) and ‘classical’ BA were determined in 102 neonates using high-performance liquid chromatography—high-resolution mass spectrometry (HPLC-HRMS). Four groups according to maturity (full term, FT vs. preterm, PT) and septic status (early-onset neonatal sepsis, EOS vs. CTR; non-septic controls) were formed: FT-CTR; (n = 47), PT-CTR (n = 22), FT-EOS (n = 20), PT-EOS (n = 13). Firstly, FT-CTR had a significant higher amount of ‘rare’ BA than PT (FT-CTR: 0.5 µmol/L, IQR: 0.3–1.3 vs. PT-CTR: 0.01 µmol/L, IQR 0.01–0.2; p

For families of $K3$ surfaces, we establish a sufficient criterion for real or complex multiplication. Our criterion is arithmetic in nature. It may show, at first, that the generic fibre of the family has a nontrivial endomorphism field. Moreover, the endomorphism field does not shrink under specialisation. As an application, we present two explicit families of $K3$ surfaces having real multiplication by $\bbQ(\sqrt{2})$ and $\bbQ(\sqrt{5})$, respectively.

Chronic liver disease places patients at increased risk of malnutrition that can be challenging to identify clinically and treat. Nutrition support is a key aspect of the management of these patients as it has an impact on their quality of life, morbidity, and mortality. There are significant gaps in the literature regarding the optimal nutrition support for patients with different types of liver diseases and the impact of these interventions on long-term outcomes. This Position Paper summarizes the available literature on the nutritional aspects of the care of patients with chronic liver diseases. Specifically, the challenges associated with the nutritional assessment of these subjects are discussed, and recently investigated approaches to determining the patients' nutritional status are reviewed. Furthermore, the pathophysiology of the malnutrition seen in the context of chronic liver disease is summarized and monitoring, as well as treatment, recommendations are provided. Lastly, suggestions for future research studies are described.

Coeliac disease (CD) and juvenile idiopathic arthritis (JIA) often coexist. This association warrants assessment for CD in patients with JIA. We evaluated the clinical relevance and cost-effectiveness of human leucocyte antigen (HLA) genotyping in first-line screening for development of CD in children with JIA.95 patients with JIA were screened for CD using CD-specific antibodies. In case of positivity, a small intestinal biopsy was performed to confirm diagnosis. In addition, HLA genotyping was performed. 110 age-matched and sex-matched Caucasian children from the same geographical area served as controls.CD was diagnosed in 4 of 95 patients with JIA (4.2%), a rate significantly higher compared with controls (p0.02) and 14 times higher than in the general population. Twenty-six patients (27.4%) had one of the variants of the risk genotypes. All four patients diagnosed with CD had a HLA-DQ2.5 genotype: one was homozygote, the remainder heterozygote. Twenty-two patients are, judging by their HLA genotypes, at risk of developing CD and require repeated serological screening. None of the 69 patients without HLA-DQ2/DQ8 genotypes had CD-specific antibodies. Screening with HLA genotyping becomes cheaper than screening without after the second determination.In our cohort of patients with JIA, lack of HLA-DQ2/DQ8 genotypes identified a majority not at risk of CD in whom repeated serological testing is unnecessary. Genotyping is nowadays the most efficient and cost-effective way to screen for CD risk in JIA.

Background Accurate analytical methods for bile acid (BA) analysis are important for clinical diagnosis in newborns, adolescents, and adults. Improvements in speed, sensitivity and simplicity enable BA profiling using high performance liquid chromatography (HPLC) together with electrospray ionization (ESI) and high-resolution mass spectrometry (HR-MS). Results We present a method, validated in different species and tissues, enabling a highly sensitive quantitative determination (in a range of 0.24 pmol/sample to 1000 pmol/sample, corresponding to 0.024–100 pmol on column) of up to 36 naturally occurring BAs from as little as 10 μl of plasma or serum, 1 ml of urine, or 10 mg of dried stool. Chromatographic separation is achieved by HPLC using a C18 reversed phase column and a water/methanol gradient. After ESI, BAs are analyzed through HR-MS using orbitrap technology in full scan mode. 30 different BAs and the corresponding internal standards are separated and analyzed in a single run. Six additional BAs are evaluated in a second run using a pentafluorophenyl (PFP) stationary phase. Conclusions This method generates detailed human and rodent BA profiles in full scan mode and accurate mass with the advantage of remarkably low required sample volume.

We show that, over every number field, the degree four del Pezzo surfaces that violate the Hasse principle are Zariski dense in the moduli scheme., Comment: 21 pages

Background: Hepatitis E virus (HEV) is endemic in large parts of the developing world. Waterborne transmission of genotypes 1 or 2 commonly causes acute hepatitis, which is usually self-limited in healthy individuals. In addition, acute HEV infections also occur outside endemic areas, mostly related to foodborne transmission of HEV genotype 3. A growing number of publications in the last decade have reported chronic infection progressing to cirrhosis in immunosuppressed patients. It has also been suggested that HEV transmission may occur via contaminated blood products. This publication aims to provide recommendations for diagnosis, prevention, and treatment of HEV infection, particularly in children after solid organ transplantation.Methods: A systematic PubMed literature search on HEV infection from 1990 to January 2016 was performed focusing on pediatric studies. The existing body of evidence was reviewed and recommendations were agreed upon following discussion and unanimous agreement by all members of the ESPGHAN Hepatology Committee during a consensus meeting in January 2016. In the absence of randomized controlled studies these recommendations were considered to be expert opinions.Key Recommendations: Immunocompetent children with increased transaminases and/or extrahepatic manifestations should be considered for testing for evidence of HEV infection. Immunocompromised children with increased aminotransferases should be repeatedly tested for HEV and may require therapeutic intervention.

Objectives: In 2017, the European Medicines Agency and the Food and Drug Administration approved the use of the fixed-dose combination of ledipasvir/sofosbuvir and of the combination of sofosbuvir and ribavirin for treatment of adolescents (12-17 years or weighing >35 kg) with chronic hepatitis C virus (HCV) genotype 1, 4, 5, and 6 and genotype 2 and 3 infections, respectively. Although trials with direct-acting antivirals are ongoing for younger children, the only available treatment in the United States and Europe for those

Given a number field $k$ and a positive integer $d$, in this paper we consider the following question: does there exist a smooth diagonal surface of degree $d$ in $\mathbb{P}^3$ over $k$ which contains a line over every completion of $k$, yet no line over $k$? We answer the problem using Galois cohomology, and count the number of counter-examples using a result of Erd\H{o}s., Comment: 14 pages

We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.

Objective Because the prevalence of obesity in children is increasing, the frequency of pediatric nonalcoholic fatty liver disease (NAFLD) is growing. A reliable noninvasive biomarker for monitoring progression of liver fibrosis would be useful. In cirrhotic persons serum bile acid (BA) levels are significantly elevated. We hypothesized that BA levels and composition in pediatric NAFLD vary depending on the stage of fibrosis. Methods Children with NAFLD were compared with controls and classified by stages of fibrosis (NAFLD-F0, n = 27; NAFLD-F≥1, n = 65) based on liver-biopsy findings. Fasted metabolic and cholestasis status was assessed by several blood tests. BA profiles were measured by tandem mass spectrometry and compared with healthy controls (n = 105). Results Compared with controls, all of the NAFLD patients were overweight and showed significantly elevated glucose, insulin, aspartate transaminase, and alanine transaminase levels. Total serum BAs were lower in nonfibrotic NAFLD children than in a control cohort (1.73 vs 3.6 μmol/L) because low glycine-conjugated BA levels were incompletely compensated by increases in taurine-conjugated or unconjugated BA. In patients with fibrotic NAFLD, BA levels were lower than in controls (2.45 vs 3.6 μmol/L) but higher than in nonfibrotic patients (2.45 vs 1.73 μmol/L), and the BA pattern resembled that of healthy controls. Fibroblast growth factor 19 levels were significantly lower in both NAFLD groups than in controls (P ≤ 0.001) and were positively correlated with ursodeoxycholic acid levels. Conclusions Our data indicate that serum BA levels decrease in early NAFLD and increase during progression to fibrosis. Given that BA levels are increased in cirrhotic adults, we postulate a continuous rise as NAFLD advances. BA may have a value as a noninvasive biomarker in pediatric NAFLD progression.

Objectives: We aimed to determine how loss of response (LOR) to adalimumab (ADA) in juvenile idiopathic arthritis (JIA) may be related to anti-ADA antibodies (AAA).Method: AAA and ADA levels were measured in 23 consecutive patients with JIA responding significantly to treatment with ADA.Results: Six out of 23 (26%) patients developed AAA and had low ADA levels. Five out of six AAA-positive patients experienced LOR. In these patients use of concomitant methotrexate (MTX) was significantly lower.Conclusions: The occurrence of AAA is a frequent event associated with LOR. Monitoring of AAA and serum ADA levels should be considered in JIA patients under ADA therapy.

We study the moduli space M ~ \widetilde {\mathscr {M}} of marked cubic surfaces. By classical work of A. B. Coble, this has a compactification M ~ \widetilde {M} , which is linearly acted upon by the group W ( E 6 ) W(E_6) . M ~ \widetilde {M} is given as the intersection of 30 cubics in P 9 \mathbf {P}^9 . For the morphism M ~ → P ( 1 , 2 , 3 , 4 , 5 ) \widetilde {\mathscr {M}} \to \mathbf {P}(1,2,3,4,5) forgetting the marking, followed by Clebsch’s invariant map, we give explicit formulas, i.e., Clebsch’s invariants are expressed in terms of Coble’s irrational invariants. As an application, we give an affirmative answer to the inverse Galois problem for cubic surfaces over Q \mathbb {Q} .

Background:Clinical presentations of Wilson's disease (WD) in childhood ranges from asymptomatic liver disease to cirrhosis or acute liver failure, whereas neurological and psychiatric symptoms are rare. The basic diagnostic approach includes serum ceruloplasmin and 24-hour urinary copper excretion. Final diagnosis of WD can be established using a diagnostic scoring system based on symptoms, biochemical tests assessing copper metabolism, and molecular analysis of mutations in the ATP7B gene. Pharmacological treatment is life-long and aims at removal of copper excess by chelating agents as D-penicillamine, trientine, or inhibition of intestinal copper absorption with zinc salts. Acute liver failure often requires liver transplantation. This publication aims to provide recommendations for diagnosis, treatment, and follow-up of WD in children.Methods:Questions addressing the diagnosis, treatment, and follow-up of WD in children were formulated by a core group of ESPGHAN members. A systematic literature search on WD using MEDLINE, EMBASE, Cochrane Database from 1990 to 2016 was performed focusing on prospective and retrospective studies in children. Quality of evidence was assessed according to the GRADE system. Expert opinion supported recommendations where the evidence was regarded as weak. The ESPGHAN core group and ESPGHAN Hepatology Committee members voted on each recommendation, using the nominal voting technique.

Paediatric autoimmune liver disease is characterised by inflammatory liver histology, circulating autoantibodies and increased levels of IgG, in the absence of a known etiology. Three conditions have a likely autoimmune pathogenesis: autoimmune hepatitis (AIH), autoimmune sclerosing cholangitis (ASC), and de novo AIH after liver transplantation. Two types of paediatric AIH are recognized according to seropositivity for smooth muscle and/or antinuclear antibody (SMA/ANA, AIH-1) or liver kidney microsomal type 1 and/or anti-liver cytosol type 1 antibodies (anti-LKM-1/anti-LC-1; AIH-2).Pertinent issues addressing the diagnosis, treatment and long term follow up were formulated by a core group of ESPGHAN members. They have commissioned the first authors with execution of this project. Initially, they have performed a systematic literature search on MEDLINE, ResearchGate and Mendeley databases over the last 30 years and produced a document focusing on prospective and retrospective studies in children. The ESPGHAN core group and ESPGHAN Hepatology Committee members voted on each recommendation, using a formal voting technique.

We describe a construction of plane quartics with prescribed Galois operation on the 28 bitangents, in the particular case of a Galois invariant Cayley octad. As an application, we solve the inverse Galois problem for degree two del Pezzo surfaces in the corresponding particular case., Comment: arXiv admin note: substantial text overlap with arXiv:1708.00071

Nonalcoholic steatohepatitis (NASH), a progressive form of nonalcoholic fatty liver disease, is one of the most common hepatic diseases in children. We conducted a randomized controlled clinical trial on children with biopsy-proven NASH based on a combinatorial nutritional approach compared with placebo. Participants were assigned to lifestyle modification plus placebo or lifestyle modification plus a mix containing docosahexaenoic acid, choline, and vitamin E (DHA–CHO–VE). Forty children and adolescents participated in the entire trial. The primary outcome was the improvement of liver hyperechogenicity. Secondary outcomes included alterations of alanine aminotransferase (ALT) and other metabolic parameters. Furthermore, changes of serum bile acids (BA) and plasma fibroblast growth factor 19 (FGF19) levels were evaluated as inverse biomarkers of disease severity. At the end of the study, we observed a significant decrease in severe steatosis in the treatment group (50% to 5%, p = 0.001). Furthermore, although the anthropometric and biochemical measurements in the placebo and DHA–CHO–VE groups were comparable at baseline, at the end of the study ALT and fasting glucose levels improved only in the treatment group. Finally, we found that BA levels were not influenced whereas FGF19 levels were significantly increased by DHA–CHO–VE. The results suggest that a combination of DHA, VE, and CHO could improve steatosis and reduce ALT and glucose levels in children with NASH. However, further studies are needed to assess the impact of a DHA and VE combination on repair of liver damage in paediatric NASH.

Objective: Inborn errors of primary bile acid (BA) synthesis are genetic cholestatic disorders leading to accumulation of atypical BA with deficiency of normal BA. Unless treated with primary BA, chronic liver disease usually progresses to cirrhosis and liver failure before adulthood. We sought to determine the prevalence of 2 common disorders, 3 beta-hydroxy-Delta(5)-C-27-steroid dehydrogenase (3 beta-HSD) and Delta(4)-3-oxosteroid-5 beta-reductase (Delta(4)-3-oxoR) deficiencies and to describe current diagnostic and treatment strategies among different European paediatric hepatology centres.Methods: A total of 52 clinical paediatric centres were approached and 39 centres in 21 countries agreed to participate in the Web-based survey. The survey comprised questions regarding general information, number of cases, diagnostic, and therapeutic management.Results: Seventeen centres located in 11 countries reported patients with inborn errors in primary BA synthesis, 22 centres never had cases diagnosed. In total, we could identify 63 patients; 55 with 3 beta-HSD and 8 with Delta(4)-3-oxoR deficiency in 21 countries. The minimum estimated combined prevalence of these diseases was 1.13 cases per 10 million (0.99 and 0.14 for 3 beta-HSD and Delta(4)-3-oxoR deficiencies, respectively). The surveyed colleagues indicated their main challenges to be the rarity of diseases and the lack of convenient laboratory facilities nearby.Conclusion: We have identified the largest cohort of patients with 3 beta-HSD or Delta(4)-3-oxoR deficiency described so far. These diseases are likely underdiagnosed mainly due to unawareness of their existence and the lack of laboratory facilities.

Background: Previous articles have identified the farnesoid X receptor (FXR) as an integral part in the formation of coated platelets. Coated platelets are preactivated platelets featuring degranulation, increased fibrinogen binding, and increased serine protease activity leading to fibrin generation. Furthermore, phosphatidylserine exposure is increased and integrin α2bβIII is inhibited - leading to a prothrombotic phenotype despite decreased platelet aggregation. We hypothesize that bile acids, as natural ligands of FXR, lead to a change of platelet phenotype and therefore play a pivotal role in the formation of coated platelets, especially in presence of cholestasis. Methods: Based on previous findings, we incubated human washed platelets of healthy adult volunteers with the synthetic FXR ligand GW4064 in various concentrations (0, 10, 20, 50, 100µM) and used flow cytometry to detect a shift in p-selectin expression, PAC-1 binding and annexin-V-binding. Moreover, we used different concentrations (0, 100, 200, 400, 600µM) of three bile acids (ursodeoxycholic acid, UDCA; chenodeoxycholic acid, CDCA; glycochenodeoxycholic acid, GCDCA) to see if natural FXR ligands induce an effect on the platelet phenotype. Results: We observed a dose dependent shift in annexin-V-binding when treating washed platelets with GW4064 as well as CDCA and GCDCA. Similarly, GW4064 led to increased p-selectin expression while increased PAC-1-binding was only detected at the highest concentration. In contrast, CDCA and GCDCA showed merely slight changes in p-selectin expression whereas PAC-1-binding seemed to be unaffected. However, none of these effects were seen when using UDCA. Conclusion: We conclude that pretreatment of washed platelets with CDCA and GCDCA initiate a dose-dependent shift towards a prothrombotic platelet phenotype. Therefore, we assume that increased levels of certain bile acids drive thrombosis in patients with cholestatic liver injury. Furthermore, a recent mouse model study suggested that platelet derived growth factor β (PDGFβ), a component of α-granula, drives liver fibrosis. Hence, in addition to their prothrombotic effects, coated platelets might exacerbate liver fibrosis. Disclosures No relevant conflicts of interest to declare.

For an arbitrary del Pezzo surface S, we compute alpha(S), which is the volume of a certain polytope in the dual of the effective cone of S, using Magma and Polymake. The constant alpha(S) appears in Peyre's conjecture for the leading term in the asymptotic formula for the number of rational points of bounded height on S over number fields.

We test the methods for computing the Picard group of a K3 surface in a situation of high rank. The examples chosen are resolutions of quartics in P 3 having 14 singularities of type A 1. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.

We report on our results concerning the distribution of the geometric Picard ranks of $K3$ surfaces under reduction modulo various primes. In the situation that $\rk \Pic S_{\overline{K}}$ is even, we introduce a quadratic character, called the jump character, such that $\rk \Pic S_{\overline\bbF_{\!\frakp}} > \rk \Pic S_{\overline{K}}$ for all good primes, at which the character evaluates to $(-1)$., Revised version, corresponding to the first half of the original submission. To appear in Research in Number Theory

We present a method to compute the geometric Picard rank of a $K3$ surface over $\bbQ$. Contrary to a widely held belief, we show it is possible to verify Picard rank $1$ using reduction only at a single prime. Our method is based on deformation theory for invertible sheaves.

We construct explicit examples of cubic surfaces over $\bbQ$ such that the 27 lines are acted upon by the index two subgroup of the maximal possible Galois group. This is the simple group of order $25 920$. Our examples are given in pentahedral normal form with rational coefficients. For such cubic surfaces, we study the discriminant and show its relation to the index two subgroup. On the corresponding parameter space, we search for rational points, discuss their asymptotic, and construct an accumulating subvariety.