Optimize the plant microbiota to increase plant growth and health

The "Groupement d'Intérêt Scientifique Biotechnologies Vertes" (GIS BV) organized on November 13th, 2018 in Paris, a scientific workshop on "Metagenomics for agro-ecosystems management and plant breeding". Thirty-four scientists, including eight from the private sector attended the workshop. General discussion was organized around the presentations related to plant, seeds and soil microbiota, and data treatment to reconstruct interaction networks. This article gathers the current French research strengths, relative to the international context and highlights the research priorities between the public and the private sectors, using plant genetics and plant-microbiota interactions for the benefit of future agricultures.

Socio-economic context, scientific challenges and opportunities

Plants live in association with a wide diverse and complex assembly of viruses and microorganisms including bacteria, archaea, oomycetes, fungi and protists [START_REF] Leach | Communication in the Phytobiome[END_REF][START_REF] Schoelz | The Role of Viruses in the Phytobiome[END_REF]. These microbial assemblages, collectively referred to as the plant microbiota, impact plant fitness through the modification of a number of traits including: biomass production [START_REF] Sugiyama | Relationships between Arabidopsis genotype-specific biomass accumulation and associated soil microbial communities[END_REF], acquisition of nutrients [START_REF] Castrillo | Root microbiota drive direct integration of phosphate stress and immunity[END_REF], flowering time [START_REF] Panke-Buisse | Selection on soil microbiomes reveals reproducible impacts on plant function[END_REF] or resistance to number of abiotic [START_REF] Lau | Rapid responses of soil microorganisms improve plant fitness in novel environments[END_REF] and biotic stresses [START_REF] Mendes | Deciphering the rhizosphere microbiome for disease-suppressive bacteria[END_REF][START_REF] Ritpitakphong | The microbiome of the leaf surface of Arabidopsis protects against a fungal pathogen[END_REF][START_REF] Haney | Rhizosphere-associated Pseudomonas induce systemic resistance to herbivores at the cost of susceptibility to bacterial pathogens[END_REF][START_REF] Kwak | Rhizosphere microbiome structure alters to enable wilt resistance in tomato[END_REF]. Hence, maximizing the plantbeneficial potential of these microbial assemblages could ultimately result in enhancing crop yield and reducing pesticides and fertilizers [START_REF] Toju | Core microbiomes for sustainable agroecosystems[END_REF].

However, management of plant microbiota composition for developing sustainable agroecosystems still remains incredibly challenging. Gaining a basic understanding of the main biological, evolutionary and ecological processes involved in the assembly and dynamics of plant microbial communities should help to deploy microbiota-based strategies to improve plant productivity. Hence, over recent years a number of research groups explored the impact of environmental factors and host genetic variation on the composition and dynamics of plant microbiota [START_REF] Bernardo | Geometagenomics illuminates the impact of agriculture on the distribution and prevalence of plant viruses at the ecosystem scale[END_REF][START_REF] Bulgarelli | Revealing structure and assembly cues for Arabidopsis root-inhabiting bacterial microbiota[END_REF][START_REF] Lundberg | Defining the core Arabidopsis thaliana root microbiome[END_REF][START_REF] Peiffer | Diversity and heritability of the maize rhizosphere microbiome under field conditions[END_REF][START_REF] Horton | Genome-wide association study of Arabidopsis thaliana leaf microbial community[END_REF][START_REF] Edwards | Structure, variation, and assembly of the root-associated microbiomes of rice[END_REF][START_REF] Carrión | Linking rhizosphere microbiome composition of wild and domesticated Phaseolus vulgaris to genotypic and root phenotypic traits[END_REF][START_REF] Fitzpatrick | Assembly and ecological function of the root microbiome across angiosperm plant species[END_REF]. Overall, these studies acknowledged an important influence of the environment on plant microbiota composition and a restricted but often significant effect of the host genotype. Gaining basic knowledge of processes involved in plant microbiota composition represents also an economic opportunity at the worldwide level for deploying biostimulant-or biocontrol-based solutions. A growing number of startups (e.g. AgBiome, Aphea.bio, Biome Makers, Concentric, Gingko Bioworks, Indigo, Pivot Bio, Trace Genomics), agrochemical (e.g. Basf, Bayer, Corteva AgriScience) and seed compagnies (e.g. Limagrain, KWS) as well as institutional platforms (e.g. Cirad-MetaHealth) are embarking on the plant microbiota adventure.

Current research strategies

Assessing the composition of plant microbiota through metagenomics-based approaches

The taxonomic structure of microorganisms communities is nowadays frequently estimated using culture-independent DNA high throughput sequencing (HTS). While most microorganisms cannot be isolated and cultivated under standard laboratory conditions, HTS technologies, including metabarcoding and shotgun metagenomics approaches, are routinely employed for assessing the taxonomic and functional profiles of microbiota. Metabarcoding relies on amplification and sequencing of a portion of gene/intergenic regions that serve as barcodes for species identification. These markers, which must be ubiquitous, are ideally composed by conserved and highly variable regions. In addition, these markers should be in single copies in all target genomes, which is usually not the case, meaning that exact quantification is still out of reach. Because of their sequence polymorphisms and ubiquities within prokaryotes and eukaryotes, the hypervariable regions of the 16S/18S rRNA genes and internal transcribed spacer (ITS) are widely employed, respectively. The success of these microbial markers is also largely due to comprehensive international publicly available databases containing many sequences of specimen [START_REF] Quast | The SILVA ribosomal RNA gene database project: improved data processing and web-based tools[END_REF][START_REF] Berney | UniEuk: Time to Speak a Common Language in Protistology![END_REF][START_REF] Abarenkov | The UNITE database for molecular identification of fungi--recent updates and future perspectives[END_REF][START_REF] Cole | The Ribosomal Database Project: improved alignments and new tools for rRNA analysis[END_REF]. Among the many obstacles for exact quantification are the so-called "compositionality issue", as sequencing only provides access to relative abundances [START_REF] Gloor | Microbiome Datasets Are Compositional: And Is Not Optional[END_REF], amplification bias, where some sequences of the markers are amplified more than others during the first step [START_REF] Pinto | PCR Biases Distort Bacterial and Archaeal Community Structure in Pyrosequencing Datasets[END_REF] and extraction bias. These obstacles can be mitigated to some extent by spiking [START_REF] Smets | A method for simultaneous measurement of soil bacterial abundances and community composition via 16S rRNA gene sequencing[END_REF], total DNA quantification [START_REF] Vandeputte | Quantitative microbiome profiling links gut community variation to microbial load[END_REF] and positive controls. The final limitation of metabarcoding is the limited phylogenetic resolution of the marker: 500 base pairs are enough to identify organisms at the genus level, but not always at the species and rarely at the strain levels. It only gives access to Operational Taxonomic Units (OTU), Amplicon Sequence Variants (ASV) or oligotypes, not directly organisms [START_REF] Eren | Minimum entropy decomposition: unsupervised oligotyping for sensitive partitioning of high-throughput marker gene sequences[END_REF][START_REF] Callahan | DADA2: High-resolution sample inference from Illumina amplicon data[END_REF][START_REF] Mahé | Swarm: robust and fast clustering method for ampliconbased studies[END_REF]. Efforts are underway to define alternative universal markers (e.g. rpoB [START_REF] Vos | A comparison of rpoB and 16S rRNA as markers in pyrosequencing studies of bacterial diversity[END_REF] and gyrB [START_REF] Barret | Emergence Shapes the Structure of the Seed Microbiota[END_REF] for eubacteria, GH63 [START_REF] Pérez-Izquierdo | A new promising phylogenetic marker to study the diversity of fungal communities: The Glycoside Hydrolase 63 gene[END_REF] and RPB2 [START_REF] Stockinger | The largest subunit of RNA polymerase II as a new marker gene to study assemblages of arbuscular mycorrhizal fungi in the field[END_REF] for dikarya) and non-universal markers (e.g. available only on some branches of the bacterial tree) with better resolutions but those efforts are hampered by the limited content of corresponding taxonomic databases. With all those limitations in mind, there is a link, however imperfect, between taxa abundance and number of sequences and metabarcoding remains a fast, cheap and relatively effective way of assessing the taxonomic composition of plant microbiota.

By contrast with bacteria and fungi, viruses do not even all encode their genomes with the same classes of nucleic acids let-alone have genes or other fragments of sequences universally found across all genomes. Therefore, virus metagenomic studies have generally relied on methodologies that firstly enrich for virus-derived nucleic acids in a sample, and then amplify and/or directly sequence these in a sequence independent manner. Consequently, plant viral metagenomic approaches have targeted five main classes of nucleic-acids: (i) total RNA or DNA; (ii) virionassociated nucleic acids (VANA) purified from viral particles; (iii) double-stranded RNAs (dsRNA); (iv) virus-derived small interfering RNAs (siRNAs) and (v) opportunistic mining of publicly accessible plant transcriptomics databases (reviewed in [START_REF] Roossinck | Plant Virus Metagenomics: Advances in Virus Discovery[END_REF]). Interestingly, virus metagenomic studies have enabled the direct testing of hypotheses relating to the impacts of host diversity, host spatial variations and environmental conditions on plant virus diversity and prevalence [START_REF] Bernardo | Geometagenomics illuminates the impact of agriculture on the distribution and prevalence of plant viruses at the ecosystem scale[END_REF][START_REF] Claverie | From Spatial Metagenomics to Molecular Characterization of Plant Viruses: A Geminivirus Case Study[END_REF][START_REF] Susi | Diverse and variable virus communities in wild plant populations revealed by metagenomic tools[END_REF].

The functional content of microbial assemblages could be either indirectly predicted according to its taxonomic profiles (e.g. PICRUSt [START_REF] Langille | Predictive functional profiling of microbial communities using 16S rRNA marker gene sequences[END_REF]; Tax4fun [START_REF] Aßhauer | Tax4Fun: predicting functional profiles from metagenomic 16S rRNA data[END_REF]; FunGuild [START_REF] Nguyen | FUNGuild: An open annotation tool for parsing fungal community datasets by ecological guild[END_REF]) or directly estimated via random shotgun DNA sequencing. Although this latter approach, coined metagenomics, produces datasets of higher complexity in comparison to metabarcoding, it offers a more robust representation of the functional profiles of a microbiota. In addition to sequence cleaning, the typical pipeline uses meta-assembly to reconstruct contigs, binning to find clusters of contigs coming from the same organism and reconstruct metagenomic species (MGS) or pangenomes (MSP) or gene prediction and clustering to reconstruct a catalog of genes present in the ecosystem. The final step involves mapping sequences to the catalog of genes or MGS to quantify the abundance of each gene / MGS in the ecosystem. Unlike well studied ecosystems such as the human gut, there is no pre-computed gene catalog for plant microbiota. Metagenomics forgoes the amplification step and therefore is not affected by amplification bias but it suffers from its own afflictions: it is expensive compared to metabarcoding, the lack of targeted amplification makes it harder to separate bacterial DNA from the host plant DNA and the catalog has millions of genes, many of which have no or low-quality annotation. Assembly-independent approaches can be used to extract specific markers or to measure dissimilarity/similarity distance. The potential of metagenomics remains powerful to find functions and metabolic pathways represented in the system.

Assessing the composition of plant microbiota, using either a taxonomic or a functional point of view, is the first step towards correlating them with plant traits of interest (e.g. plant yield, resistance to plant pathogens). These studies can also provide useful information to isolate candidates taxa or genes associated with the trait of interest.

Reinoculation of synthetic communities or microbiota fraction to assess the impact of microbiota on plant fitness

Correlation between plant microbiota structure and host phenotypes provide useful indications for discovering which member(s) of the microbial assemblages influence specific plant traits. Reconstructing synthetic communities (SynCom) that are composed of defined microbial strains is an interesting approach to infer causal relationship between microbiota membership and host phenotype. This experimental strategy was recently employed to increase phosphate concentration in Arabidopsis thaliana shoots [START_REF] Paredes | Design of synthetic bacterial communities for predictable plant phenotypes[END_REF] or for plant protection against plant pathogenic fungi [START_REF] Durán | Microbial Interkingdom Interactions in Roots Promote Arabidopsis Survival[END_REF]. The pre-requisite for performing SynCom reconstruction is the availability of a culture collection that is representative of the microbial diversity [START_REF] Bai | Functional overlap of the Arabidopsis leaf and root microbiota[END_REF]. In most cases, obtaining a good coverage of this diversity is challenging, especially when the sampled habitat possesses a high microbial richness. An alternative approach is therefore to use washes of plant tissues (e.g. roots, leaves, and seeds), rhizosphere soil suspensions, or soil spore extractions as representatives of plant microbiota composition. Such microbiota inocula are then inoculated in soil or sprayed on surface-sterilized plant seeds before monitoring the trait of interest. Such experimental designs highlighted for instance plant protection mediated by leaf or rhizosphere microbiotas against bacterial plant pathogens [START_REF] Kwak | Rhizosphere microbiome structure alters to enable wilt resistance in tomato[END_REF][START_REF] Berg | Nutrient-and Dose-Dependent Microbiome-Mediated Protection against a Plant Pathogen[END_REF]. Similarly, transferring tiny amounts of disease suppressive soil to a conducive soil conferred plant protection to fungal pathogens [START_REF] Siegel-Hertz | Comparative Microbiome Analysis of a Fusarium Wilt Suppressive Soil and a Fusarium Wilt Conducive Soil From the Châteaurenard Region[END_REF].

Scientific and technical bottlenecks

How to predict the microbial traits involved in plant growth and the corresponding plant traits that would be affected?

While the importance of a wide range of microbial processes on soil nutrient cycling was deeply investigated [START_REF] Yao | Community proteogenomics reveals the systemic impact of phosphorus availability on microbial functions in tropical soil[END_REF][START_REF] Fierer | Cross-biome metagenomic analyses of soil microbial communities and their functional attributes[END_REF], the functional potential of the microbial communities associated with plants remains a challenge. The roles of specific functions (e.g. nitrogen fixation, phosphate solubilization, phytohormone synthesis, and control of ethylene levels)

were generally demonstrated independently. Random metagenome and metatranscriptome sequencing approaches provide an interesting starting point for predicting the whole functional diversity. However, many microbial genes families do not have established biological activity. Microbiota effect on plant growth and plant fitness has to be specified by traits amenable to measurement in large environmental variations, but robust investigations still remain rare [START_REF] Li | Legacy of land use history determines reprogramming of plant physiology by soil microbiome[END_REF].

What are the mechanisms for plant recognition of microbial assemblage processions?

The way in which a plant genotype determines the composition and structure of its microbiota is relatively unknown compared with the interactions of human genetic variations with its microbiome [START_REF] Awany | Host and Microbiome Genome-Wide Association Studies: Current State and Challenges[END_REF]. Unfortunately, biological and methodological difficulties limit any mechanism transposition from one system to another [START_REF] Beilsmith | Genome-wide association studies on the phyllosphere microbiome: Embracing complexity in host-microbe interactions[END_REF]. Biological difficulties include reduced host genotype diversity of both cases and interactions of uncontrolled environmental factors with host genotypes and microbes. Technical difficulties lie in the transformation of molecular data into phenotypes and in the different statistical approaches used according to the system. Anyway, genome-wide associations studies (GWAS) include statistical models that compare and quantify how the effects of genotype on phenotype vary across environmental conditions to understand how factors may shape the microbiome [START_REF] Porter | Multivariate simulation framework reveals performance of multi-trait GWAS methods[END_REF]. GWAS of plant-microorganisms interactions initially focused on binary studies involving a plant genotype and a single plant pathogen [START_REF] Bartoli | Genome-Wide Association Studies In Plant Pathosystems: Toward an Ecological Genomics Approach[END_REF]. An analogy to the functioning of the plant's immune system conceptualized how this, by discriminating pathogens from other microorganisms, determines the composition and structure of the microbiota, but this approach remains restricted in scope, although it involves molecular signalling between the host-plant and microorganisms, particularly through microbe-associated molecular patterns (MAMPs) and MAMP-triggered immunity (MTI) [START_REF] Hacquard | Interplay Between Innate Immunity and the Plant Microbiota[END_REF]. On the other side, mapping microbiota traits in natural situations is complex because environmental variations mask relationships between host genes and microbiota traits; therefore, it is preferable to create synthetic communities composed of the most abundant microorganisms found in natural situations and of increasing complexity to then, test how plant genotypes shape the microbiota [START_REF] Beilsmith | Genome-wide association studies on the phyllosphere microbiome: Embracing complexity in host-microbe interactions[END_REF]. The discovery of a genetic basis in host plant for interactions with microbiotas suggests new opportunities to exploit natural genetic variation in plant crops to enhance our understanding of beneficial plant-microbiota interactions and develop agroecological strategies for disease control or/and plant growth and development in agriculture.

Heritability of the plant microbiota.

Although its effect is more limited than environmental fluctuations, the plant genotype is a significant driver of plant microbiota composition. For instance, host genetics significantly affects the abundance of bacterial taxa associated to the maize rhizosphere is [START_REF] Walters | Large-scale replicated field study of maize rhizosphere identifies heritable microbes[END_REF]. This broad-sense heritability (H2) is probably linked to host genes that are either related to plant metabolites [START_REF] Sasse | Feed Your Friends: Do Plant Exudates Shape the Root Microbiome?[END_REF] and/or linked to plant immunity [START_REF] Hacquard | Interplay Between Innate Immunity and the Plant Microbiota[END_REF][START_REF] Carvalhais | Linking Jasmonic Acid Signaling, Root Exudates, and Rhizosphere Microbiomes[END_REF][START_REF] Lebeis | Salicylic acid modulates colonization of the root microbiome by specific bacterial taxa[END_REF]. Whether heritable plantassociated bacterial taxa are transmitted from the environment (i.e. horizontally) or from maternal plant to its progeny (i.e. vertically) remains to be investigated. Despite a probable limited vertical transmission in plants in comparison to horizontal transmission [START_REF] Leff | Plant domestication and the assembly of bacterial and fungal communities associated with strains of the common sunflower, Helianthus annuus[END_REF], a fraction of the plant microbiota could be transmitted in clonal offsprings of Glechoma hederacea [START_REF] Vannier | A microorganisms' journey between plant generations[END_REF] or in seeds of various plant species [START_REF] Barret | Emergence Shapes the Structure of the Seed Microbiota[END_REF][START_REF] Klaedtke | Terroir is a key driver of seed-associated microbial assemblages[END_REF][START_REF] Johnston-Monje | Conservation and Diversity of Seed Associated Endophytes in Zea across Boundaries of Evolution, Ethnography and Ecology[END_REF][START_REF] Rezki | Assembly of seed-associated microbial communities within and across successive plant generations[END_REF].

Studying the relative impact of vertical versus horizontal transmission could have important implications for the design of agronomical practices. Vertical transmission might be considered for the selection of desirable microorganisms through plant breeding, and horizontal transmission be integrated into agro-ecological approaches (e.g. selection of key production areas or agricultural practices) for the benefit of crops.

Prediction of microbial interaction through design of co-occurence networks

Plant microbiota are complex systems with hundreds of actors that interact with each other and are affected by environmental conditions. Networks are a simple way to model those interactions to find groups of taxa that interact preferentially with each other, hubs and keystone (systematically important) species [START_REF] Layeghifard | Disentangling Interactions in the Microbiome: A Network Perspective[END_REF]. Six keystone taxa were identified among plantassociated taxa [START_REF] Banerjee | Keystone taxa as drivers of microbiome structure and functioning[END_REF]. Unfortunately, interactions are not directly observed and must be reconstructed from the footprints they leave in abundance data. Most network inference methods are based on co-occurrence or co-abundance data and rely on some variant of correlation to reconstruct ecological interactions. Ecological interactions such as mutualism, commensalism, and competition can reasonably be inferred from co-abundance data but others such as amensalism, and syntrophy are almost impossible to recover [START_REF] Weiss | Correlation detection strategies in microbial data sets vary widely in sensitivity and precision[END_REF]. Many methods based on correlation thresholding [START_REF] Friedman | Inferring Correlation Networks from Genomic Survey Data[END_REF] or graphical models [START_REF] Kurtz | Sparse and compositionally robust inference of microbial ecological networks[END_REF] were proposed to infer interactions. However, they do not account for environmental variables so that shared habitats preferences can be mistaken for direct interactions [START_REF] Vacher | Learning Ecological Networks from Next-Generation Sequencing Data[END_REF]. Recent methods control for environmental variables [START_REF] Biswas | Learning Microbial Interaction Networks from Metagenomic Count Data[END_REF] and integrate organisms assessed with different markers (such as fungi and bacteria) into the same network [START_REF] Chiquet | Variational inference for sparse network from count data[END_REF]. It is however important to remember that network reconstruction framed that way is in essence a statistical problem with severe limitations attached. For optimal performance, the number of samples should be roughly similar to the number of taxa in the network: this requires either (i) large sample sizes or (ii) a focus on dominant taxa, excluding de facto rare but potentially interesting taxa from the network. Finally, no matter how sophisticated are the methods, inferred edges may not represent ecological interactions and/or miss genuine ones [START_REF] Freilich | Species co-occurrence networks: Can they reveal trophic and non-trophic interactions in ecological communities?[END_REF]. Whenever feasible, experimental validations (for example, based on microorganisms isolated through culturomics) should be performed to validate the strongest edges.

Progress must be made in the isolation and culture of microorganisms representing the microbial diversity associated with plants and in the evaluation of their environmental impact Despite variability in post-inoculation plant growth promoting effects [START_REF] Verbruggen | Mycorrhizal fungal establishment in agricultural soils: factors determining inoculation success[END_REF], the production and market of microorganisms know a considerable increase and should account for more than $ 10 billion in 2025 [START_REF] Sessitsch | The contribution of plant microbiota to economy growth[END_REF]. Current limitations for efficient production of stable microbial inoculants include limited large-scale cultivability and shelf life of some taxa (e.g. mycorrhizal fungi; [START_REF] Rosikiewicz | Cost-efficient production of in vitro Rhizophagus irregularis[END_REF][START_REF] Lalaymia | Maintenance and preservation of ectomycorrhizal and arbuscular mycorrhizal fungi[END_REF]) as well as extensive data requirements for the registration of new strains [START_REF] Sessitsch | Microbiome Applications from Lab to Field: Facing Complexity[END_REF]. This latter obstacle currently hampers the development of complex inoculants composed of multiple strains, although these complex inoculants showed better efficiency than single strains [START_REF] Alori | Microbial Inoculants for Soil Quality and Plant Health[END_REF].

While the range and sustainability of inoculummediated plant benefits (biomass, yield, and survival) are the main demand for end-users [START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let's Benefit from Past Successes[END_REF], their environmental impact are poorly considered whereas it remains a critical issue. Three levels of environmental impacts were identified for mycorrhizal fungi, but this also stands for other microorganisms [START_REF] Alori | Microbial Inoculants for Soil Quality and Plant Health[END_REF][START_REF] Rodriguez | The role of community and population ecology in applying mycorrhizal fungi for improved food security[END_REF]: First, the alteration of composition and structure of native microbial community. Second, the exchange of genetic material with native community, and third the persistence and/or spread of inoculants, increasing consequently the first two impacts.

Cartography of ongoing international initiatives

Over recent years, a number of international initiatives emerged in the field of plant microbiota. The most widely recognized international action is probably the Phytobiomes Alliance (http://www.phytobiomesalliance.org/Pages/defa ult.aspx), which is an industry-academic collaborative initiative composed of more than 20 sponsors. While studying plant microbiota interactions is a key priority of Phytobiomes Alliance, other aspects of the plant phytobiome, i.e. targeting or not the plant immediate environment including micro-and macroorganisms [START_REF] Leach | Communication in the Phytobiome[END_REF]. A more recent initiative, the Ag Microbiomes Research Coordination Network, was funded by the National Science Foundation (https://agmicrobiomercn.umn.edu/). The goal of Ag Microbiomes RCN is to promote crossdisciplinary collaborations in the field of the Plant and Soil microbiome. Others international actions in plant microbiomes were recently released and included; they are, for instance, the EU-funded project Microbiome Support (https://www.eufic.org/en/collaboration/article/t he-microbiome-saga-what-does-research-need-todo-better), the Australian Microbiome Initiative (https://www.australianmicrobiome.com/) and the UK Plant Microbiome Initiative (https://www.cabi.org/news-andmedia/2017/cabi-and-rothamsted-researchlaunch-uk-plant-microbiome-initiative/). Another initiative is the working group on Plants and Microbiomes that is part of the European Plant Science Organization (EPSO) network. EPSO is an independent academic organization federating more than 220 public research organizations in Europe and beyond. In 2017, the working group published a report defining a strategy for plant microbiome research in Europe. A second working group meeting in 2019 is to define the needs and to provide advice to current and future EU framework programs.

Overview of the French public and private research on the topic

The PhytoMic network was created in 2016. It is supported by the INRA (French National Institute for Agronomical Research) divisions i.e. EA (Environment and Agronomy), EFPA (Forest, Grassland and Freshwater Ecology), and SPE (Plant Health and Environment) as well as the metaprogram MEM (Meta-omics and microbial ecosytems). The network is currently composed of 20 research units interested in plant microbiota. The primary objective of the network is to bring together the skills (e.g. plant genetics, community ecology, microbiology, plant pathology and agronomy) required for a comprehensive understanding of the plant microbiota. The primary objective of the network is to create a unify taskforce to address the scientific and societal challenges emerging from the phytobiome concept and provide innovations for sustainable agriculture. The network relies on a wide range of experts (i.e. ecophysiologists, plant physiologists, breeders, plant pathologists, molecular ecologists, microbiologists, virologists, entomologists and computer scientists) investigating the soil-plantatmosphere continuum and the whole plant system (rhizosphere, endosphere and phyllosphere).

Conclusion: strategic research targets for public-private research, at French and European levels

Over the past few years a growing number of studies highlighted that environmental selection and plant genotype partly drove the structure of the plant microbiota. Identifying at a finer grain resolution the management practices and/or host genes involved in this selection required a high level of replications over space and time along with a wide range of accessions. Public-private partnerships (PPP) should provide opportunities for developing such ambitious experimental on important agronomical crops. For instance, private partner provides access to experimental plots and inbred lines, while public partner provides expertises in genomics/metagenomics. This type of PPP would potentially identify suit of genes/practices that change the structure of plant microbiota. The impact of such changes on host fitness should be latter evaluated. This could be performed in controlled or semi-controlled conditions through emerging public plant phenotyping platforms that can monitor, in a highthroughput manner, number of plant traits such as germination rate, plant architecture or chlorophyll content.

Acting on this idea to develop PPP in plant microbiome basic and applied knowledge, the March 2019 workshop was one major step and must be followed by partnership implementation. This paper is endorsed by the strategic committee of the GIS BV.

The

  PhytoBioM network was created in 2018. It is supported by the LabEx AGRO (coordinated by the University of Montpellier and Agropolis Fondation). The network is currently composed of 14 research units in close interaction with institutional structures for research and training in Southern countries IRD's (Research Institute for Development) international joint laboratories, CIRAD's (Agricultural Research and International Cooperation for Development) platforms in partnership, INRA's international laboratories).

Annex 1. List of French laboratories and implication in research funded projects

	Unit	City	Habitat	Plant	Topic
			Soil
	AGIR	Toulouse	Spermosphere Agricultural Crops	Abiotic stress -Biotic Stress
			Soil	Model species, Agricultural	Mutualism (AMF) -Biotic stress -
	Agroecologie Dijon	Rhizosphere	Crops	Biogeochemical cycles -Biocontrol
			Carposphere
	BFP	Bordeaux	Phyllosphere	Agricultural Crops	Biotic stress (viruses)
			Phyllosphere	Tropical and
			Endosphere	Mediterreanean Agricultural	Biotic stress (Xanthomonas ; Magnaporthe ;
	BGPI	Montpellier	Rhizosphere	Crops	viruses) -Mutualism (AMF) -Biocontrol
	BIOGECO	Bordeaux	Phyllosphere	Forest	Abiotic stress -Biotic stress -
	BIOGER	Grignon	Residuesphere Agricultural Crops	Biotic stress (Zymoseptoria, Leptosphaeria)
				Model species, Agricultural
	Ecobio	Rennes	Rhizosphere	Crops	Mutualism (AMF, Rhizobia)
			Soil	Tropical and Mediterrean
	Eco&Sols	Montpellier	Rhizsophere	Agricultural Crops	Abiotic stress -Biogeochemical cycles
	GDEC	Clermont	Ear	Agricultural Crops	Biotic stress (Fusarium)
	IAM	Nancy	Rhizosphere	Forest	Mutualism (EMF) -Abiotic stress
					Abiotic stress -Biotic stress (Leptosphaeria
	IGEPP	Rennes	Rhizosphere	Agricultural Crops	-Delia; Plasmodiophora; nematodes)
			Rhizosphere		Mutualism-Abiotic stress -Biotic stress -
	IPME	Montpellier	Phyllosphere	Agricultural Crops	Biocontrol
	IPS2	Saclay	Rhizosphere	Model species	Mutualism (Rhizobia)
			Seed
	IRHS	Angers	Spermosphere Agricultural Crops	Biotic stress (Xanthomonas, Alternaria)
		Sophia-
	ISA	Antipolis	Rhizosphere	Agricultural Crops	Biotic stress (Phytophtora)
					Biotic stresses -Biogeochemical cycles -
	LEM	Lyon	Rhizosphere	Agricultural Crops	Biocontrol
			Rhizosphere		Abiotic stress -Biotic stress (Xanthomonas -
	LIPM	Toulouse	Phyllosphere	Model species	Ralstonia)
	LRSV	Toulouse	Rhizosphere	Model species	Biotic stress (Aphanomyces -Phytophtora)
				Mediterranean and Tropical
	LSTM	Montpellier	Rhizosphere	plants	Mutualism (Rhizobia -AMF/EMF)
	SAVE	Bordeaux	Phyllsophère	Agricultural Crops	Biotic stress


B Karadeniz
email: busrakaradeniz@gtu.edu.tr

H Mourtada
email: hussein.mourtada@imj-prg.fr

C Pl Énat

M Tosun
email: mtosun@gsu.edu.tr

THE EMBEDDED NASH PROBLEM OF BIRATIONAL MODELS OF RATIONAL TRIPLE SINGULARITIES

Keywords:

Introduction

Given a variety X defined over an algebraically closed field of characteristic 0, we are often not able to exhibit an explicit resolution of its singularities; on the other hand there are infinitely many resolutions of singularities of X giving extra information which is not intrinsic to the singularity. The need for understanding the information which is common to all the resolutions of singularities of a given space X led Nash (in [START_REF] Nash | Arc structure of singularities[END_REF]) to study the arc space of X. See also [START_REF] De Fernex | The space of arcs of an algebraic variety, Algebraic Geometry[END_REF][START_REF] Plénat | The Nash problem and its solution: a survey[END_REF] for more details. This paper follows this line of thoughts. The difference here is that we are interested in the embedded resolutions of singularities of X ⊂ A n . For this purpose, we replace the arc space X ∞ of X with the jet schemes of X: the arc space X ∞ of X is the space of germs of formal curves drawn on X. The jet schemes are a family of finite dimensional schemes indexed by integers which approximate the infinite dimensional arc space; for m ∈ N, the m-th jet scheme X m of X, can be thought of as the space of arcs in the ambient space A n whose "contact" with X is greater or equal to m + 1; this gives the intuition why these schemes should detect information about embedded resolutions of singularities. The main question considered in this paper is: can we construct an embedded resolution of singularities from the jet schemes of X ⊂ A n ? More precisely, we ask the following much less optimistic question:

( ) Can one construct an embedded resolution of singularities of X ⊂ A n from the irreducible components of the space X Sing m of jets centered at the singular locus of X ⊂ A n ? This question is studied in [START_REF] Mourtada | Jet schemes and minimal generating sequences of divisorial valuations in dimension 2[END_REF][START_REF] Mourtada | Jet schemes of rational double point surface singularities, Valuation Theory in Interaction[END_REF][START_REF] Lejeune-Jalabert | Jet schemes and minimal embedded desingularization of plane branches[END_REF][START_REF] Mourtada | Jet schemes and minimal toric embedded resolutions of rational double point singularities[END_REF]. In [START_REF] Mourtada | Jet schemes and minimal toric embedded resolutions of rational double point singularities[END_REF], the authors proved that the irreducible components of the jet schemes centered at the singular locus of a rational double point surface singularity (known also as "simple singularities" in the literature) give a minimal embedded resolution by a birational toric modification of the ambient space. Equivalently, a certain natural family of the irreducible components of the jet schemes of X centered at the singular point 0 X 0 m is in bijection with the divisorial valuations whose center is a toric divisor on every toric embedded resolution; this bijection is actually a conceptual correspondence since one can associate with any irreducible component of X 0 m a divisorial valuation centered at the origin of A n (see [START_REF] Ein | Contact loci in arc spaces[END_REF]). In general, such a statement is hopeless: indeed, even for an irreducible plane curve singularity (say, for the cusp {y 2 -x 3 = 0} ⊂ A 2 ), the irreducible components of the jet schemes centered at the origin give divisorial valuations which do not appear in the minimal embedded resolution of the curve singularity (in that case, the minimal embedded resolution makes sense and is unique).

The answer to ( ) is no in general. Indeed, consider the three-dimensional variety defined by

X = {x 2 + y 2 + z 2 + w 5 = 0} ⊂ A 4 .
It has an isolated singularity at the origin 0. On the one hand, by a direct computation, we see that the jet schemes X 0 m centered at 0 are irreducible for every m ≥ 1. On the other hand, we have two exceptional (irreducible) divisors that appear on every embedded resolution of the singularity (at least those which correspond to the two essential divisors appearing in the abstract resolution of the origin 0 ) of X; these are the divisors associated with the monomial valuations on k[x, y, z, w] defined by the vectors (1, 1, 1, 1) and (2, 2, 2, 1). The valuation associated with the vector (2, 2, 2, 1) does not correspond to any of the schemes X 0 m with m ≥ 1. Note that this example is one of the counterexamples to the Nash problem given in [START_REF] Johnson | Arc spaces of cA-type singularities[END_REF]; note also that the Nash correspondence is bijective in dimension 2 [START_REF] De Fernex | Terminal valuations and the Nash problem[END_REF][START_REF] Fernandez De Bobadilla | The Nash problem for surfaces[END_REF] but there are many counter-examples in higher dimension ( [START_REF] Ishii | The Nash problem on arc families of singularities[END_REF][START_REF] De Fernex | Three-dimensional counter-examples to the Nash problem[END_REF]). This suggests that a reasonable frame to study the question ( ) is the surface singularities.

In this paper we study the question ( ) for a family of hypersurface singularities whose normalizations are rational triple point singularities (RTP-singularities, for short). These hypersurfaces are classified in [START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF] and are called the non-isolated forms of RTP-singularities. We prove that, for this class of singularities, the answer to ( ) is positive. When X is of that type, we determine again a natural family of irreducible components of X Sing m , m ≥ 1 whose associated divisorial valuations are monomial, hence defined by some vectors in N 3 . For all of the nonisolated forms of RTP-singularities except when X is of type B k-1,2l-1 , we show that these vectors give a regular subdivision Σ of the dual Newton fan of X and hence a nonsingular toric variety Z Σ ; since our singularities are Newton non-degenerate [START_REF] Varchenko | Zeta-function of monodromy and Newton's diagram[END_REF][START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF][START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF], this gives a birational toric morphism Z Σ -→ A 3 which is an embedded resolution of X ⊂ A 3 ; the irreducible components of the exceptional divisor correspond to the natural set of irreducible components of X Sing m . When X is of type B k-1,2l-1 , we again build a toric embedded resolution from the irreducible components of the jet schemes which does not factor through the toric map associated with the dual Newton fan (such resolutions of non-degenerate singularities also appear when one considers an embedded resolution in family; work in progress of Leyton-Alvarez, Mourtada and Spivakovsky). This again shows mysteriously that the jet schemes tell us something about the "minimality" of the embedded resolution, as in the case of rational double point singularities.

The paper is organized as follows: Section 2 present a reminder on RTP-singularities. Section 3 is devoted to jet schemes and how one can associate a divisorial valuation with a component of the jet schemes; it also contains a summary of the approach to the embedded resolutions which will be constructed in the sequel. Each of the remaining sections is devoted to a class of RTP-singularities (given in the table of contents above): we compute each of the jet schemes and present the results in the jet graph (see Section 3). We then give the toric embedded resolution which comes from the jet schemes. We give the explicit computations with all details for the classes E 6,0 and A k-1,l-1,m-1 . For the other classes, except a subclass of the type B, we proceed similarly, so we present here only the results of the computations. The case B k-1,2l-1 with k ≥ l is treated in detail as its behavior is completely different from the other cases. This is related to the fact that the abstract toric resolution of B k-1,2l-1 which is obtained from a subdivision of the two dimensional cones of the dual Newton fan is not minimal [START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF].

Acknowledgements. We are grateful to the two referees for their comments and corrections which greatly improved the article. The third author would like also to thank the UMI fibonacci (Pisa) and the Lama (University of Savoie-Montblanc (Chambéry)) for their hospitality during the preparation of this work.

RTP-singularities

Let X denote a germ of a surface (X, 0) ⊂ (C N , 0) having a singularity at 0. We say that the singularity of X is rational if H 1 ( X, O X ) = 0 where π : X -→ X is a resolution of X. This definition does not depend on the resolution π. It is well known that the rational singularities of complex surfaces have nice combinatorial properties which can be computed via their resolutions. In [START_REF] Artin | On isolated rational singularities of surfaces[END_REF], the rational singularities of multiplicity 3 are classified by their dual graphs associated with the irreducible components of the minimal resolutions. For short, we call RTP-singularities this class of rational singularities. They are among the surface singularities defined in C 4 and, each of which is defined by three equations given in [START_REF] Tjurina | Absolute isolation of rational singularities, and triple rational points[END_REF]. The classification problem of rational singularities of multiplicity m ≥ 3 is well studied in [START_REF] Lê | Combinatorics of rational surface singularities[END_REF] and [START_REF] Stevens | On the classification of rational surface singularities[END_REF].

In [START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF], the authors obtain the equations of a class of hypersurfaces in C 3 having nonisolated singularities obtained by projecting the equations of RTP-singularities to a generic hyperplane in C 4 and, they call them the non-isolated forms of RTP-singularities since the normalizations of these hypersurfaces in C 3 are exactly the RTP-singularities. They also show that:

Theorem 2.1. The RTP-singularities are non-degenerate with respect to their Newton polyhedron. In particular, they can be resolved by a toric birational map Z -→ C 4 .

In [START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF], the dual graph of the minimal resolution for all RTP-singularities, except those of type B k-1,2l-1 for k ≥ l (see Section 6) are constructed by refining the dual Newton fan of the corresponding non-isolated forms of RTP-singularity (see also [START_REF] Oka | Non-degenerate complete intersection singularity[END_REF][START_REF] Varchenko | Zeta-function of monodromy and Newton's diagram[END_REF]). In the case of the nonisolated form of a rational singularity of type B k-1,2l-1 with k ≥ l, the resolution obtained by the subdivision of the corresponding dual Newton fan is not minimal: consider the vectors R := (2l-2, 2, 2k +1), Q := (2k -l+2, 1, 2k -l+2), P := (l-1, 1, l-1), V := (2k -l, 1, 2k -l+1) and U := (l -1, 1, l) coming out in the subdivision of the dual Newton fan of that singularity:

-2 Q R P Q R -1 -3 -2 -2 -2 -2 V P U -2 -2 -2 -2 -2 -2 -2 -2
Figure 1. Dual Newton fan of a B k-1,2l-1 singularity (with k ≥ l), and its dual (abstract) resolution graph Using [START_REF] Oka | Non-degenerate complete intersection singularity[END_REF], one can compute the self-intersections of the irreducible components of the exceptional divisors corresponding to these vectors; they are given by the number decorating the dual graph given on the right hand side. We omit the genus decorations which are all 0 in this case. The exceptional component corresponding to the vector Q has self-intersection (-1); by Castelnuovo's criterion, (cf. for example [START_REF] Hartshorne | Algebraic Geometry[END_REF], chapter V), that component can be contracted to a nonsingular point without creating singularities. If we continue to contract each (-1)-curve and neighboring components accordingly we obtain a (-3)-curve on the segment [QR] and the dual graph of the minimal resolution of the RTP-singularity of type B k-1,2l-1 , k ≥ l.

Jet schemes

Let k be an algebraically closed field of arbitrary characteristic and X be a k-algebraic variety. For m ∈ N, the jet scheme X m is the scheme representing the functor

F m : k-Schemes → Sets Spec(A) → Hom k Spec A[t]/(t m+1 ) , X
where A is a k-algebra. The closed points of X m are in bijection with the k[t]/(t m+1 ) points of X. By definition, we have X 0 = X. Moreover, for m, p ∈ N with m > p, we have a canonical projection π m,p : X m -→ X p which is induced by the surjection A[t]/(t m+1 ) -→ A[t]/(t p+1 ). These morphisms are affine and verify π m,p • π q,m = π q,p for p < m < q; they define a projective system whose limit is a scheme that we denote X ∞ and which is called the arc space of X. Let us denote the canonical projection π m,0 : X m -→ X 0 by π m and, the canonical morphisms X ∞ -→ X m by Ψ m .

We show here for a surface X = {f = 0} ⊂ k 3 (since the varieties that we are considering are defined this way) that the functor of the jet schemes is representable; this explains also how one determines jet schemes. We have

X = Spec k[x, y, z] (f ) .
For a k-algebra A, an element γ in F m (Spec(A)) corresponds to a k-algebra homomorphism

γ * : k[x, y, z] (f ) -→ A[t] (t m+1 )
.

The data of such a γ is equivalent to the data of

γ * (x) = x(t) = x 0 + x 1 t + • • • + x m t m ∈ A[t]/(t m+1 ), γ * (y) = y(t) = y 0 + y 1 t + • • • + y m t m ∈ A[t]/(t m+1 ), γ * (z) = z(t) = z 0 + z 1 t + • • • + z m t m ∈ A[t]/(t m+1 ); such that f (x(t), y(t), z(t)) = F 0 + F 1 t + • • • + F m t m + • • • = 0 mod (t m+1 ). Here, for i ≥ 0, F i is simply the coefficient of t i in the expanding of f (x(t), y(t), z(t)).
Hence, the data of such a γ is equivalent to the data of x j , y j , z j ∈ A with j = 0, . . . , m such that F i (x 0 , y 0 , z 0 , . . . , x i , y i , z i ) = 0 with i = 0, . . . , m. This is equivalent to determining an A-point of the scheme

X m := Spec K[x i , y i , z i ; i = 0, . . . , m] (F 0 , . . . , F m ) ,
which then represents the functor F m and, is by definition the m-th jet scheme of X.

From now on, we assume that X is a surface in

C 3 defined by {f (x, y, z) = 0} and Y is a subvariety of X. Let m ∈ N We denote by X Y m := π -1 m (Y ).
We consider a special type of the irreducible components of X Y m , m ∈ N where Y is the singular locus of X or Y ⊂ X is a curve contained in a coordinate hyperplane of C 3 . To such Y, we associate a divisorial valuation over C 3 with an irreducible component C m ⊂ X Y m in the following way.

Let ψ a m : C 3 ∞ -→ C 3 m be the truncation morphism associated with the ambient space C 3 , here the exponent "a" stands for ambient map . The morphism ψ a m is a trivial fibration, hence

ψ a m -1 (C m ) is an irreducible cylinder in C 3 ∞ . Let η be the generic point of ψ a m -1 (C m ). By Corollary 2.6 in [5], the map ν Cm : C[x, y, z] -→ N defined by ν Cm (h) = ord t h • η is a divisorial valuation on C 3 . To each irreducible component C m of X Y
m , let us associate a vector, called the weight vector, in the following way:

v(C m ) := (ν Cm (x), ν Cm (y), ν Cm (z)) ∈ N 3 .

Now, we want to characterize the irreducible components of X Y

m that will allow us to construct an embedded resolution of X: For p ∈ N, we consider the following cylinder in the arc space:

Cont p (f ) = {γ ∈ C 3 ∞ ; ord t f • γ = p}. Definition 3.1.
Let X : {f = 0} be a surface in C 3 and let Y be a subvariety of X. (i) The elements of the set:

EC(X) := {Irreducible components C m of X Y m such that ψ a m -1 (C m ) ∩ Cont m+1 f = ∅ and v(C m ) = v(C m-1 ) for any component C m-1 verifying π m,m-1 (C m ) ⊂ C m-1 , m 1} are called the essential components for X.
(ii) the elements of the set of associated valuations:

EV (X) := {ν Cm , C m ∈ EC(X)}, are called embedded-valuations for X.
This means that the elements of EV (X) appear in the embedded toric resolution of X. We will be interested in a subset of EV (X), which gives us an embedded resolution. In the following sections, in order to determine such a subset when X is a non-isolated form of an RTP-singularity, we will study the m-th jet schemes of X, for m ≤ l with l large enough. We will encode the structure of these jet schemes by a levelled graph whose vertices correspond to the irreducible components of X Y m for an integer m; two vertices at the level m and m -1 are joined by an edge if the transition morphism π m,m-1 sends the corresponding components one into the other [START_REF] Mourtada | Jet schemes of complex plane branches and equisingularity[END_REF]. An element of EV (X) corresponding to a component C m ∈ EC(X) is actually a monomial (or toric) valuation (see proposition 2.3 in [START_REF] Mourtada | Jet schemes and minimal toric embedded resolutions of rational double point singularities[END_REF]) and is defined by the vector

v(C m ) = (a, b, c): this means that, for h = {(i,j,k)} a (i,j,k) x i y j z k ∈ C[x, y, z] we have: ν Cm (h) = min {(i,j,k)|a (i,j,k) =0} {ai + bj + ck}.
By subdividing the first quadrant of R 3 using the vectors v(C m ) for some C m ∈ EC(X), we obtain a fan Σ whose support is the first quadrant of R 3 and whose one dimensional cones are generated by these v(C m )'s. Note that one can obtain different fans from a set of vectors in R 3 , depending on the way one relies the vertices and, some of them may not be regular, but here we are interested in finding a regular fan. Hence we have a proper birational map µ Σ : Z Σ -→ C 3 where Z Σ is smooth and the irreducible components of the exceptional divisor of µ Σ correspond to the vectors v(C m ) that we consider. More precisely, the divisorial valuations corresponding to the irreducible components of the exceptional divisor of µ Σ are exactly the ν Cm associated with the components C m that we consider. We will find such a regular fan Σ for a non-isolated form X of an RTP-singularity which is not of the type B k-1,2l-1 (i.e we will construct Σ using the vectors of type v(C m )) that refines the dual Newton fan of X ⊂ C 3 . Thanks to Varchenko's theorem [START_REF] Varchenko | Zeta-function of monodromy and Newton's diagram[END_REF], this gives that µ Σ is an embedded resolution of X ⊂ C 3 . On the other hand, for a B k-1,2l-1 -singularity, we cannot apply Varchenko's theorem because there is no Σ refining the dual Newton fan as described above; nevertheless we build a regular fan Σ satisfying the properties above and, we prove by studying the total transform of our singularity by µ Σ that µ Σ : Z Σ -→ C 3 is an embedded resolution of the B k-1,2l-1 singularity.

RTP-singularities of type E 6,0

The singularity of X ⊂ C 3 defined by the equation:

f (x, y, z) = z 3 + y 3 z + x 2 y 2 = 0
is called E 6,0 -type singularity. Its dual Newton fan is given in Figure 2.

In this section, we compute explicitly the m-th jet schemes (for m ≤ 18) and we determine a subset of EV (X) which gives a regular subdivision of the dual Newton fan as explained in the previous section. We represent the irreducible components as a graph in Figure 3, where we also weight the vertex associated with a component C m by the vector v(C m ) also defined in the previous section. For a component C m which projects by the maps π m,m-1 given in Section 3 on a monomial component (i.e. a component whose associated valuation is monomial) C m-1 , which is not itself monomial; we also weight the associated vertex by the unique non-monomial equation which, together with the hyperplane coordinates C m-1 , defines C m . That helps for the computations of the irreducible components in the process. Here we do not pay much attention to the edges since they are not relevant for the problem at hand. The arrows in Figure 3 correspond to a component C m such that the inverse image of a dense open set in it gives an irreducible component for every n ≥ m. First let us fix some notations:

(1,0,2) (5,4,6) (0,3,2)
Figure 2. Dual Newton fan of E 6,0 singularity

Notation: Let ( * ) f ( m i=0 x i t i , m i=0 y i t i , m i=0 z i t i ) = i=m i=0 F i t i mod(t m+1 )
We know that (e.g. [START_REF] Mourtada | Jet schemes and minimal toric embedded resolutions of rational double point singularities[END_REF]) the m-th jet scheme X m is defined by the ideal

I m = (F 0 , F 1 , . . . , F m ) ⊂ C[x i , y i , z i ; i = 0, . . . , m].
4.1. Jet Schemes of E 6,0 . For m ≥ 1, we will determine the irreducible components of the space of m-jets that projects on the singular locus of X, i.e. the irreducible components of

X Sing m := π -1 m,0 (V (y 0 , z 0 )) ⊂ X m ⊂ Spec(C[x i , y i , z i ; i = 0, . . . , m]) = C 3
m ; here V (I) denotes the variety defined by an ideal I and C 3 m is the m-th jet scheme of the affine three dimensional space C 3 ; we insist here that when considering X Sing m for a given m, the symbol V (I) designates the variety defined by an ideal I in C 3 m . Recall that π m 0 : X m -→ X 0 = X. We also insist on the fact that we consider only the reduced structure of these schemes.

For m = 1, we have X Sing

1 = V (y 0 , z 0 ) ⊂ Spec(C[x i , y i , z i ; i = 0, 1]
) because, if we put y 0 = z 0 = 0 in the equation ( * ) we get F 0 = F 1 = 0 modulo the ideal (y 0 , z 0 ). Hence, X Sing 1 consists of a unique irreducible component, denoted by C 1,1 . The weight vector of C 1,1 is (0, 1, 1).

For m = 2, we have X Sing 2 = π -1 2,1 (C 1,1 ); this uses the fact π 2,0 = π 1,0 •π 2,1 .
A direct computation using the equation ( * ) gives:

F 2 = x 2 0 y 2 1 mod (y 0 , z 0 ). Hence X Sing 2 = V (y 0 , z 0 , x 2 0 y 2 1 ) ⊂ Spec(C[x i , y i , z i ; i = 0, 1, 2]) = C 3 2 .
We deduce that X Sing 2 has two irreducible components C 2,1 := V (y 0 , z 0 , x 0 ) and C 2,2 := V (y 0 , z 0 , y 1 ) both are sent via π 2,1 into C 1,1 ; there weight vectors are respectively (1, 1, 1) and (0, 2, 1). These vectors are represented in Figure 3 at the levels m = 1 and m = 2.

For m = 3, using the fact π 3,0 = π 2,0 • π 3,2 , it is sufficient to study π -1 3,2 (C 2,j ) with j = 1, 2 to understand X Sing 3 .

• To find π -1 3,2 (C 2,1 ), we compute F 3 modulo the ideal (x 0 , y 0 , z 0 ) and we obtain:

F 3 = z 3 1 mod(x 0 , y 0 , z 0 ); Hence we obtain that C 3,1 := π -1 3,2 (C 2,1 ) = V (x 0 , y 0 , z 0 , z 1 ) is irreducible. • Similarly, we obtain that C 3,2 := π -1 3,2 (C 2,2 ) = V (y 0 , y 1 , z 0 , z 1 ) is irreducible. So we have X Sing 3 = C 3,1 ∪ C 3,2
where C 3,1 and C 3,2 are both irreducible and clearly there is no inclusions between them: indeed, C 3,1 is included in V (x 0 ) but C 3,2 is not and, C 3,2 is included in V (y 1 ) but C 3,1 is not. We conclude that C 3,1 and C 3,2 are the irreducible components of X Sing For m = 4, as in the previous case, it is sufficient to consider π -1 4,3 (C 3,j ), with j = 1, 2. As the computations go almost in the same way, we just announce what we obtain:

• To determine π -1 4,3 (C 3,1 ) we compute F 4 modulo the ideal (x 0 , y 0 , z 0 , z 1 ). We have

F 4 = x 2 1 y 2 1 mod (x 0 , y 0 , z 0 , z 1 ). Hence, π -1 4,3 (C 3,1 ) has 2 irreducible components C 4,1 = V (x 0 , y 0 , y 1 , z 0 , z 1 ) and C 4,2 = V (x 0 , y 0 , x 1 , z 0 , z 1 ). • Similarly we have π -1 4,3 (C 3,2 ) = C 4,1 ∪ C 4,3 where C 4,3 = V (y 0 , y 1 , y 2 , z 1 , z 0 ). Then we get X Sing 4 = C 4,1 ∪ C 4,2 ∪ C 4,3
which is a decomposition into irreducible varieties. Using a similar argument as in the case of m = 3, we conclude that there are no mutual inclusions between these components; hence this is the decomposition into irreducible components. The corresponding weight vectors of C 4,1 , C 4,2 and C 4,3 are respectively (1, 2, 2), (2, 1, 2) and (0, 3, 2). Figure 3 encodes also this information.

For m = 5, we have

X Sing 5 = π -1 5,4 (C 4,1 ) ∪ π -1 5,4 (C 4,2 ) ∪ π -1 5,4 (C 4,3 ) ⊂ C 3 5 .
• To determine π -1 5,4 (C 4,1 ), we compute F 5 modulo (x 0 , y 0 , y 1 , z 0 , z 1 ) that we find to be 0. We deduce, that C 5,1 := π -1 5,4 (C 4,1 ) = V (x 0 , y 0 , y 1 , z 0 , z 1 ) is irreducible. A small attention here is needed: The varieties C 4,1 and C 5,1 are not the same; they are defined by the same equations but in different rings; they actually define the same valuation on C 3 (see proposition 2.3 in [START_REF] Mourtada | Jet schemes and minimal toric embedded resolutions of rational double point singularities[END_REF]).

• Computing F 5 modulo the ideal (x 0 , x 1 , y 0 , z 0 , z 1 ), we find

F 5 = y 3 1 z 2 = 0. So π -1 5,4 (C 4,2
) is the union of V (x 0 , x 1 , y 0 , y 1 , z 0 , z 1 ) and C 5,2 := V (x 0 , x 1 , y 0 , z 1 , z 0 , z 2 ).

• As for π -1 5,4 (C 4,1 ), computing F 5 modulo (y 0 , y 1 , y 2 , z 0 , z 1 ) we find zero. This gives that

C 5,3 := π -1 5,4 (C 4,3 ) = V (y 0 , y 1 , z 0 , z 1 , z 2 )is irreducible. Hence we obtain X Sing 5 = C 5,1 ∪ C 5,2 ∪ V (x 0 , x 1 , y 0 , y 1 , z 0 , z 1 ) ∪ C 5,3 .
Since V (x 0 , x 1 , y 0 , y 1 , z 0 , z 1 ) is included in C 5,1 , the decomposition

X Sing 5 = C 5,1 ∪ C 5,2 ∪ C 5,3
is the decomposition into the irreducible components. Moreover, the weight vectors of C 5,j for j = 1, 2, 3 are (1, 2, 2), (2, 1, 3) and (0, 3, 2) respectively.

THE EMBEDDED NASH PROBLEM OF BIRATIONAL MODEL OF RATIONAL TRIPLE SINGULARITIES 9

For m = 6, we have

X Sing 6 = π -1 6,5 (C 5,1 ) ∪ π -1 6,5 (C 5,2 ) ∪ π -1 6,5 (C 5,3 ) ⊂ C 3 6 . • To determine π -1
6,5 (C 5,1 ), we compute F 6 modulo the ideal (x 0 , y 0 , y 1 , z 0 , z 1 ) and we find C 6,1 := π -1 6,5 (C 5,1 ) = V (x 0 , y 0 , y 1 , z 0 , z 1 , z 3 2 + x 2 1 y 2 2 ) ⊂ C 3 6 . Notice that C 6,1 is isomorphic to the product of an affine space and the hypersurface defined by {z 3 2 + x 2 1 y 2 2 = 0}; this hypersurface is a Hirzebruch-Jung singularity which is well known to be an irreducible quasi-ordinary singularity [START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF]; in particular C 6,1 is irreducible. Actually, we will see that C 6,1 will give rise to an irreducible component of X Sing 6 whose weight vector is same as the weight vector associated with C 5,1 , so it is not an essential component (see definition above): the divisorial valuation associated with it is not monomial while a divisorial valuation associated with an essential component is monomial. Before we continue to study on X Sing 6 , let us consider π -1 m,6 (C 6,1 ) for m ≥ 7: For this, we will stratify C 6,1 into its regular locus and its singular locus which are defined respectively by x 1 = z 2 = 0 and y 2 = z 2 = 0. The inverse images π -1 7,6 (C 6,1 ∩{x 1 = z 2 = 0}) and π -1 7,6 (C 6,1 ∩{y 2 = z 2 = 0}) will give the irreducible components of X Sing 7 looking like the irreducible components that we have studied before which are the essential components, so give the new weight vectors. The inverse image of the regular part of C 6,1 with respect to π m,6 , with m ≥ 7 is equal to π -1 m,6 (C 6,1 ∩{z 2 = 0}); this latter is defined in C 3 m ∩{z 2 = 0} by the ideal generated by x 0 , y 0 , y 1 , z 0 , z 1 , z 3 2 + x 2 1 y 2 2 and F j = c j z 3 z j-3 + H j (x 1 , . . . , x j-5 , y 2 , . . . , y j-4 , z 3 , . . . , z j-3 ), c j ∈ C * for 7 ≤ j ≤ m. The functions F j are linear as we can invert c j z 3 = 0. Then the Zariski closure π -1 m,6 (C 6,1 ∩ {z 2 = 0}) is irreducible and, is actually an irreducible component of X Sing m for every m ≤ 7. Note that the weight vector of π -1 m,6 (C 6,1 ∩ {z 2 = 0}) is (1, 2, 2) which is same as the one for C 6,1 and for C 5,1 ; hence they don't give an essential component. They are encoded in Figure 3 by the dashed arrow which starts at the vertex weighted by the vector (1, 2, 2) and the equation z 3 2 + x 2 1 y 2 2 = 0.

• To determine π -1 6,5 (C 5,2 ), we compute F 6 modulo the ideal (x 0 , x 1 , y 0 , z 0 , z 1 , z 2 ) and we find that F 6 = y 2 1 (z 2 y 1 + x 2 2 ). So π -1 6,5 (C 5,2 ) is the union of C 6,2 := V (x 0 , x 1 , y 0 , y 1 , z 0 , z 1 , z 2 ) and C 6,3 := V (x 0 , x 1 , y 0 , z 1 , z 0 , z 2 , z 3 y 1 + x 2

2 ) which are both irreducible. We note that π -1 m,6 (C 6,3 ) is irreducible for every m ≥ 7 and gives rise to an irreducible component of X Sing m for every m ≥ 7. The irreducibility of the inverse image results from the fact that C 6,3 is the product of an affine space and an A 1 -singularity and the jet schemes of such singularity are irreducible [START_REF] Mustata | Jet schemes of locally complete intersection canonical singularities[END_REF][START_REF] Mourtada | Jet schemes of normal toric surfaces[END_REF] (what applies here for A 1 is also true for any rational singularity). The components of π -1 m,6 (C 6,3 ) are not the essential components, they are associated with non-monomial valuations and they have the same weight vector, namely [START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF][START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF]. They are encoded in Figure 3 (to the most right of the graph) by the dashed arrow which starts at the vertex weighted by the vector (2, 1, 3) and the equation x 2 2 + z 3 y 1 = 0.

• To determine π -1 6,5 (C 5,3 ), we compute F 6 modulo the ideal (y 0 , y 1 , y 2 , z 0 , z 1 ) and we find that F 6 = z 3 2 +x 2 0 y 3 3 . Hence C 6,4 := π -1 6,5 (C 5,3 ) = V (y 0 , y 1 , y 2 , z 0 , z 1 , z 3 2 +x 2 0 y 2 3 ) is irreducible. By the same argument as in the case of π -1 m,6 (C 6,1 ), the inverse images π -1 7,6 (C 6,4 ∩ {x 0 = z 2 = 0}) and π -1 7,6 (C 6,4 ∩ {y 3 = z 2 = 0}) will give rise to the irreducible components of X Sing

7

; the Zariski closure π -1 m,6 (C 6,4 ∩ {z 2 = 0}) is irreducible and is actually an irreducible component of X Sing m for every m ≥ 7. This is encoded in Figure 3 by the dashed arrow starting at the vertex weighted by the vector (0, 3, 2) and the equation z 3 2 + x 2 0 y 2 3 .

To summarize, we obtain X Sing

6 = C 6,1 ∪ C 6,2 ∪ C 6,3 ∪ C 6,4
where each C 6,j for j = 1 . . . , 4 is irreducible. Obviously, C 6,2 ⊂ C 6,1 and, using the same argument as in the case of m = 3, we verify that there is no inclusion among the remaining C 6,j 's. Hence we get the irreducible decomposition

X Sing 6 = C 6,1 ∪ C 6,3 ∪ C 6,4
with the respective weight vectors (1, 2, 2), (2, 1, 3) and (0, 3, 2).

For m = 7, by the above discussions, we have a stratification

X Sing 7 = π -1 7,6 (C 6,1 ∩ {x 1 = z 2 = 0}) ∪ π -1 7,6 (C 6,1 ∩ {y 2 = z 2 = 0}) ∪ π -1 7,6 (C 6,1 ∩ {z 2 = 0})∪ π -1 7,6 (C 6,3 ) ∪ π -1 7,6 (C 6,4 ∩ {y 3 = z 2 = 0}) ∪ π -1 m,6 (C 6,4 ∩ {z 2 = 0})
which is the decomposition into irreducible components; indeed, on the one hand using the same argument as for m = 3, there is no inclusions between π -1 7,6 (C 6,3 ) and the other components; on the other hand, the other components are clearly not equal, this means that there are no inclusions between them because they are irreducible and they have the same dimension (actually codimension 7 in C 3 7 ). Note that the codimension is easy to compute since the equations are either hyperplane coordinates in C 3 7 or we consider the closure of a constructible set which is defined by hyperplane coordinates and by linear equations. The weight vectors are respectively (2, 2, 3), [START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF], (1, 2, 2), (2, 1, 3), (0, 4, 3) and (0, 3, 3). Moreover we have π

-1 7,6 (C 6,4 ∩ {x 0 = z 2 = 0}) = π -1 7,6 (C 6,1 ∩ {y 2 = z 2 = 0}
). We should also note that although C 6,2 is not an irreducible component, its inverse image π -1 7,6 (C 6,2 ) which is equal to π -1 7,6 (C 6,1 ∩ {y 2 = z 2 = 0}) gives an irreducible component.

We have gone through the arguments which allow to determine all the irreducible components of X Sing m for m ≤ 18. This is encoded in Figure 3. Note that 18 is the quasi-degree of the weighted homogeneous polynomial defining our singularity. One last important thing is that the axis Y = {x = z = 0} is drawn on our singularity. We determine the essential components of X Y m , m ≥ 0, we find

V (x 0 , z 0 ) ⊂ X 0 ⊂ C 3 0 and V (x 0 , z 0 , z 1 ) ⊂ X 1 ⊂ C 3
1 whose weight vectors are respectively (1, 0, 1), (1, 0, 2).

To conclude, the essential components are the irreducible components of X Z m (where Z is the singular locus of X or Z = Y is the y-axis) whose defining equations are hyperplane coordinates and, their associated valuations are monomial and determined with their weight vectors. Hence we get the graph in figure 3 for the jet schemes. Proposition 4.1. For an E 6,0 -singularity, the monomial valuations associated with the vectors (0, 1, 1), (0, 2, 1), (1, 1, 1), (0, 3, 2), (1, 1, 2), (1, 2, 2), (2, 1, 2), (2, 1, 3), (2, 2, 3), (3, 2, 3), [START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF][START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF], [START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF], [START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Ein | Contact loci in arc spaces[END_REF], [START_REF] Ein | Contact loci in arc spaces[END_REF][START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF][START_REF] De Fernex | The space of arcs of an algebraic variety, Algebraic Geometry[END_REF] belong to EV (X).

THE EMBEDDED NASH PROBLEM OF BIRATIONAL MODEL OF RATIONAL TRIPLE SINGULARITIES 11

(0,1,1)

(1,1,1) (1,1,2) (2,1,2) (0,2,2) (0,3,2) (0,3,2) (1,2,2) (2,2,3) (3,2,3) (3,2,4) (3,3,4) (3,3,4) (4,3,5) (5,4,6) (4,4,6)
(3,5,6) (2,2,6) (0,8,6) (0,4,3) (0,5,4) (1,4,3) (2,4 ,4) (2,5,5) (1,5,4) (1,0,1) (1,0,2) 0 9
(2,1,3) (0,2,1)

z 3 + y 3 z + y 2 x 2 = 0 x 2 + yz = 0 x 2 y 2 = 0 z 3 = 0 z -iy = 0 z + iy = 0 z 3 + x 2 y 2 = 0 (1,2,2) Figure 3
. Jets schemes of E 6,0 4.2. Toric Embedded Resolution of E 6,0 . Now we are ready to announce the main result for the surface X of type E 6,0 -singularity.

Theorem 4.2. There exists a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 such that the components of the exceptional divisor of µ Σ correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplane). Moreover this yields a construction (not canonical) of µ Σ .

Proof. By [START_REF] Varchenko | Zeta-function of monodromy and Newton's diagram[END_REF][START_REF] Oka | Non-degenerate complete intersection singularity[END_REF][START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF] (see also [START_REF] Mourtada | Jet schemes and minimal toric embedded resolutions of rational double point singularities[END_REF] for a summary), an embedded resolution of X ⊂ C can be obtained by constructing a regular subdivision of the dual Newton fan of X ⊂ C 3 . The dual Newton fan Σ for E 6,0 is presented in Figure 2 .

In Figure 4, we give a regular subdivision Σ where the rays (cones of dimension 1) are the lines supported by the vectors given in proposition 4.1. To see that this is a regular subdivision, it is sufficient to show that each cone is regular (means that the determinant of the matrix whose columns are any three vectors generating a cone of Σ equals 1). Moreover the 1-dimensional cones (rays) are in bijective correspondence with the components of the exceptional divisors.

(0,0,

(1,1,2) (0,2,1) (0,1,1) (2,1,2) (1,0,0) (2,1,3) (1,0,2) 1) (5,4,6) (0,3,2) (0,1,0) (1,0,1) (3,2,3) (3,3,5) (2,2,3)
. An embedded resolution of E 6,0

RTP-singularities of type

A k-1,l-1,m-1
The singularity of X ⊂ C 3 defined by the equations:

• k ≥ ≥ m, z 3 + xz 2 -(x + y k + y + y m )y k z + y 2k+ = 0, • k = < m, z 3 + (x -y k )z 2 -(x + y k + y m )y k z + y 2k+m = 0. is called A k-1,l-1,m-1 -type singularity
where k, , m ≥ 1.

Jet Schemes and toric Embedded resolution of

A k-1,l-1,m-1 when k = l ≤ m.
The singular locus is {y = z = 0}. So we compute the jets schemes over {y = z = 0}. The graph representing the irreducible components of the jet schemes of A k-1,l-1,m-1 is on the figure 5 Theorem 5.1. With preceding notation, the monomial valuations associated with the vectors

• (0, 1, 1), (0, 1, 2), . . . (0, 1, k + m) • (s, 1, s), . . . , (s, 1, m + k -s) 1 ≤ s ≤ k -1 • (k, 1, k), . . . , (k, 1, m)
belong to EV (X). Moreover, these vectors give a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 (in the neighborhood of the origin) such that the components of the exceptional divisor of µ Σ correspond to the monomial valuations defined by them; hence they correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplanes).

THE EMBEDDED NASH PROBLEM OF BIRATIONAL MODEL OF RATIONAL TRIPLE SINGULARITIES 13

(0,1,

(0,1,2k-1) (1,1,2k-1) (0,1,2k) (0,1,k+m) (k,1,m) (k,1,k) (k-l,1,k-l) (1,1,k-1) (2,1,k-1) 2) (0,1,1) (0,1,1) (1,1,1) (1,1,2) (2,1,2) (0,1,2) (0,1,3) (0,1,k-1) (0,1,k-1) (0,1,k) (0,2,k+1) (0,2,k+2) (0,2,k+2) (2,2,k+1) (1,2,k+1) (k+1,2,k+1)
• for all 0 ≤ s ≤ k, we have (s, 1, k) ∈ [(0, 1, k), (k, 1, k)] • for all 0 ≤ s ≤ k, we have (s, 1, k + m -s) ∈ [(0, 1, k + m), (k, 1, m)]: k 0 s 1 1 1 m k + m k + m -s = k 0 s 0 1 1 -k k + m -s = 0
• the vectors (α, 1, l + α + 1) for all 0 ≤ α ≤ k are aligned, for each 0 ≤ l ≤ k.

C 2 C 3 C 4 C 1 C 5 (0,1,k+m) (0,1,k) (k,1,m) (k,1,k) (0,1,7) (0,1,3) (3,1,3) (3,1,4)
k k 1 1 1 0 β β + 1 0 = 1.
Decomposition of C 2 : The cone C 2 contains the vectors (1, 1, 1), . . . , (k, 1, k) which are on the skeleton. For 0 ≤ α ≤ k -1 we have:

1 α α + 1 0 1 1 0 α α + 1 = 1.
Decomposition of C 3 : To decompose the cone C 3 , we first add successively an edge between the vectors (k -1, 1, k), (k -2, 1, k -2), (k -3, 1, k), . . . with the last vector being (0, 1, k) if k is odd and with (0, 1, 0) if k is even. Then we obtain that the vectors (α, 1, α), . . . , (α, 1, k) are in the same triangles (see Figure 7). Now let us add those vectors and the vectors on the associated edges successively. • Case 1: for α ≤ β ≤ k -1 we have

(1,1,k) (2,1,k) (s,1,k) (k,1,k) (2,1,2) (1,1,1) (0,1,0) (s+1,1,s+1) (s,1,s) (s-1,1,s-1) (s-1,1,k) (s,1,k) (s+1,1,k) (s,1,s) (0,1,k)
α -1 α α 1 1 1 k β β + 1 = 1 and α + 1 α α 1 1 1 k β β + 1 = 1
• Case 2:

α + 1 α α 1 1 1 α + 1 β β + 1 = 1 and α -1 α α 1 1 1 α -1 β β + 1 = -1
Decomposition of C 4 : The cone C 4 is decomposed by adding successively the edges between the vectors (k, 1, m), (k -1, 1, k), (k -2, 1, m + 2), . . . with the last vector being (1, 1, k) if k is odd and with (1, 1, k + m -1) if k is even. Then let us add successively the vectors and the associated edges (s, 1, α)

for k ≤ α ≤ k + m -s. (k,1,m) (k-1,1,m+1) (0,1,k+m) (k,1,k) (k-1,1,k) (0,1,k) (s+1,1,k) (s,1,k) (s-1,1,k) (s,1,k) (s,1,k+m-s) (s+1,k,k+m-s-1) (s,1,k+m-s) (s-1,1,k+m-s+1)
≤ β ≤ k + m -s, s -1 s s 1 1 1 k β β + 1 = 1 and s + 1 s s 1 1 1 k β β + 1 = 1 or s + 1 s s 1 1 1 k + m -s -1 β β + 1 = 1 and s -1 s s 1 1 1 k + m -s + 1 β β + 1 = 1
Decomposition of C 5 : The cone C 5 contains the vectors (s, 1, k + m -s) for 0 ≤ s ≤ k which are on the skeleton. For 0 ≤ α ≤ k, we have:

0 α α + 1 0 1 1 1 k + m -α k + m -1 -α = 1, k 1 0 1 0 0 m 0 1 = 1.

Jet Schemes and toric Embedded resolution of

A k-1,l-1,m-1 when k ≥ l ≥ m.
The graph representing the irreducible components of the jet schemes of A k-1,l-1,m-1 projecting on the singular locus {y = z = 0} is given by Figure 9 below.

(0,1,2) (0,1,1) (0,1,1) (1,1,1) (1,1,2) (2,1,2) (0,1,2) (0,1,3) (0,1,k-1) (0,1,k-1) (0,1,k) (0,2,k+1) (0,2,k+2) (0,2,k+2) (2,2,k+1) (1,2,k+1) (1,1,k-1) (2,1,k-1) 0
(m,1,k) (m,1,k+1) (0,1,k+l) (m+k,2,m+k) (if m+k odd) (0,1,,k+m-1) (1,1,k+m-1) (m-3,2,k+1) 2k+1 2k+m-2 2k+m-1 2k+m (0, 2, k + 2 + [ m-2 2 ] + [ 1-m 2 ]) ( k+m 2 , 1, k+m (m+1,1,k-1)
)if k + m even ([ k+m 2 , 1, [ k+m 2 )if k + m odd (m, 1, k + -m) ( -1, 2, k + 1) Figure 9. Jets schemes of A k-1,l-1,m-1 Theorem 5.2. Let X ⊂ C 3 be a surface of type A k-1,l-1,m-1 with k ≥ l ≥ m. 2
The monomial valuations associated with the vectors:

• (0, 1, 1), (0, 1, 2), . . . (0, 1, k + l) • (s, 1, s), . . . , (s, 1, l + k -s) 1 ≤ s ≤ m -1 • (m, 1, m), . . . , (m, 1, k + l -m) • (m + r, 1, m + r), . . . , (m + r, 1, k -r) with 1 ≤ r ≤ E( k-m
2 ) belong to EV (X). Moreover, these vectors give a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 (in the neighborhood of the origin) such that the components of the exceptional divisor of µ Σ correspond to the monomial valuations defined by the vectors; hence they correspond to irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplanes).

THE EMBEDDED NASH PROBLEM OF BIRATIONAL MODEL OF RATIONAL TRIPLE SINGULARITIES 17

Proof. As above, we first study the positions of the vectors given in theorem 5.2:

• (m + r, 1, k -r) ∈ [(m + k, 2, m + k), (m, 1, k)]: m + k m + k m 1 2 1 k + r m + k k = r k m 0 1 1 -r m k = r k m 0 1 1 0 m + k k + m = 0 • (α, 1, k + l -α) ∈ [(m, 1, k + l -m), (0, 1, k + l)] for 0 ≤ α ≤ m : m 0 α 1 1 1 k + l -m k + l k + l -α = m 0 α 0 1 0 -m k + l -α = 0
If m+k 2 ∈ Z, then the dual fan can be decomposed in the same way as for the case A k-1,k-1,m-1 . Otherwise, we have to show the subcones containing the vector (m + k, 2, m + k) are regular. In

this case E( k-m 2 ) = k-m-1 2 and (m + E( k-m 2 ), 1, m + E( k-m 2 )) = ( k+m-1 2 , 1, k+m-1 2
). We have : The singularity of X ⊂ C 3 defined by the equations:

k+m-1 2 k+m-1 2 m + k 1 1 2 k+m+1 2 k+m-1 2 m + k = 0 k+m-1 2 m + k 0 1 2 1 k+m-1 2 m + k = 1 and 1 k+m-1 2 m + k 0 1 2 0 k+m-1 2 m + k = 1. (0,1,k) (0,1,k+l) (m,1,k) (m+k,
• m = 2 , z 3 + xz 2 -(y k+1 + y )y k z -xy 2k+1 = 0, • m = 2 -1, z 3 + (x -y -1 )z 2 -y 2k+1 z -xy 2k+1 = 0. is called B k-1,m -type singularity with k ≥ 2 and m ≥ 3.
In the case where m = 2l, the jet schemes and the toric embedded resolution behaves as in the case of A m,k,l ; so, let's just present the jet graph presenting the irreducible components of the jets schemes projecting over the singular locus {y = z = 0} and the axis {x = z = 0} included in X: belong to EV (X). Moreover there exists a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 such that the irreducible components of the exceptional divisor of µ Σ correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplane). This yields a construction of µ Σ (not canonical). (2,0,1)

(1,0, The computations are similar to the case B k-1,2l ; the associated vectors with the jet schemes give a subdivision of the dual Newton fan, thus an embedded resolution, of the singularity. Theorem 6.3. Let X be of type B k-1,2l-1 for l ≤ k. The monomial valuations associated with the vectors

• (0, 1, 1), (0, 1, 2), . . . , (0, 1, k + 1)

• (1, 1, 1), . . . , (1, 1, k + 1) • . . . • (l -1, 1, l -1), . . . , (l -1, 1, k + 1)

• (l, 1, k + 1)

• (0, 2, 2k + 1) . . . (2l -2, 2, 2k + 1) belong to EV (X). Moreover there exists a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 such that the irreducible components of the exceptional divisor of µ Σ THE EMBEDDED NASH PROBLEM OF BIRATIONAL MODEL OF RATIONAL TRIPLE SINGULARITIES 21 correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplane). This yields a construction of µ Σ (not canonical). Remark 6.4. The set of vectors above does not contain the vector Q = (2k -l + 2, 1, 2k -l + 2), thus the decomposition obtained by these vectors will not be a regular decomposition of the dual Newton fan of the singularity.

Proof. Consider the polygons J = [(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 0, 1), (l, 1, k + 1), (2l -2, 2, 2k + 1), (l -1, 1, l -1)] and K = [(1, 0, 0), (1, 0, 1), (l, 1, k + 1), (2l -2, 2, 2k + 1), (l -1, 1, l -1)] in the dual Newton fan of the singularity. In J, the vectors obtained from the jet schemes give a regular subdivision of this polygon (following the computations of B k-1,2l ). As J is a sub-polygon of the fan, the strict transform of X is regular on these charts. In K, we find a subdivision by adding an edge from (1, 0, 0) to (l, 1, k + 1), another edge from (1, 0, 0) to (l -1, 1, s) for l -1 ≤ s ≤ k and another edge from (l, 1, k + 1) to (l -1, 1, k). In this way, we obtain a regular subdivision of K. locus and the intersection of X with the coordinate hyperplane). This yields a construction (not canonical) of µ Σ .

The tree representing the irreducible components of the jets schemes projecting on the singular locus {x = z = 0} and the axe {y = z = 0} included in X is the following:

n = 3k n = 3k -1 n = 3k + 1

3 .

 3 Their associated weight vectors are respectively (1, 1, 2) and (0, 2, 2).

+ 3 + xz 2 -y k z 2 -Figure 5 .

 3225 Figure 5. Jets schemes of A k-1,k-1,m-1

Figure 6 .

 6 Figure 6. Dual Newton fan of A k-1,k-1,m-1 and an embedded resolution for A 2,2,3

Figure 7 .

 7 Figure 7. Decomposition of the cone C 3 and of its two types of subcones

Figure 8 .

 8 Figure 8. Decomposition of the cone C 4 and of its two types of subcones

Figure 10 . 2 6.

 102 Figure 10. Dual Newton fan of A k-1,l-1,m-1 with k ≥ l ≥ m, and a resolution of A 5,4,2

  1, 1, k + 1) (l, 1, k + 1) (l -1, 1, l -1) (l -1, 1, l)

Figure 14 .

 14 Figure 14. Jets schemes of B k-1,2l-1

l > k + 1 l = k + 1 Figure 15 .

 1115 Figure 15. Dual Newton fan of B k-1,2l-1 for l > k + 1 (resp. l = k + 1) and an embedded resolution

Figure 16 .

 16 Figure 16. Dual Newton fan of B k-1,2l-1 with l ≤ k and it is with the vectors of Theorem 6.3

  THE EMBEDDED NASH PROBLEM OF BIRATIONAL MODEL OF RATIONAL TRIPLE SINGULARITIES 3

This work is supported by the projects "PHC Bosphore no.42613UE", "TUBITAK no.118F320" and "Galatasaray University no.16.504.001BAP". The second named author is also partially supported by Project ANR-LISA-17-CE40-0023.

B.KARADENIZ, H. MOURTADA, C.PL ÉNAT, AND M. TOSUN

(1,0,1) (0,1,1) (0,1,1) (0,1,2) (0,1,2) (1,1,1) (2,1,2)

(1,1,2)

if k even (0,1,k) (0,1,k) (0,1,k+1) (0,2,k+1)

(1,2,2k+1) (0,1,1) (0,1,1) (0,1,2) (0,1,2)

(1,0,1) • (0, 1, 1), (0, 1, 2), . . . , (0, 1, k + 1)

• (1, 1, 1), . . . , (1, 1, k + 1 • . . . • (l, 1, l), . . . , (l, 1, k + 1)

• (l + 1, 1, l + 1), . . . , (l + 1, 1, k -1)

• (l + 2, 1, l + 2), . . . , (l + 1, 1, k -2)

• . . .

• (E((l + k)/2), 1, E((l + k)/2)), and (E((l + k)/2), 1, E((l + k)/2) + 1) if k + l is odd.

• (0, 2, 2k + 1) . . . (2l -l, 2, 2k + 1)

belong to EV (X). Moreover, these vectors give a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 (in the neighborhood of the origin) such that the components of the exceptional divisor of µ Σ correspond to the monomial valuations defined by them; hence they correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplanes).

The vectors given in the theorem allows us to decompose the corresponding dual Newton fan into regular subcones and find an embedded resolution of the singularity.

(1,0,1) (k+l,1,k+l) In the case of B k-1,2l-1 , there is an amazing subclass (see Section 2 below) for which the jet schemes give a resolution which is not a subdivision of the dual Newton fan of the singularity. So this case needs to be treated in details. There are two sub-cases to be considered which are the cases k + 1 ≤ l and k ≥ l. Let us first treat the case k + 1 ≤ l: we start by computing the irreducible components of the jet schemes projecting on the singular locus {y = z = 0} and the axe {x = z = 0} included in X. And, by computing the associated vectors we obtain the figure 14: Theorem 6.2. Let X be of type B k-1,2l-1 with k + 1 ≤ l. The monomial valuations associated with the vectors:

• (0, 1, 1), (0, 1, 2), . . . , (0, 1, k + 1)

Since K is not compatible with the dual Newton fan, we cannot use Varchenko's theorem to deduce the smoothness of the strict transform of X in the charts corresponding to the subdivision of K by the toric map. So, we should prove this fact:

• For this, let us first consider the cone [(1, 0, 0), (l -1, 1, s), (l -1, 1, s + 1)] for l -1 ≤ s < k; the monoidal transformation corresponding to it is:

Then the total transform of B k-1,2l-1 is defined by:

The strict transform is smooth and transversal to the exceptional divisors defined by y 1 = 0 and z 1 = 0. • Now let us consider the cone [(1, 0, 0), (l-1, 1, k), (l, 1, k+1)]; the monoidal transformation corresponding to it is:

Then the total transform of B k-1,2l-1 is :

The strict transform is smooth and transversal to the exceptional divisors defined by y 1 = 0 and z 1 = 0. • Finally let us consider the cone [(1, 0, 0), (1, 0, 1), (l, 1, k+1)]; the monoidal transformation corresponding to it is:

Then the total transform of B k-1,2l-1 is :

The strict transform is smooth and transversal to the exceptional divisors defined by y 1 = 0 and z 1 = 0. The singularity of X ⊂ C 3 defined by the equation:

-type singularity where k ≥ 1 and ≥ 2. For k = 3q -1, we obtain the jet graph given in Figure 18 which represents the irreducible components of the jet schemes of C k-1,l+1 projecting on the singular locus {y = z = 0}. • for k = 3q -1 and l = 2p

)p) belong to EV (X). Moreover there exists a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 such that the irreducible components of the exceptional divisor of µ Σ correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplane). This yields a construction (not canonical) of µ Σ .

The embedded resolutions are represented on the figure below. The singularity of X ⊂ C 3 defined by the equation:

is called D k-1 -type singularity with k ≥ 1. The jet graph is given in Figure 20 where the irreducible components of the jet schemes of D k-1 projecting on the singular locus {(y = z = 0} and the axe {x = z = 0} included in X: Theorem 8.1. Let X be a surface singularity of type D k-1 . The monomial valuations associated with the following vectors belong to EV (X). Moreover there exists a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 such that the irreducible components of the exceptional divisor of µ Σ correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplane). This yields a construction (not canonical) of µ Σ .

• (1, 0, 1), (1, 0, 2)

• (0, 1, 1), (0, 1, 2)...(0, 1, k)

• (1, 1, 1), ..., (1, 1, k), (2, 2, 2k + 1)), (1, 1, k + 1)

When k is odd, we should add two more vectors:

These vectors placed in the dual Newton fan give the regular subdivision:

(1,1,1) The singularity of X ⊂ C 3 defined by the equation:

is called E 7,0 -type singularity. The singular locus is {y = z = 0}. Theorem 9.1. Let X be a surface singularity of type E 7,0 . The monomial valuations associated with the vectors: {(0, 1, 1), (0, 2, 1), (0, 1, 2), (0, 1, 3), (1, 1, 1), (1, 1, 2), (1, 2, 2), [START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF][START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF], [START_REF] Sharland | Nonisolated forms of rational triple singularities[END_REF][START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF][START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF], [START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF][START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF], [START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF], [START_REF] Aroca | Torical modification of Newton non-degenerate ideals[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Ein | Contact loci in arc spaces[END_REF], [START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF], [START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF][START_REF] Ein | Contact loci in arc spaces[END_REF], [START_REF] Artin | On isolated rational singularities of surfaces[END_REF][START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF][START_REF] De Fernex | The space of arcs of an algebraic variety, Algebraic Geometry[END_REF], [START_REF] Cobo | Jet schemes of quasi-ordinary surface singularities[END_REF][START_REF] Ein | Contact loci in arc spaces[END_REF][START_REF] De Fernex | Three-dimensional counter-examples to the Nash problem[END_REF], [START_REF] Ein | Contact loci in arc spaces[END_REF][START_REF] De Fernex | The space of arcs of an algebraic variety, Algebraic Geometry[END_REF][START_REF] De Fernex | Terminal valuations and the Nash problem[END_REF]} belong to EV (X). There exists a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 such that the irreducible components of the exceptional divisor of µ Σ correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplane). Moreover this yields a construction (not canonical) of µ Σ .

Following almost the same process as in the case of E 6,0 , we continue until m = 22 to obtain the following jet graph: 10. Jet Schemes and Toric Embedded Resolution of E 0,7

The singularity of X ⊂ C 3 defined by the equation:

is called E 0,7 -type singularity. The singular locus is {y = z = 0}. The jet graph representing the irreducible jet schemes is obtained as: Theorem 10.1. Let X be a surface of type E 0,7 . The monomial valuations associated with the vectors {(0, 1, 1), (

} belong to EV (X). There exists a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 such that the irreducible components of the exceptional divisor of µ Σ correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplane). Moreover this yields a construction (not canonical) of µ Σ .

( The singularity of X ⊂ C 3 defined by the equation:

Theorem 11.1. Let X be a surface singularity of type F k-1 . The monomial valuations associated with the vectors:

• (0, 1, 1), . . . , (0, 1, k)

3 ) and (c, 2, d) = ( 4k+4 3 , 2, 4k+7 3 ) if k = 3m + 2 for m ∈ N * belong to EV (X). Moreover there exists a birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 such that the irreducible components of the exceptional divisor of µ Σ correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplane). Moreover this yields a construction (not canonical) of µ Σ .

The jet graph representing the irreducible components of the jet schemes projecting on the singular locus is given by: Theorem 12.1. Let X be a surface of type H n . The monomial valuations associated with the vectors:

1

• (0, 2, 1), (1, 2, 2), . . . (2k, 2, 2k + 1)

• (0, 3, 2), (1, 3, 3), . . . (3k -1, 3, 3k + 1)

• (1, 0, 1), (1, 0, 2), (2, 0, 1)

• (1, 1, 1), (2, 1, 2), . . . (k, 1, k) belong to EV (X). Moreover there exists a toric birational map µ Σ : Z Σ -→ C 3 which is an embedded resolution of X ⊂ C 3 such that the irreducible components of the exceptional divisor of µ Σ correspond to the irreducible components of the m-th jet schemes of X (centered at the singular