[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117957-en":3,"doc-seo-117957-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},117957,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","MACHINE LEARNING DETECTS TERMINAL SINGULARITIES - Research","Algebraic varieties arise from systems of polynomial equations and may contain singularities; the paper targets terminal singularities within Q-Fano varieties, a central class in algebraic geometry. Classification of Q-Fano varieties is largely unknown, motivating a machine-learning approach to test whether specific 8-dimensional, positively curved toric varieties with Picard rank two are Q-Fano. A neural network classifier achieves 95% accuracy and sketches the dimension-eight Q-Fano landscape, with structure suggested by bounded regions under the quantum period and stratification by Fano index. Building on the ML analysis, the work derives and proves a global combinatorial criterion for terminal singularities and provides new evidence for future conjectures.","arXiv :2310 .20458v1 [math .AG] 31 Oct 2023  \nMACHINE LEARNING DETECTS TERMINAL SINGULARITIES  \nTOM COATES , ALEXANDER M. KASPRZYK , AND SARA VENEZIALE   \nAbstract. Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties:  \npositively curved shapes which have Q-factorial terminal singularities. Q-Fano varieties are of fundamental importance in geometry as they are ‘atomic pieces’ of more complex shapes – the process of breaking a shape into simpler pieces in this sense is called the Minimal Model Programme.  \nDespite their importance, the classification of Q-Fano varieties remains unknown. In this paper we demonstrate that machine learning can be used to understand this classification. We focus on eightdimensional positively-curved algebraic varieties that have toric symmetry and Picard rank two, and develop a neural network classifier that predicts with 95% accuracy whether or not such an algebraic variety is Q-Fano. We use this to give a first sketch of the landscape of Q-Fano varieties in dimension eight.  \nHow the neural network is able to detect Q-Fano varieties with such accuracy remains mysterious, and hints at some deep mathematical theory waiting to be uncovered. Furthermore, when visualised using the quantum period, an invariant that has played an important role in recent theoretical developments, we observe that the classification as revealed by ML appears to fall within a bounded region, and is stratified by the Fano index. This suggests that it may be possible to state and prove conjectures on completeness in the future.  \nInspired by theML analysis, we formulate and prove anew global combinatorial criterion for a positively curved toric variety of Picard rank two to have terminal singularities. Together with the first sketch of the landscape of Q-Fano varieties in higher dimensions, this gives strong new evidence that machine learning can be an essential tool in developing mathematical conjectures and accelerating theoretical discovery.  \n1. Introduction  \nSystems of polynomial equations occur throughout mathematics and science; see e.g. [4, 23, 25, 43] . Solutions of these systems define shapes called algebraic varieties. Depending on the equations involved, algebraic varieties can be smooth (as in Figure 1(a)) or have singularities (as in Figures 1(b) and 1(c)) . In this paper we show that machine learning methods can detect a class of singularities called terminal singularities.  \n(a) 􀁇2 + 􀁈2 = 􀁉 2 + 1 (b) 􀁇2 + 􀁈2 = 􀁉 2 (c) 􀁇2 + 􀁈2 = 􀁉 3  \nFigure 1 . Algebraic varieties in R3 with different defining equations.  \nA key class of algebraic varieties are Fano varieties: positively curved shapes that are basic building blocks in algebraic geometry. Fano varieties are ‘atomic pieces’ of more complex shapes, in the sense of the Minimal Model Programme [11,33,35] . Running the Minimal Model Programme–that is, breaking an algebraic variety 􀀭 into atomic pieces – involves making birational transformations of 􀀭 . These are  \n2020 Mathematics Subject Classification. 14J45 (Primary); 68T07 (Secondary) . Key words and phrases. Fano varieties, terminal singularities, machine learning. 37th Conference on Neural Information Processing Systems (NeurIPS 2023) .  \n2 T. COATES, A. M. KASPRZYK, AND S. VENEZIALE  \nmodifications on subsets with zero volume (and codimension at least one), and can either introduce or remove singularities. The building blocks that emerge from this process are not necessarily smooth: they satisfy a weaker condition called Q-factoriality,1 and can have mild singularities called terminal singularities [48] . Fano varieties that are Q-factorial and have terminal singularities are called Q-Fano varieties.  \nThe classification of Q-Fano varieties is therefore a long-standing problem of great importance [6, 20, 34, 41, 42] – one can think of this as building a Periodic Tab","cbCaim6WUMBSxTzs","https://ap.wps.com/l/cbCaim6WUMBSxTzs","pdf",6002173,1,20,"English","en",105,"# Introduction\n## Algebraic varieties and singularities\n## Fano and Q-Fano varieties\n## Toric varieties and weight matrices\n## The computational challenge of terminal singularities\n## Paper approach and contributions","[{\"question\":\"What problem does the paper address about Q-Fano varieties?\",\"answer\":\"It addresses the lack of knowledge about how to classify Q-Fano varieties, focusing on terminal singularities as the key barrier.\"},{\"question\":\"Which class of varieties is studied for the machine-learning classifier?\",\"answer\":\"The classifier targets 8-dimensional positively curved toric varieties with toric symmetry and Picard rank two.\"},{\"question\":\"How does the paper connect machine learning results to deeper structure?\",\"answer\":\"After classification, visualization using the quantum period shows that the revealed classification lies in a bounded region and is stratified by the Fano index.\"}]","MACHINE LEARNING DETECTS TERMINAL SINGULARITIES - Research | PDF",1785680524,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"machine-learning-detects-terminal-singularities-research","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/machine-learning-detects-terminal-singularities-research/117957/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address about Q-Fano varieties?","Question",{"text":75,"@type":76},"It addresses the lack of knowledge about how to classify Q-Fano varieties, focusing on terminal singularities as the key barrier.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which class of varieties is studied for the machine-learning classifier?",{"text":80,"@type":76},"The classifier targets 8-dimensional positively curved toric varieties with toric symmetry and Picard rank two.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper connect machine learning results to deeper structure?",{"text":84,"@type":76},"After classification, visualization using the quantum period shows that the revealed classification lies in a bounded region and is stratified by the Fano index.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,126,129,133],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":29,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":21,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":21,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":106,"slug":136},19,"General","general"]