{"product_id":"degeneration-of-abelian-varieties-in-depth-mathematical","title":"Degeneration of Abelian Varieties - In-depth Mathematical","description":"\u003cp\u003eIn this review of Degeneration of Abelian Varieties the bottom line is clear: this is a specialized, rigorous treatment intended for researchers and advanced graduate students who need a coherent account of degeneration theory and compactifications of moduli spaces. The book stands out for connecting the theory of degenerations with concrete applications to compactifications and diophantine problems, and this review highlights that connection as the single biggest reason to consult the volume. Readers seeking an approachable introduction will find the material demanding, but those needing a modern survey will appreciate its depth.\u003c\/p\u003e\n\u003ch2\u003eKey Features\u003c\/h2\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cstrong\u003eTheory of degenerations:\u003c\/strong\u003e Presents a focused account of how degenerations of abelian varieties are described by maps G - S where the generic fibre is an abelian variety, giving a framework for classification in controlled families.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eCompactification focus:\u003c\/strong\u003e Explains how the study of degenerations informs the construction of compactifications of moduli spaces, making the book useful for researchers working on moduli problems.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eSemi-abelian reduction:\u003c\/strong\u003e Details the stable reduction theorem and why semi-abelian families suffice for many compactification questions, clarifying a central technical simplification.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eApplications to diophantine problems:\u003c\/strong\u003e Connects geometric results to arithmetic applications, showing how compactifications produced from degenerations can play a role in diophantine investigations.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eSurvey style:\u003c\/strong\u003e Functions as a modern survey in mathematics, collecting important results and viewpoints that guide further reading and research.\u003c\/li\u003e\n\u003c\/ul\u003e\n\u003ch2\u003eWho It's For\u003c\/h2\u003e\n\u003cp\u003eThis book is aimed at mathematicians working in algebraic geometry, arithmetic geometry, and anyone specifically studying moduli of abelian varieties or compactification techniques. Advanced graduate students who already have background in scheme theory and abelian varieties will find the exposition valuable for research preparation.\u003c\/p\u003e\n\u003cp\u003eIt is not intended as a first introduction to abelian varieties; those seeking elementary or highly expository treatments should look elsewhere for a gentler, example-driven approach before tackling this survey.\u003c\/p\u003e\n\u003ch2\u003ePros \u0026amp; Cons\u003c\/h2\u003e\n\u003cp\u003e\u003cstrong\u003ePros\u003c\/strong\u003e\u003c\/p\u003e\n\u003cul\u003e\n\u003cli\u003eComprehensive treatment of degenerations linking geometric and arithmetic perspectives.\u003c\/li\u003e\n\u003cli\u003eClear emphasis on compactifications, useful for researchers in moduli theory.\u003c\/li\u003e\n\u003cli\u003eExplains the stable reduction theorem and the role of semi-abelian families in a single place.\u003c\/li\u003e\n\u003c\/ul\u003e\n\u003cp\u003e\u003cstrong\u003eCons\u003c\/strong\u003e\u003c\/p\u003e\n\u003cul\u003e\n\u003cli\u003eDemands significant background in schemes and abelian varieties, so it is not accessible to beginners.\u003c\/li\u003e\n\u003c\/ul\u003e\n\u003ch2\u003eSpecifications\u003c\/h2\u003e\n\u003ctable\u003e\n\u003ctr\u003e\n\u003ctd\u003eTitle\u003c\/td\u003e\n\u003ctd\u003eDegeneration of Abelian Varieties\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eSeries\u003c\/td\u003e\n\u003ctd\u003eErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eScope\u003c\/td\u003e\n\u003ctd\u003eTheory of degenerations and compactifications of moduli spaces\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eKey theorem discussed\u003c\/td\u003e\n\u003ctd\u003eStable reduction theorem and semi-abelian families\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eApplications\u003c\/td\u003e\n\u003ctd\u003eCompactifications and diophantine problems\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eIntended audience\u003c\/td\u003e\n\u003ctd\u003eResearchers and advanced graduate students in algebraic geometry\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003c\/table\u003e\n\u003ch2\u003eOur Verdict\u003c\/h2\u003e\n\u003cp\u003eDegeneration of Abelian Varieties is a valuable, high-level survey for specialists who need a concentrated discussion of degenerations, compactifications, and their arithmetic consequences. Its treatment of the stable reduction theorem and semi-abelian families makes it good value for researchers seeking a single reference that links technical results to moduli and diophantine applications.\u003c\/p\u003e\n\u003ch2\u003eFrequently Asked Questions\u003c\/h2\u003e\n\u003cp\u003e\u003cstrong\u003eIs this book suitable for beginners?\u003c\/strong\u003e\u003cbr\u003eNo. The book assumes familiarity with scheme theory and abelian varieties, so beginners should consult introductory texts first.\u003c\/p\u003e\n\u003cp\u003e\u003cstrong\u003eDoes it cover compactifications in detail?\u003c\/strong\u003e\u003cbr\u003eYes. A major focus is how degenerations contribute to constructing compactifications of moduli spaces of abelian varieties.\u003c\/p\u003e\n\u003cp\u003e\u003cstrong\u003eAre arithmetic applications discussed?\u003c\/strong\u003e\u003cbr\u003eYes. The survey explains how compactifications arising from degenerations have applications to diophantine problems.\u003c\/p\u003e","brand":"Gerd Faltings, Ching-Li Chai","offers":[{"title":"Default Title","offer_id":48680606400731,"sku":"364208088X","price":126.45,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0724\/1043\/1707\/files\/61LamjNcYJL._SL1187.jpg?v=1778712292","url":"https:\/\/gearmusthave.com\/products\/degeneration-of-abelian-varieties-in-depth-mathematical","provider":"GearMustHave","version":"1.0","type":"link"}