introduction to algebraic geometry - dolgachev

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1、Systemsofalgebraicequations1Lecture1.SYSTEMSOFALGEBRAICEQUATIONSThemainobjectsofstudyinalgebraicgeometryaresystemsofalgebraicequationsandtheirsetsofsolutions.Letkbeaeldandk[T;:::;T]=k[T]bethealgebraofpolynomialsinn1nvariablesoverk.Asystemofalgebraicequationsoverkisanexp

2、ressionfF=0gF2SwhereSisasubsetofk[T].WeshalloftenidentifyitwiththesubsetS.nLetKbeaeldextensionofk.AsolutionofSinKisavector(x;:::;x)2Ksuchthat1nforallF2SF(x;:::;x)=0:1nLetSol(S;K)denotethesetofsolutionsofSinK.LettingKvary,wegetdierentsetsofnsolutions,eachasubsetofK.Fore

3、xample,letS=fF(T;T)=0g:12beasystemconsistingofoneequationintwovariables.Then2Sol(S;Q)isasubsetofQanditsstudybelongstonumbertheory.ForexampleoneofthemostbeautifulresultsofthetheoryistheMordellTheorem(untilveryrecentlytheMordellConjecture)whichgivesconditionsfornitenessof

4、thesetSol(S;Q):2Sol(S;R)isasubsetofRstudiedintopologyandanalysis.Itisaunionofanitesetand2analgebraiccurve,orthewholeR,orempty.Sol(S;C)isaRiemannsurfaceoritsdegenerationstudiedincomplexanalysisandtopology.Allthesesetsaredierentincarnationsofthesameobject,ananealgebraic

5、varietyoverkstudiedinalgebraicgeometry.OnecangeneralizethenotionofasolutionofasystemofequationsbyallowingKtobeanycommutativek-algebra.RecallthatthismeansthatKisacommutativeunitaryringequippedwithastructureofvectorspaceoverksothatthemultiplicationlawinKisabilinearmapKK!K

6、.Themapk!Kdenedbysendinga2ktoa
1isanisomorphismfromktoasubeldofKisomorphictoksowecanandwewillidentifykwithasubeldofK.ThesolutionsetsSol(S;K)arerelatedtoeachotherinthefollowingway.Let:K!Lbeahomomorphismofk-algebras,i.eahomomorphismofringswhichisidenticalonk.Wecannnne

7、xtendittothehomomorphismofthedirectproducts:K!L.Thenweobtainforanya=(a;:::;a)2Sol(S;K),1nn(a):=((a);:::;(a))2Sol(S;L):1nThisimmediatelyfollowsfromthedenitionofahomomorphismofk-algebras(checkit!).Letsol(S;):Sol(S;K)!Sol(S;L)12Lecture1bethecorrespondingmapofthesolut

8、ionsets.Thefollowingpropertiesareimmediate:(i)sol(S;id)=id;whereiddenotestheidentitymapofasetA;KASol(S;