Enumerative Algebraic Geometry of Conics AMM awarded paper

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1、EnumerativeAlgebraicGeometryofConicsAndrewBashelor,AmyKsir,andWillTraves1.INTRODUCTION.In1848JakobSteiner,professorofgeometryattheUniver-sityofBerlin,posedthefollowingproblem[19]:Givenfiveconicsintheplane,arethereanyconicsthataretangenttoallfive?Ifso,howmany

2、arethere?Problemsthataskforthenumberofgeometricobjectswithgivenpropertiesareknownasenumera-tiveproblemsinalgebraicgeometry.Thetoolsdevelopedtosolvetheseproblemshavebeenusedinmanyothersituationsandrevealdeepandbeautifulgeometricphenom-ena.Inthisexpositorypa

3、per,wedescribethesolutionstoseveralenumerativeproblemsinvolvingconics,includingSteiner’sproblem.Theresultsandtechniquespresentedherearenotnew;rather,weusetheseproblemstointroduceanddemonstrateseveralofthekeyideasandtoolsofalgebraicgeometry.Theproblemswedis

4、cussarethefollowing:Givenppoints,llines,andcconicsintheplane,howmanyconicsaretherethatcontainthegivenpoints,aretangenttothegivenlines,andaretangenttothegivenconics?Itisnotevenclearapriorithatthesequestionsarewell-posed.Theanswersmaydependonwhichpoints,line

5、s,andconicswearegiven.Nineteenthandtwentiethcenturygeometersstruggledtomakesenseofthesequestions,toshowthatwiththeproperinterpretationtheyadmitcleananswers,andtoputthesubjectofenumerativealgebraicgeometryonafirmmathematicalfoundation.Indeed,Hilbertmadethise

6、ndeavorthesubjectofhisfifteenthchallengeproblem.Enumerativeproblemshavealonghistory:manysuchproblemswereposedbytheancientGreeks.Enumerativegeometryisalsocurrentlyoneofthemostactiveareasofresearchinalgebraicgeometry,mainlyduetoarecentinfluxofideasfromstringth

7、eory.Forinstance,mirrorsymmetryandGromov-Wittentheoryaretwohotmathematicaltopicslinkedtoenumerativegeometry;bothareasdevelopedrapidlybecauseoftheirconnectiontotheoreticalphysics.Whilewewillnotdiscussthesesubjectsexplicitlyinthemainpartofthispaper,manyofthe

8、ideasandtechniquesweintroducearefundamentaltothesemoreadvancedtopics.Inthenextsectionwegivebasicdefinitionsofwhatwemeanbya“conic,”andintroduceamodulispaceofallconics.Foreachconditionimposedontheconicswearecoun