Basic Algebraic Geometry 2 2nd ed.

Books2and3correspondtoChap.V-IXofthefirstedition.Theystudyschemesandcomplexmanifolds,twonotionsthatgeneraliseindifferentdirectionsthevarietiesinprojectivespacestudiedinBook1.Introducingthemleadsalsotonewresultsinthetheoryofprojectivevarieties.Forexample,itiswithintheframeworkofthetheoryofschemesandabstractvarietiesthatwefindthenaturalproofoftheadjunctionformulaforthegenusofacurve,whichwehavealreadystatedandappliedinChap.IV,2.3.Thetheoryofcomplexanalyticmanifoldsleadstothestudyofthetopologyofprojectivevarietiesoverthefieldofcomplexnumbers.Forsomequestionsitisonlyherethatthenaturalandhistoricallogicofthesubjectcanbereasserted;forexample,differentialformswereconstructedinordertobeintegrated,aprocesswhichonlymakessenseforvarietiesoverthe(maior)complexfields.ChangesfromtheFirstEdition