A Short Introduction to Projective Geometry

AShortIntroductiontoProjectiveGeometryVectorSpacesoverFiniteFieldsWeareinterestedonlyinvectorspacesoffinitedimension.Toavoidanotationaldifficultythatwillbecomeapparentlater,wewillusethewordrank(oralgebraicdimension)forthedimension(numberofvectorsinanybasis)ofthevectorspace.Theorem:AranknvectorspaceoverGF(q)hasqnvectors.RecalltheimportantrankformulaforsubspacesUandWofthevectorspaceV:rank(U,W)=rank(U)+rank(W)-rank(U∩W)VectorSpacesLetV=V(n,q)denotearanknvectorspaceoverGF(q).Arank1subspaceofVconsistsofallthescalarmultiplesofagivenvector,thusthereareqvectorsinsuchasubspace(includingthezerovector).Bytherankformula,thejoinofanytwodistinctrank1subspaceshasrank2,sincetheycanonlyintersectinthezerovectorwhichasasubspacehasrank0.Ifweexaminetwodistinctrank2subspaces,UandW,wenoticethatthereareseveralpossibilitiesfortherankoftheirjoin.Ifn=3thenrank(U,W)=3andtherankformulasaysthattheymustintersectinarank1subspace.Ifn3,thentherearetwopossibilities,eitherthejoinhasrank4(whentheirintersectionisjustthezerovector)orrank3(iftheyintersectinarank1subspace.)Theorem:ThevectorspaceV(n+1,q)has(qn+1-1)/(q-1)=qn+qn-1+...+q+1rank1subspaces.ProjectiveGeometriesAprojectivegeometryisageometricstructureconsistingofvarioustypesofobjects(points,lines,planes,etc.)andtherelationsbetweenthemwhichsatisfiesasetofaxioms.Here,wewillnotdevelopthesubjectaxiomatically(asisdoneinM6221)butwillsettleforanalgebraicconstructionstartingwithavectorspacewhichwillgiveastructurethatsatisfiesthe(unstated)axioms.WewillstartwiththevectorspaceV(n+1,q)andconstructthegeometricstructurePG(n,q)calledtheprojectivegeometryofdimensionnoverGF(q).ProjectiveGeometryTheworddimensionisusedhereintheclassicalgeometricsenseinwhichlineshave1dimension,planeshave2dimensions,etc.Thisuseofthetermisdifferentfrom(butrelatedto)thealgebraicdimensionofvectorspaces(rank).Sinceinthistreatmentbothgeometriesandvectorspacesappeartogether,itisinevitablethatconfusionwillariseunlessoneisverycareful.Weshallalwaysusethetermdimensioninitsgeometricsense,sometimesusingprojectivedimension(orgeometricdimension)foradditionalemphasis.DefinitionGiventhevectorspaceV(n+1,q),wedefinePG(n,q)asfollows:TheobjectsofPG(n,q)consistof:points,whicharetherank1subspacesofV(n+1,q).lines,whicharetherank2subspacesofV(n+1,q).planes,whicharetherank3subspacesofV(n+1,q)....i-spaces,whicharetheranki+1subspacesofV(n+1,q)....hyperplanes,whicharetheranknsubspacesofV(n+1,q).IncidenceTherelationshipbetweentheobjectsofPG(n,q)iscalledincidenceandisdefinedbycontainmentofthecorrespondingsubspaces.Theincidencerelationismeanttobesymmetric,sowesaythatapointisincidentwithaline(thepointisontheline)orthatalineisincidentwithapoint(thelinepassesthroughthepoint)iftherank1subspaceiscontainedintherank2subspace.Ingeometricnotation,todenotethatapointPisincidentwithalinel,wewouldwritePIlorlIP.ProjectiveGeometryWecannowrephrasestatementsaboutvectorspacesintermsofthegeometricobjectsoftheprojectivegeometry.Forinstance,thestatementmadeearlierabouttwodistinctrank1subspacesbecomestwodistinctpointsdetermineauniqueline.Thestatementsaboutrank2subspacesbecome,inPG(2,q)everytwodistinctlinesmeetatauniquepoint,whileinhigherdimensionalprojectivespacestwodistinctlineswhichmeet,lieinauniqueplaneandiftheydonotmeet(areskew),lieinaunique3-space(solid).ExampleLetourfieldbeGF(4)whoseelementsare0,1,aanda2.Recallthatthisisafieldofcharacteristic2,so1+1=0andthata2=a+1=1/a.Withrespecttothestandardbasis,thevectorsofV(3,4)(43=64intotal)arerepresentedby3-tuplesoverGF(4),suchas:(0,1,0),(a,0,0),(a,a2,1),and(a2,1,a).Now,arank1subspaceofthisvectorspacecontaininganon-zerovectorconsistsofthezerovectorandthreenon-zerovectors.Forinstance,(a,a2,1)={(0,0,0),(a,a2,1),(a2,1,a),(1,a,a2)}.Arank2subspaceconsistsofalllinearcombinationsoftwovectorswhicharenotinthesamerank1subspace(i.e.,arelinearlyindependent).Thus,therewillbe16vectorsinarank2subspace,thezerovectorand15non-zerovectors.Ifavectorisinthissubspace,thenallofitsscalarmultiplesareaswell,sothe15non-zerovectorsaredividedupinto5setsofsize3,andthereare5rank1subspacescontainedinarank2subspace.ExampleContinuedForinstance,therank2subspacecontaining(0,1,0)and(a,0,0)consistsofthevectorsA(0,1,0)+B(a,0,0)=(Ba,A,0)asAandBrunthroughGF(4).Wegetthefollowingvectors:(0,0,0)(a,0,0),(a2,0,0),(1,0,0)(0,1,0),(a,1,0),(a2,1,0),(1,1,0)(0,a,0),(a,a,0),(a2,a,0),(1,a,0)(0,a2,0),(a,a2,0),(a2,a2,0),(1,a2,0)Andwhenwereorganizethislistbygroupingthescalarmultiplestogetherweget:(0,0,0),(1,0,0),(0,1,0),(1,1,0),(1,a,0),(1,a2,0)PG(2,4)TheprojectivegeometryPG(2,4)thenconsistsof21points(rank1subspaces)and21lines(rank2subspaces).Eachlinecontains5pointsandeachpointiscontainedin5lines.Allthepointsandlinesarecontainedin1plane,sowecallthisgeometryaprojectiveplaneoforder4.Notethatinthiscasethehyperplanesofthegeometryarelines.PG(2,q)Thesameargumentscanbegeneralizedtodifferentfields,sowegetthefollowingcounts:1)ThenumberofpointsofPG(2,q)isq2+q+1.2)ThenumberoflinesofPG(2,q)isq2+q+1.3)ThenumberofpointsonalineinPG(2,q)isq+1.4)ThenumberoflinesthroughapointinPG(2,q)isq+1.F