Feynman motives of banana graphs(7)

We consider the infinite family of Feynman graphs known as the "banana graphs" and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern-Schwartz-MacPherson classes, using the c

BANANAMOTIVES5

thoughweseeinLemma1.2belowthatitiswellde nedalsoonthegeneralpointofΣn,itslocusofindeterminaciesbeingonlythesingularitysubschemeofΣn.

LetG(C)denotetheclosureofthegraphofC.ThenG(C)isasubvarietyofPn 1×Pn 1withprojections

(1.17)G(C)????π2 π1 ?? ? CPn 1______Pn 1ThestandardCremonatransformationofPn 1isthemap 1(1.15)C:(t1:···:tn)→.tnThisisaprioride nedawayfromthealgebraicsimplexofcoordinateaxes n 1ti=0} Pn 1,(1.16)Σn={(t1:···:tn)∈P|i

ingcoordinates(s1:···:sn)forthetargetPn 1,thegraphG(C)hasequations

(1.18)t1s1=t2s2=···=tnsn.

Inparticular,thisdescribesG(C)asacompleteintersectionofn 1hypersurfacesinPn 1×Pn 1withequationstisi=tnsn,fori=1,...,n 1.

Proof.Theequations(1.18)clearlycutoutG(C)overtheopensetU Pnwhereallt-coordinatesarenonzero.Sinceeverycomponentofaschemede nedbyn 1equationshascodimension≤n 1,itsu cestoshowthatequations(1.18)de neasetofcodimension>n 1overthecomplementofU.Nowassumethatatleastoneofthet-coordinatesequal0.Withoutlossofgenerality,supposetn=0.Intersectingwiththelocusde nedby(1.18)determinesthesetwithequations

t1s1=···=tn 1sn 1=tn=0,

whichhascodimensionn>n 1,aspromised.

ItisnothardtoseethatthevarietyG(C)hassingularitiesincodimension3.Itisnonsingularforn=2,3,butsingularforn≥4.

TheopensetUasaboveisthecomplementofthedivisorΣnof(1.16).TheinverseimageofΣninG(C)canbedescribedeasily.Itconsistsofthepoints

((t1:···:tn),(s1:···:sn))

suchthat

{i|ti=0}∪{j|sj=0}={1,...,n}.

Thislocusconsistsof2N 2componentsofdimensionn 2:onecomponentforeachnonemptypropersubsetIof{1,...,n}.ThecomponentcorrespondingtoIisthesetofpointswithti=0fori∈Iandsj=0forj∈I.

Thesituationforn=3iswellrepresentedbythefamouspictureofFigure1.Thethreezero-dimensionalstrataofΣ3areblownupinG(C)asweclimbthediagramfromthelowerlefttothetop.Thepropertransformsoftheonedimensionalstrataareblowndownaswedescendtothelowerright.Thehorizontalrationalmapisanisomorphismbetweenthecomplementsofthetriangles.TheinverseimageofΣ3consistsof23 2=6components,asexpected.

Ofcoursethesituationiscompletelysymmetric:thealgebraicsimplex(1.16)maybe 1 1(Σn)=π2(Σn).embeddedinthetargetPnaswell(withequationisi=0).Onehasπ1

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