Minkowski sums and Hadamard products of algebraic varieties

0.41 MB·

by  Netanel Friedenberg

Minkowski sums and Hadamard products of
algebraic varieties
NetanelFriedenberg,AlessandroOneto,andRobertL.Williams
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AbstractWestudyMinkowskisumsandHadamardproductsofalgebraicvarieties.
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Specifically we explore when these are varieties and examine their properties in
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termsofthoseoftheoriginalvarieties.ThisprojectwasinspiredbyProblem5on

] Surfacesin[13].
G
A
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h 1 Introduction
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m
Inalgebraicgeometrywehaveseveralconstructionstobuildnewalgebraicvarieties
[ fromgivenones.Examplesofclassical,well-studiedconstructionsarejoins,secant

1 varieties,rationalnormalscrolls,andSegreproducts.Inthesecases,itisveryinter-
v estingtounderstandgeometricproperties,e.g.,thedimensionandthedegree,ofthe
1 varietyconstructedintermsofthoseoftheoriginalvarieties.Inthischapterwefo-
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cusontheMinkowskisumandtheHadamardproductofalgebraicvarieties.These
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NetanelFriedenberg
.
1 YaleUniversity,
0 10HillhouseAve-Ste442
7 POBox208283
1 NewHaven,CT06520-8283,UnitedStates
: e-mail:[email protected]
v
i AlessandroOneto
X INRIASophiaAntipolisMe´diterrane´e,
r 2004RoutedeLucioles,
a 06902SophiaAntipolis,France,
e-mail:[email protected]
RobertL.Williams
TexasA&MUniversity,DepartmentofMathematics
Mailstop3368
CollegeStation,TX77843-3368UnitedStates
e-mail:[email protected]
1
2 NetanelFriedenberg,AlessandroOneto,andRobertL.Williams
areconstructedbyconsideringtheentry-wisesumandmultiplication,respectively,
ofpointsonthevarieties.Duetothenatureoftheseoperations,thereisaremarkable
differencebetweentheaffineandtheprojectivecase.
The entry-wise sum is not well-defined over projective spaces. For this reason,
weconsideronlyMinkowskisumsofaffinevarieties.However,inthecaseofaffine
cones, the Minkowski sum corresponds to the classical join of the corresponding
projectivevarieties.Conversely,wefocusonHadamardproductsofprojectivevari-
etiesand,inparticular,ofvarietiesofmatriceswithfixedrank.Thisisbecausethese
Hadamardproductsparametrizeinterestingproblemsrelatedtoalgebraicstatistics
andquantuminformation.
Ouroriginalmotivatingquestionwasthefollowing.
Question1.1.Which properties do the Minkowski sum and the Hadamard product
havewithrespecttothepropertiesoftheoriginalvarieties?Inparticular,whatare
theirdimensionsanddegrees?
Wenowintroducetheseconstructions.Weworkoveranalgebraicallyclosedfieldk.
Wewilladdextraassumptionsonkwhenneeded.Weusethenotationk×:=k\{0}.
Definition1.2.LetX,Y ⊂An beaffinevarieties.WedefinetheMinkowskisumof
X and Y, denoted X+Y, as the Zariski closure of the image of X×Y under the
entry-wisesummationmap
φ : An×An → An,
+
((a ,...,a ),(b ,...,b ))(cid:55)→(a +b ,...,a +b )
1 n 1 n 1 1 n n
Note that taking the Zariski closure of φ (X×Y) is necessary to construct an
+
algebraicvariety,asexplainedinExample3.1.
As far as we know there is no literature about Minkowski sums of varieties. We
computethedimensionanddegreeofMinkowskisumsofgenericaffinevarieties.
Theorem3.9.Let X,Y ⊂An be varieties. Then, for X andY in general position,
dim(X+Y)=min{dim(X)+dim(Y),n}.
Corollary3.12.Supposekhascharacteristicotherthan2.LetX,Y ⊂An bevari-
eties whose projective closures X,Y ⊂Pn are contained in complementary linear
subspaces; equivalently, X,Y are contained in disjoint affine subspaces which are
notparallel.Thenforgenericα ∈k×,deg(αX+Y)=deg(X)deg(Y).
AcrucialobservationinourcomputationsisthattheMinkowskisumofaffineva-
rietiesdisjointatinfinitycanbedescribedintermsofthejoinoftheirprojectiviza-
tions, see Proposition 3.5 and Remark 3.6. This is a construction inspired by the
combinatorialCayleytrickusedtoconstructMinkowskisumsofpolytopes.
Definition1.3.LetX,Y ⊂Pnbeprojectivevarieties.WedefinetheHadamardprod-
uctofX andY,denotedbyX(cid:63)Y,astheZariskiclosureoftheimageofX×Y under
themap
MinkowskisumsandHadamardproductsofalgebraicvarieties 3
φ : Pn×Pn (cid:57)(cid:57)(cid:75) Pn,
(cid:63)
([a :...:a ],[b :...:b ])(cid:55)→[a b :a b :...:a b ].
0 n 0 n 0 0 1 1 n n
Let {x ,...,x } be the homogeneous coordinates over Pn. The map φ is not de-
0 n (cid:63)
fined over the union of coordinate spaces HI×HIc, where I ⊂{0,...,n}, Ic is its
complement,andH isthelinearspacedefinedby{x =0|i∈I}.
I i
Thus,theHadamardproductofprojectivevarietiesX,Y ⊂Pnis
X(cid:63)Y :={p(cid:63)q : p∈X, q∈Y, p(cid:63)qisdefined}⊂Pn,
where p(cid:63)q:=[p q :...: p q ]isthepointobtainedbyentry-wisemultiplication
0 0 n n
of the points p=[p :...: p ] and q=[q :...:q ]. Also in this construction the
0 n 0 n
operationofclosureiscrucial,asweshowinExample4.1.
In [1], the authors studied the geometry of Hadamard products, with a particular
focusonthecaseoflinearspaces.Thisworkhasbeencontinuedin[2].
Inparticular,weareinterestedinstudyingHadamardproductsofvarietiesofma-
trices.TheHadamardproductofmatricesisaclassicaloperationinmatrixanalysis
[7].Itsmostrelevantpropertyisthatitisclosedonpositivematrices.TheHadamard
productoftensorsappearedmorerecentlyinquantuminformation[8]andinstatis-
tics[4,11].Inthelatter,theauthorsstudiedrestrictedBoltzmannmachineswhich
are statistical models for binary random variables where some are hidden. From
a geometric point of view, this reduces to studying Hadamard powers of the first
secantvarietyofSegreproductsofcopiesofP1.Aninterestingquestionistounder-
standhowtoexpressmatricesasHadamardproductsofsmallrankmatrices.Wecall
theseexpressionsHadamarddecomposition.WedefineHadamardranksofmatri-
cesbyusingamultiplicativeversionoftheusualdefinitionsusedforadditivetensor
decompositions.ThestudyofHadamardranksisrelatedtothestudyofHadamard
powersofsecantvarietiesofSegreproductsofprojectivespaces.
InSection4,wefocusinparticularonthedimensionoftheseHadamardpowers.
We define the expected dimension and, consequently, we define the expected r-
th Hadamard generic rank, i.e., the expected number of rank r matrices needed to
decomposethegenericmatrixofsizem×nastheirHadamardproduct.Itis
(cid:24)dimP(Mat )−dim(X )(cid:25) (cid:24) mn−(m+n−1) (cid:25)
exp.Hrk◦(m,n)= m,n 1 = .
r dim(X )−dim(X ) r(m+n−r)−m−n+1
r 1
WeconfirmthisiscorrectforsquarematricesofsmallsizeusingMacaulay2.
Thepaperisstructuredasfollows.InSection2,wepresentsomeexplicitcom-
putationsofthesevarieties.WeusebothMacaulay2[5]andSage[12].Thesecom-
putations allowed us to conjecture some geometric properties of Minkowski sums
andHadamardproductsofalgebraicvarieties.InSection3,weanalyzeMinkowski
sums of affine varieties. In particular, we prove that, under genericity conditions,
thedimensionoftheMinkowskisumisthesumofthedimensionsandweinvesti-
gatethedegreeoftheMinkowskisum.InSection4,westudyHadamardproducts
4 NetanelFriedenberg,AlessandroOneto,andRobertL.Williams
and Hadamard powers of projective varieties. In particular, we focus on the case
of Hadamard powers of projective varieties of matrices of given rank. We intro-
ducethenotionofHadamarddecompositionandHadamardrankofamatrix.These
conceptsmaybeviewedasthemultiplicativeversionsofthewell-studiedadditive
decompositionoftensorsandtensorranks.
2 Experiments
Problem5onSurfacesin[13]askedthefollowing:
ComputetheMinkowskisumandtheHadamardproduct
oftworandomcirclesinR3.Tryothercurves.
InordertocomputeMinkowskisumsandHadamardproductsofcirclesandother
curves,weusedthealgebrasoftwaresMacaulay2andSagetoobtainequationsand
nice graphics. These also aided our general understanding of the geometric prop-
ertiesoftheseconstructions.Viaeliminationtheory,wecancomputetheidealsof
MinkowskisumsandHadamardproducts.ThisisthescriptinMacaulay2todoso.
R = QQ[z_1..z_n,
x_1..x_n,y_1..y_n];
I = ideal( ... ); -- ideal of X in variables x_i;
J = ideal( ... ); -- ideal of Y in variables y_i;
---- construct the ideals of graphs of the maps
---- phi_+ and phi_star
S = I + J + ideal(z_1-x_1-y_1,...,z_n-x_n-y_n);
P = I + J + ideal(z_1-x_1*y_1,...,z_n-x_n*y_n);
Msum = eliminate(toList{x_1..x_n | y_1..y_n}, S);
Hprod = eliminate(toList{x_1..x_n | y_1..y_n}, P);
With Sage, we produced graphics of the real parts of Minkowski sums and
HadamardproductsofcurvesinA3.Thisisthescriptweused.
A.<x1,x2,x3,y1,y2,y3,z1,z2,z3>=QQ[]
I=( ... )*A # ideal of X in the variables x;
J=( ... )*A # ideal of Y in the variables y;
# construct the ideals defining the graphs of the maps
# phi_+ and phi_star
S = I + J + (z1-(x1+y1),z2-(x2+y2),z3-(x3+y3))*A
P = I + J + (z1-(x1*y1),z2-(x2*y2),z3-(x3*y3))*A
MSum = S.elimination_ideal([x1,x2,x3,y1,y2,y3])
HProd = P.elimination_ideal([x1,x2,x3,y1,y2,y3])
# Assuming we get a surface, take the one generator of
# each ideal.
MSumGen=MSum.gens()[0]
HProdGen=HProd.gens()[0]
MinkowskisumsandHadamardproductsofalgebraicvarieties 5
# We plot these surfaces.
# Because MSumGen and HProdGen are considered as elements of A,
# which has 9 variables, they take 9 arguments.
var(’z1,z2,z3’)
implicit_plot3d(MSumGen(0,0,0,0,0,0,z1,z2,z3)==0,
(z1, -3, 3), (z2, -3,3), (z3, -3,3))
implicit_plot3d(HProdGen(0,0,0,0,0,0,z1,z2,z3)==0,
(z1, -3, 3), (z2, -3,3), (z3, -3,3))
In Figures 1, 2, 3 and 4 are some of the pictures we obtained. These experiments
gaveusafirstideaaboutthepropertiesofMinkowskisumsandHadamardproducts.
Note2.1.ThefactthatX+Y andX(cid:63)Y aretheclosuresofimagesofX×Y under
themapsφ andφ immediatelygivesusthat
+ (cid:63)
dim(X+Y),dim(X(cid:63)Y)≤dim(X)+dim(Y)
andthatifX andY areirreducible,soareX+Y andX(cid:63)Y.
Also,thefactthatX+Y andX(cid:63)Y arelinearprojectionsofX×Y ⊂An×AnandX×
Y ⊂Pn×Pn ⊂Pn2+2n, respectively, leads us to expect other geometric properties
ofX+Y andX(cid:63)Y.BecausetheprojectionofavarietyZ⊂PN ingenericposition
fromalinearspaceLwithdim(Z)+dim(L)<N−1isgenericallyone-to-one,we
na¨ıvelyexpectthat,forX andY ingeneralposition,withdim(X)+dim(Y)<n,
dim(X+Y),dim(X(cid:63)Y)=dim(X)+dim(Y),
deg(X+Y)=deg(X×Y)=deg(X)deg(Y), (X×Y ⊂An)
deg(X(cid:63)Y)=deg(X×Y)=(cid:0)dim(X)+dim(Y)(cid:1)deg(X)deg(Y), (X×Y ⊂Pn2+2n).
dim(X)
Theseexpectations,however,donotfollowdirectlyfromtheprojectionsofthevari-
etiesingeneralpositionbecause,evenforX andY ingeneralposition,X×Y isnot
ingeneralposition.Hence,weneedfurtheranalysisasinthefollowingsections.
Fig.1 Minkowskisumofacircleofradius1inthex,y-planeandacircleofradius2inthex,z-
plane.Thisisadegreefoursurface.
6 NetanelFriedenberg,AlessandroOneto,andRobertL.Williams
Fig.2 Minkowskisumofthetwoparabolasx=y2 inthex,y-planeandy=z2 inthey,z-plane.
Thisisadegreefoursurface.
Fig.3 Minkowskisumofthetwistedcubicwiththeunitcircleinthex,y-plane,y,z-plane,and
x,z-planefromlefttoright,respectively.Eachoftheseareadegreesixsurface.
Fig.4 TherealpartoftheHadamardproductof(left)theunitcircleinthez=1planeandthe
unitcircleinthey=1planeand(right)thecirclesx2+(y+z)2=1,z−y=1andx2+(z−y)2=
1,y+z=1.Thesurfaceontheleftisofdegreefour,andthesurfaceontherightisofdegreetwo.
3 Minkowskisumofaffinevarieties
Recall the definition of the Minkowski sum of two affine varieties X andY as the
closureoftheimageofX×Y underthemap
φ : An×An → An,
+
((a ,...,a ),(b ,...,b ))(cid:55)→(a +b ,...,a +b )
1 n 1 n 1 1 n n
MinkowskisumsandHadamardproductsofalgebraicvarieties 7
Theoperationofclosureisneededinordertogetanalgebraicvariety.Indeed,we
can give an example where X+ Y :=φ (X×Y)={p+q| p∈X,q∈Y}, the
set +
setwiseMinkowskisumofX andY,isnotclosed.
Example3.1.In the affine plane A2 with coordinates {x,y}, consider the plane
curves X ={xy=1} andY ={xy=−1}. We claim that φ (X×Y) contains the
+
torus(k×)2,soX+Y =A2.Forif(α,β)∈(k×)2and(p,q)∈X×Y thenφ (p,q)=
+
(α,β) if and only if we can write p and q in the forms p = (α+a, 1 ) and
α+a
q=(−a,1)forsomescalara(cid:54)=0,−α suchthatβ = 1 +1.Clearingdenomina-
a α+a a
torsinthislastexpression,wefindtherequirementisthatβa(α+a)=a+(α+a),
i.e.aisazeroofthequadraticpolynomial f (t)=βt2+(βα−2)t−α.Notethat
α,β
f (0)=−α and f (−α)=βα2−βα2+2α−α =α.Soifweletabeazero
α,β α,β
of f thenwith pandqasabovewehaveφ (p,q)=(α,β).
α,β +
OntheotherhandX+ Y isnotallofA2.Ifthecharacteristicofthebasefieldis
set
not2,thenX+ Ydoesnotcontaintheorigin(thoughitdoescontainthepunctured
set
axes). In characteristic 2 we find that X+ Y contains no point of the punctured
set
axes{x=0}\{(0,0)}and{y=0}\{(0,0)}.
OneofourmaintoolsforprovingresultsabouttheMinkowskisumisanalter-
nativedescriptionofitintermsofthejoinofthetwovarieties.
ForX,Y subvarietiesofAn orPn,weletJ (X,Y)bethesetwisejoinofX and
set
Y,i.e.,theunionofthelinesconnectingdistinctpointsx∈X andy∈Y.Thisspace
isusuallynotclosedanditsZariskiclosureJ(X,Y)istheclassicaljoinofX andY.
Our analysis of the Minkowski sum of affine algebraic sets X andY via a join
willinvolvehyperplanespositionedasinLemma3.2below.Foranintuitivesense
ofthestatementofthelemma,onemayconsiderthecasewhereL,M, andNarethe
projectivizationsofparallelaffinehyperplanes.
Lemma3.2.Let L,M,N be three distinct hyperplanes in Pn with E :=L∩M =
L∩N =M∩N. Say X ⊂M and Y ⊂N are nonempty disjoint subvarieties. Let
Xa=X\E,Ya=Y\E,∂X =X∩E,and∂Y =Y∩E.Then:
(i)J(X,Y)=J (X,Y),
set
(ii)J(X,Y)∩L=(J (Xa,Ya)∩L)∪J (∂X,∂Y)∪∂X∪∂Y,and
set set
(iii)J(X,Y)∩L\E=J (Xa,Ya)∩L.
set
Inparticular,ifX andY havepositivedimensionthen
J(X,Y)∩L=(J (Xa,Ya)∩L)∪J (∂X,∂Y).
set set
Proof. (i) Because X and Y are disjoint, we have J (X,Y) is Zariski closed, so
set
J(X,Y)=J (X,Y)[seeExample6.17onp.70of[6]].
set
(ii)Fromthefirstpart,wehave
J(X,Y)=J (Xa,Ya)∪J (Xa,∂Y)∪J (∂X,Ya)∪J (∂X,∂Y).
set set set set
SotogettheclaimedexpressionforJ(X,Y)∩Litsufficestoshow
8 NetanelFriedenberg,AlessandroOneto,andRobertL.Williams
(a) J (Xa,∂Y)∩L,J (∂X,Ya)∩L⊂∂X∪∂Y and
set set
(b) J (∂X,∂Y)∪∂X∪∂Y ⊂L.
set
(a) BysymmetryitisenoughtoshowthatJ (Xa,∂Y)∩L⊂∂Y.
set
Say x∈Xa and y∈∂Y. So y∈L but x∈/ L. Thus, the line between x and y
intersectsLinexactly{y}⊂∂Y.
(b) WeshowthatJ (∂X,∂Y)∪∂X∪∂Y ⊂E.
set
Bydefinition,∂X,∂Y ⊂E.So,becauseE isalinearspace,foranyx∈∂X and
y∈∂Y,thelinebetweenxandyiscontainedinE.
(iii)First,notethatbecauseJ (∂X,∂Y)∪∂X∪∂Y ⊂E,wehave
set
(J(X,Y)∩L)\E⊂(J (Xa,Ya)∩L)\E.
set
Hence,wejustneedtoshowthatJ (Xa,Ya)∩LisdisjointfromE.
set
Considering any x∈Xa and y∈Ya, it suffices to show that the line (cid:96) between
xandydoesnotmeetE.Ifweassume,towardsacontradiction,thatthereissome
z∈(cid:96)∩E,thenzandx wouldbedistinctpointsonthehyperplaneM,sotheline(cid:96)
betweenthemwouldbecontainedinM.Buty∈Ya⊂N\E =N\M,so(cid:96)cannot
becontainedinM.
Finally,ifX andY arepositivedimensionalthen∂X =X∩Land∂Y =Y∩Lare
nonempty,so∂X,∂Y ⊂J (∂X,∂Y). (cid:116)(cid:117)
set
Our alternative description of the Minkowski sum will give us cases in which
X+ Y is already closed. Recall from Example 3.1 that for the two plane curves
set
X ={xy=1}andY ={xy=−1},X+ Y isnotZariskiclosed.Notethatinthis
set
example,X andY haveacommonasymptote,orequivalently,thattheirprojective
closuresmeetatthelineatinfinity.Wewillseethatwhenthecharacteristicofthe
basefieldisnot2,allcaseswhereX+ Y isnotclosedshareananalogousproperty.
set
Definition3.3.Let X,Y ⊂An be varieties and denote the projective closures of X
andY inPn byX andY,respectively.LetH ={[x :···:x ]∈Pn|x =0}bethe
0 0 n 0
hyperplaneat∞.WesaythatX andY aredisjointatinfinityifX∩Y∩H =0/.
0
Remark3.4.If X andY are disjoint at infinity then dim(X∩Y)<1, thus dimX+
dimY ≤n.
Proposition3.5.Assumethatthecharacteristicofkisnot2.SupposeX,Y ⊂Anare
varietiesthataredisjointatinfinity.Letz ,z bedistinctscalarsandletX˜,Y˜ ⊂Pn+1
0 1
betheprojectiveclosuresofX×{z }andY×{z },respectively.Letx ,x ,...,x ,z
0 1 0 1 n
bethecoordinatesonPn+1.
IfweidentifyS={z= z0+z1x }⊂Pn+1withPnandS\H withAn,then:
2 0 0
(i)J (X˜,Y˜)=J(X˜,Y˜);
set
(ii) 1(X+Y)=J(X˜,Y˜)∩S\H ;
2 0
(iii)X+ Y =X+Y,namely,X+ Y isZariskiclosed.
set set
MinkowskisumsandHadamardproductsofalgebraicvarieties 9
Proof. (i)LetE={x =0}⊂Pn+1bethehyperplaneat∞inPn+1.Notethat
0
(cid:26) (cid:27)
z +z
E∩{z=z x }=E∩ z= 0 1x =E∩{z=z x }={z=0,x =0},
0 0 0 1 0 0
2
which is identified with H . Therefore the statement that X and Y are disjoint at
0
infinityisequivalentto
X˜∩Y˜∩E=0/.
On the other hand, X˜ \E =X×{z } and Y˜ \E =Y ×{z }, and so we see that
0 1
X˜∩Y˜ =0/. So, by Lemma 3.2 applied to S={z= z0+z1x }, X˜ ⊂{z=z x }, and
2 0 0 0
Y˜ ⊂{z=z x },wefindthat
1 0
J (X˜,Y˜)=J(X˜,Y˜) and J(X˜,Y˜)∩S\H =J (X×{z },Y×{z })∩S.
set 0 set 0 1
(ii)&(iii)Foranyx∈X andy∈Y,thelinebetweenthepoints(x,z )∈X×{z }
0 0
and (y,z )∈Y×{z } meets the affine hyperplane S\E ={z= z0+z1} in exactly
1 1 2
the point (x+y,z0+z1). So, we have shown that J(X˜,Y˜)∩S\H = 1(X+ Y). In
2 2 0 2 set
particular,becauseJ(X˜,Y˜)isclosed,thistellsusthatX+ Y isaclosedsubsetof
set
S\H ∼=An.Hence, 1(X+Y)= 1(X+ Y)=J(X˜,Y˜)∩S\H . (cid:116)(cid:117)
0 2 2 set 0
Remark3.6.Wecalltheconstruction 1(X+Y)=J(X˜,Y˜)∩S\H theCayleytrick,
2 0
astheunderlyingideaisexactlythesameasthatoftheCayleytrickusedtoconstruct
Minkowskisumsofpolytopes.
Asaconsequenceofthefollowinglemma,ifwerestricttothecaseswithdimX+
dimY ≤n, then the hypothesis that X andY are disjoint at infinity is a genericity
condition.
Lemma3.7.LetX,Y ⊂PnbevarietieswithdimX+dimY <n.Then,wehavethat
theset{g∈GL |gX∩Y =0/}isanonemptyopensubsetofGL .Thatis,for
n+1 n+1
genericg∈GL ,gX andY donotintersect.
n+1
Proof. First,notethatforanypoint p∈Pn thestabilizerof pinGL hasdimen-
n+1
sionn2+n+1.ThisisbecauseanytwopointstabilizersinGL areconjugateand
n+1
thestabilizerof[1:0:0:···:0]∈Pn isthesetofallg∈GL withfirstcolumn
n+1
(cid:2) (cid:3)T
oftheform ∗00··· 0 ,whichhasdimension(n+1)n+1.
LetZ={(g,x,y)∈GL ×X×Y |gx=y}whichisasubvarietyofGL ×
n+1 n+1
X×Y.Letπ :Z→GL andπ :Z→X×Y betherestrictionsofthecanonical
1 n+1 2
projectionsfromGL ×X×Y.Notethatπ issurjective,becauseforanyx,y∈Pn
n+1 2
thereexistssomeg∈GL takingxtoy.Further,thefiberoveranypoint(x,y)∈
n+1
X×Y isaleftcosetofapointstabilizerinGL andsohasdimensionn2+n+1.
n+1
Thus,dimZ=n2+n+1+dimX+dimY <n2+2n+1=dimGL .
n+1
BecauseX×Y isprojective,theprojectionGL ×X×Y →GL isaclosed
n+1 n+1
map,soπ (Z)isaclosedsubsetofGL .Sincedimπ (Z)≤dimZ<dimGL ,
1 n+1 1 n+1
π (Z)isaproperclosedsubsetofGL .So,
1 n+1
{g∈GL |gX∩Y =0/}=GL \π (Z)
n+1 n+1 1
10 NetanelFriedenberg,AlessandroOneto,andRobertL.Williams
isanonemptyopensubsetofGL . (cid:116)(cid:117)
n+1
Now,weclaimthatifX,Y ⊂AnarevarietieswithdimX+dimY ≤n,then
forgeneralg∈GL ,gX andY aredisjointatinfinity.
n
Toseethis,notethat,consideringH ={x =0},
0 0
dim(X∩H )+dim(Y∩H )≤dimX−1+dimY−1<dimX+dimY−1≤n−1.
0 0
TheactionofGL onAnextendstoanactiononPn,andtheidentificationH ∼=Pn−1
n 0
isGL -equivariant.Sowehave
n
gX∩Y∩H =(gX∩H )∩(Y∩H )=g(X∩H )∩(Y∩H ).
0 0 0 0 0
ByLemma3.7,forgeneralg∈GL thisisempty.
n
Remark3.8.We could have used the group of affine transformations Aff =An(cid:111)
n
GL . Indeed, shifting an affine variety does not change the part at infinity of its
n
projectiveclosure.
When a result holds under the same conditions as Lemma 3.7, i.e., if we fix X
andY thenitholdsforgX andY,forgeneralg∈Aff ,weshallsaythattheresult
n
holdsforX andY ingeneralposition.
WearenowreadytocomputethedimensionofMinkowskisums.Basedonthe
examples in Section 2, it seems that for dimX+dimY ≤n, we get dim(X+Y)=
dimX+dimY.Thisdoeshappengenerically.
Theorem3.9.Let X,Y ⊂An be varieties. Then for X and Y in general position,
dim(X+Y)=min{dimX+dimY,n}.
Proof. AsobservedinNote2.1,wehavethat,foranyX,Y ⊂An,
dim(X+Y)≤min{dim(X)+dim(Y),n}.
So,weneedtoprovetheconverseforX,Y ingeneralposition.
Letk=dim(X)andl=dim(Y).
Notethatbecause(X+v)+Y =(X+Y)+vforanyvectorv,itsufficestoshow
that for general g∈GL , dim(gX+Y)≥min{dim(X)+dim(Y),n}. We consider
n
thecasedim(X)+dim(Y)≤n,thecasedim(X)+dim(Y)≥nbeinganalogous.
Notethatbyjustlookingatfull-dimensionalirreduciblecomponentsofX andY,
wemayassumewithoutlossofgeneralitythatX andY areirreducible.
WedenotebyT (X)thetangentspacetothevarietyX atthepoint p.
p
Fornowfixg∈GL .If(p,q)∈An×Anthen
n
(dφ ) :T An×T An→T An
+ (p,q) p q p+q
issimplytheadditionmapφ ,andsoweseethatifp∈gX andq∈Y thenT (gX)+
+ p
TY ⊆T (gX+Y).Sotoconcludethatdim(gX+Y)≥dim(X)+dim(Y)itsuf-
q p+q
ficestoshowthatthereisadensesubsetΞ ofgX+Y suchthatforeachξ ∈Ξ there
exist p∈gX andq∈Y withξ =p+qandT (gX)∩TY =0,forthen
p q