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The Chern-Mather class of the multiview variety 7 1 Corey Harrisand Daniel Lowengrub 0 2 January26, 2017 n a J Abstract 4 2 The multiview variety associated to a collection of N cameras records which sequences of imagepointsinP2N canbeobtainedbytakingpicturesofagivenworldpointx ∈ P3 with ] the cameras. In order to reconstruct a scene from its picture under the different cameras it G isimportanttobeabletofindthecriticalpointsofthefunctionwhichmeasuresthedistance A between a general point u ∈ P2N and the multiview variety. In this paper we calculate a . specificdegree3polynomialthatcomputesthenumberofcriticalpointsasafunctionofN. In h ordertodothis,weconstructaresolutionofthemultiviewvariety,anduseittocomputeits t a Chern-Matherclass. m [ 1 Introduction 1 v Suppose that a collection of cameras are used to generate images of a scene. The problem of 9 7 triangulationistodeducetheworldcoordinatesofanobjectfromitspositionineachofthecamera 0 images. Ifweassume thatthe imagepointsaregivenwithinfinite precision, thentwocameras 7 sufficetodeterminetheworldpoint. However,duetothemanysourcesofnoiseinrealimages 0 such as pixelization and distortion, there typically will not be an exact solution and we will . 1 insteadtrytofindaworldpointwhosepictureis“ascloseaspossible”totheimagepoints. 0 More precisely, suppose the cameras are C1,...,CN and the image points are p1,...,pn ∈ 7 R2. Thegoalistofindaworldpointq∈R3 thatminimizestheleastsquareserror 1 : v N Xi error(q)= X(Ci(q)−pi)2. i=1 r a Oneapplicationistheproblemofreconstructingthe3Dstructureofatouristattractionbased on millions of online pictures. It is difficult to obtain the precise configuration of any single camera,soitwouldnotmakesensetouseonlyasmallsubsetofthemanddisregardtherest. A betterapproachistosolveanoptimizationproblemwhichincorporatesasmanyofthecameras aspossible. Thistechniquewasusedin[1]toreconstructtheentirecityofRomefromtwomillion onlineimages. Since the camera function C : R3 R2 is not linear, the standard method for solving the i triangulation problem is to first find th→e critical points of error(q) (e.g, with gradient descent), andthenselecttheonewiththesmallesterror.Inordertogaugethedifficultyofthisproblem,it isimportanttobeabletopredictthenumberofcriticalpointsthatweexpecttofindforagiven configurationofcameras. The goal of this paper is to give an explicit expression for the number of critical points of error(q) as a function of the number of cameras N. In fact, we compute this expression for a variation of the problem in which we allow the world points to take complex values, and we allowthesepointstobeintheprojectivespaceP3 asopposedtotheaffinespaceC3. Ourmain C resultisthatthenumberofcriticalpointsoferror(q)ispolynomialinthenumberofcameras. 1 Theorem1. Thenumberofcriticalpointsoferror(q)onP3 isequalto C p(N)=6N3−15N2+11N−4 whereNisthenumberofcameras. Note that our reformulation of the problem only increases the number of possible critical points. Onecansolvetheoriginalproblembyfirstfindingthesepoints,andthendiscardingthe onesthatarenotinR3. Forasimilarreason,thepolynomialp(N)isanupperboundonthenumberofcriticalpoints intheclassicaltriangulationproblem. In[10],adetailedinvestigationoftheLagrangemultiplier equationswhichdefinethecriticalpointsisusedtocomputethenumberofsuchpointsforN≤7. Basedontheseresults,itwasconjecturedin[4,Conjecture3.4]thatthenumberofpointsshould growasthefollowingpolynomial: 9 21 q(N)= N3− N2+8N−4. 2 2 Wenotethatourupperboundp(N)isfairlyclose. Inordertocomputethenumberp(N),wetakeaslightlydifferentperspectiveonthefunction error(q). BycombiningthecamerasC :R3 R2 weobtainarationalmap i → φ:R3 R2N. → Afterpassingtothecomplexnumbersandtakingtheprojectiveclosureweobtainarational map φ:P3C P2CN. → The image of this map is a three-dimensional variety MV ⊂ P2N which is known as the N multiviewvariety. Wecannowinterprettheerrorfunctionerror(q)asmeasuringthedistancebe- tweenapointq∈ P2N andMV . Withthisformulation,thenumberofcriticalpointsisknown N astheEuclideandistancedegreeofthevarietyMV .ThenotionofEDdegreewasintroducedin[4], N andtheauthorsremarkin[4,ex3.3]thatthetriangulationproblemwastheiroriginalmotivation forthisconcept. Inparticular,byusingresultsfrom[4]weproveinsection5thatthisnumbercanbecomputed intermsoftheChern-MatherclasscM(MV ). Ingeneral,theChern-Matherclassonlyprovides N an upper bound on the ED degree, but in the proof of theorem 4 we show that this inequality canbepromoted toanequalityfor reasonsspecific to the multiviewvariety. One advantageof thisapproachisthatitdependsonlyonthegeometricpropertiesofMV ,andnotonthespecific N featuresofthedefiningequations. Anotheradvantageisthatitreducesmostofthedifficultyto localcalculationsonMV . N OnecommonwayofcalculatingtheChern-MatherclassofasingularvarietyXistofirstfind aresolution X˜ −f X → and then analyze the singularities of f in order to compare the Chern class c(X˜) to the Chern- MatherclasscM(X) Inoursituation,itisnaturaltobuildaresolutionofMV byresolvingtherationalmapφ. In N section3,weconstructsucharesolution φ˜ :P˜3 MV N → andcalculateitsChowringandChernclass. In order to compare the Chern class of P˜3 to the Chern-Mather class of MV , we use the N theory of higher discriminants which was introduced in [9]. One aspect of this theory is that it specifieswhichpartsofthesingularlocusofXweneedtounderstandinordertorelatec(X˜)to cM(X). Aprecisestatementisgiveninproposition5. 2 Asweshowinproposition6,thehigherdiscriminantsofφ˜ aresurprisinglynice. Specifically, itturnsoutthatinordertocalculatecM(MV ),weonlyhavetocomputetheEulerobstruction N ofasinglepointx∈MV . N Moreover, in section 5.2 we show that after intersecting MV with a hyperplane at x, the N resultingsurfacesingularity(S,x)istaut. Inparticular,theEulerobstructionEu (x)isdeter- MVN minedbytheresolutiongraphofxinS. ThisallowsustousetheenumerativepropertiesofP˜3 thatareworkedoutinsection3tocomputeEu (x). MVN Inthefinalsection,weputthesepiecestogetherandobtainthepolynomialp(N). 2 Definitions and notation Let P be a 3×4 matrix with values in R. We consider each row l as an affine function on R3. Explicitly, l sends a vector v = (x,y,z) to the dot product of l and (x,y,z,1). We denote these functionsbyf,gandh. ThematrixPdefinesarationalmapφ :R3 99KR2: P v7 (f(v)/h(v),g(v)/h(v)) → which corresponds to the operation of mapping the “world coordinates” R3 to the “image coordinates” R2. In other words, it describes the process of taking a picture of the world with a camerawhoseparametersareencodedinP. It is not hard to prove that this description of a camera is equivalent to the pinhole camera model. Inparticular,thecamerahasapositioncalledthecameracenterandispointinginacertain direction.Theplanedefinedbythecameracenteranddirectioniscalledthecameraplane. Itturns outthatwiththeabovenotation,thecameraplaneistheplanedefinedbytheideal(h),andthe camera center is the point defined by (f,g,h). For the purposes of this paper, this observation willbetakenasadefinition. Now, suppose that we have a collection of cameras P ,...,P . By taking a picture of the 1 N worldwitheachofthecameras,weobtainarationalmap: φ ×···×φ :R3 99KR2×···×R2 =∼ R2N P1 PN Thismapclearlyextendstothecomplexnumbers,givingusarationalmapfromC3 99KC2N. Furthermore,byclearingthedenominatorsinthedefinitionofthemapsφ weobtainarational Pi map φ:P3 99KP2N C C definedby φ([x:y:z:w])=(f1h2...hN :g1h2...hN :···:h1...hN−1gN :h1...hN). (1) Theschemetheoreticimageofthismapiscalledthemultiviewvarietyassociatedtothecameras P ,...,P . 1 N Example1. Considerthefollowingthreecameras: 1 0 0 1 0 1 0 1 1 0 0 1 P1 =0 0 1 1, P2 =0 0 1 1, P3 =0 1 0 1. 0 1 0 0 1 0 0 0 0 0 1 0       Theassociatedrationalmapis φ([x:y:z:w])=[(x−w)xz:(z−w)xz:(y−w)yz:(z−w)yz:(x−w)xy:(y−w)xy:xyz]). 3 L 12 H 2 H 1 q 2 q 1 p 123 L 23 q 3 L 13 H 3 Figure1: Schematicofthreecameras Wesaythatacollectionofcamerasisingeneralpositionifthehyperplanesdefinedbythelinear functions{f ,g ,h ,...,f ,g ,h }associatedtotherowsofthecameramatricesareingeneral 1 1 1 N N N position. Finally, we will use the following notation throughout the paper (see figure 1). The camera planeofthei-thcamerawillbedenotedbyH andthecenterofthei-thcamerawillbedenoted i by q . Also, we define L = H ∩H for all 1 ≤ i < j ≤ N, and p = H ∩H ∩H for all i ij i j ijk i j k 1≤i<j<k≤N. 3 A resolution of the multiview variety InthissectionwedescribearesolutionofthemultiviewvarietyassociatedtoNcamerasingen- eralposition. Itisobtainedasaniteratedblowupalongsmoothcenters. Wethenapplystandard theoremstocomputeapresentationoftheChowringoftheresolution,andidentifyacoupleof importantringelements. LetP ,...,P becameramatricesforacollectionofNcamerasingeneralposition,andlet 1 N φ:P3 99KP2N bethecorrespondingrationalmap.WedenotetheassociatedmultiviewvarietybyMV ⊂P2N. N Proposition1. ThebaselocusBofφisthereducedschemesupportedontheunionofthecameracenters q ,...,q andthelinesL =H ∩H forall1≤i<j≤N. 1 N ij i j Proof. It can be seen directly from the equations of φ (equation 1) that B is supported on the cameracenters union the lines L . We will show that the scheme structure of B is the reduced ij structure on this set. By a strategic choice of coordinates on P3, we can assume that h = x, 1 h =yandh =z. 2 3 We now analyze the scheme structure of B in a neighborhood of the point p = (x,y,z). 123 Firstofall,recallthatthei-thcameracontributesthetwoequationsf · h andg · h i Qj6=i j i Qj6=i j totheidealofB. Byourgenericityassumptions,allofthef ’s,alloftheg ’s,andh fori ≥ 4areinvertiblein i i i someZariskineighborhoodofp . Thisimpliesthatinaneighborhood ofp , theidealofB 123 123 hastheform: (xy,xz,yz). 4 Thus, the ideal defined by this scheme is reduced and supported on the coordinate axes. The sameargumentshowsthatallofthelinesL inthebaselocushavethereducedschemestructure. ij Asimilarargumentimpliesthepointsq arereduced. i 3.1 Constructing a resolution of φ InthissectionweconstructaresolutionofMV intwostages.First,weblowupP3atthepoints N q ,...,q andatthepointsp forall1≤i<j<k≤N. Thisgivesusamap 1 N ijk b :Y P3. 1 1 → LetL˜ ⊂ Y denotethepropertransformofL . Note thatthesepropertransformsaredisjoint ij 1 ij linesinY . 1 Forthesecondstep,weblowupeachofthelinesL˜ andobtainaresolution ij b :Y Y 2 2 1 → LetusdenoteY byP˜3, and denotethe composition b ◦b byπ. Sincethe pullbackof the 2 1 2 base locus π−1(B) is a Cartierdivisor on P˜3, there exists a canonicalmap P˜3 −ψ Bl P3 which B fitsintothefollowingdiagram: → P˜3 ψ //Bl P3 ❋❋❋❋❋❋π❋❋❋"" B(cid:15)(cid:15)b❍❍❍❍❍B❍l❍B❍φ## P3 ❴ ❴ ❴ ❴//P2N φ werebistheblowupmapandBl φistheresolutionoftherationalmapφ. B Finally,wedefineφ˜ =Bl φ◦ψ. SinceP˜3 issmooth,wethusobtainthefollowingresolution B ofMV : N P˜3 (cid:15)(cid:15)π▲▲▲▲▲▲φ▲˜▲▲▲%% P3 ❴ ❴ ❴//MV ⊂P2N N φ Byanabuseofnotation,wewillsometimesthinkofφ˜ asamaptoP2N,andothertimesasa maptoMV . N 3.2 The Chow ringof P˜3 Since P˜3 is an iterated blowup of P3 along smooth centers, we can use standard theorems to computeitsChowring. Wewilluseastatementin[6]whichwestatehereforconvenience. Theorem 2. [6, Appendix, Thm.1] Let X −i Y be a closed embedding ofsmoothschemes. Let Y˜ be the blowup ofY along Xand let X˜ denotetheex→ceptionaldivisor. Supposethemapi∗ : A•(Y) A•(X) is surjective. Then,A•(Y˜)isisomorphicto → A•(Y)[T] (P(T),T ·ker(i∗)) whereP (T)∈A•(Y)[T]isadegreedpolynomialwhoseconstanttermis[X],andwhoserestrictionto X/Y XistheChernpolynomialofN . Inotherwords, X/Y i∗PX/Y(T)=Td+c1(NX/Y)Td−1+···+cd−1(NX/Y)T +cd(NX/Y). The isomorphism is induced by the map f∗ : A•(Y) A•(Y˜), and by sending −T to the class of the exceptionaldivisor. → 5 ThepolynomialP iscalledthePoincare´polynomialofXinY. X/Y Byapplyingtheorem2firsttoY −b−1 P3 andthentoP˜3 = Y −b−2 Y wefindthatA•(P˜3)is 1 2 1 aquotientofthepolynomialalgebra → → A=Z[{h}∪{Q } ∪{P } ∪{T } ]. i 1≤i≤N ijk 1≤i<j<k≤N ij 1≤i<j≤N The meaning of the generators is as follows. Let q˜ ∈ P˜3 denote the exceptional divisor of i the camera center q , p˜ ∈ P˜3 the exceptional divisor of the point p , and L˜ ⊂ P˜3 the i ijk ijk ij exceptionaldivisorofthelineL . ij Then,wehavethefollowingidentitiesinA•(P˜3): [q˜ ]=−Q , [p˜ ]=−P , [L˜ ]=−T . i i ijk ijk ij ij Inthenextsection,wewillneedtoevaluatethedegreemap deg:A3(P˜3) Z. → SinceP˜3 isirreducible,A3(P˜3)hasrankone. Inaddition, deg(h3) = 1. This meansthatcalcu- latingthedegreemapisequivalenttoexpressingeverymonomialα ∈ A3(P˜3)asamultiple of h3: α=deg(α)·h3. To simplify the calculation, note that product of two generators that correspond to disjoint subschemesofP˜3iszero.Forexample,Q ·P =0foralli,j,kandl. i jkl Thus, the maindifficultyis dealingwith self intersections such asT3. In orderto dealwith ij these,wewillcalculatethePoincare´polynomialsofq ⊂Y ,p ⊂Y andL ⊂Y .Bytheorem i 1 ijk 1 ij 2 2,thiswillgiveusrelationsinvolvingtheselfintersections,whichinthiscaseturnouttosuffice forthedegreecalculation. Sinceq ⊂Y isapoint,itsPoincare´ polynomialis i 1 P (Q )=Q3+h3, qi/Y1 i i andsimilarly, P (P )=P3 +h3. pijk/Y1 ijk ijk Finally, note thatL ⊂ Y is a line that passes through N−2 blown up points. We deduce ij 1 fromthisthat PLij/Y2(Tij)=Ti2j−2(N−3)hTij+h2+ X Pi2jk. k∈/{i,j} 3.3 The Chernclass of the resolution In this section we compute c(P˜3) as an element of A•(P˜3) and find its pushforward to P2N (proposition4). Ourmaintoolwillbethefollowingproposition. Proposition2. [5,Example15.4.2]LetY beasmoothschemeandX ⊂ Y beaclosedsmoothsubscheme withcodimensiond. Considerthefollowingblowupdiagram. X˜ j //Y˜ g f (cid:15)(cid:15) (cid:15)(cid:15) // X Y i Supposethatc (N ) = i∗c forsomec ∈ Ak(Y), and thatc(X) = i∗αforsomeα ∈ A•(Y). Let k X/Y k k η=c (O (X˜)). Then, 1 Y˜ c(Y˜)−f∗c(Y)=f∗(α)·β 6 where d d β=(1+η)X(1−η)if∗cd−i−Xf∗cd−i. i=0 i=0 OnetakeawayofthispropositionisthattheChernclassoftheblowupalongadisjointunion ofsubvarietiesisobtainedbysummingovercontributionsfromtheindividualcomponents. Proposition3. TheChernclassoftheresolutionP˜3isequalto c(P˜3)=(1+h)4+ X αi+ X βij+ X γijk 1≤i≤N 1≤i<j≤N 1≤i<j<k≤N where α =(1−Q )(1+Q )3−1, i i i β =(1+h)2·[(1−T )((1+T )(−2(N−3)h)+(1+T )2)−(1−2(N−3)h)], ij ij ij ij γ =(1−P )(1+P )3−1. ijk ijk ijk Proof. Ourstrategywillbetouseproposition2tocomputethecontributions totheChernclass ofeachofthevarietiesthatareblownupduringtheconstructionofP˜3. Wefirstapplyproposition2tothesituationwhereY =P3andX=q forsomei.Inthiscase, i wecantakec =1,c =0fork>0andα=1. Byproposition2theblowupatq willcontribute 0 k i α =(1−Q )(1+Q )3−1. i i i Similarly,γ representsthecontributionfromtheblowupofthepointp . ijk ijk Finally, we compute the contribution from the blowup along a line f : L ֒ Y . Since L ij 1 ij passesthrough N−2 of the blown up points in Y , a quick calculationshows t→hat we cantake 1 c = 1, c = −2(N−3)h, and the rest to be zero. In addition, since L =∼ P1, we can take 0 1 ij α=(1+h)2. ThisimpliesthatthecontributioncomingfromL is ij β =(1+h)2·[(1−T )((1−η)(−2(N−3)h)+(1+T )2)−(1−2(N−3)h)]. ij ij ij Wenowcomputethepullbackofc1(OP2n(1))inA•(P˜3)alongthemapφ˜. Lemma1. Thepullbackofc1(OP2n(1))toP˜3is φ˜∗(c1(OP2n(1)))∩[P˜3]=N·h+2· X Pijk+ X Qi+ X Tij. 1≤i<j<k≤N 1≤i≤N 1≤i<j≤N Proof. Itiswellknown(e.g[5,4.4])thatifLisalinebundleonX,V ⊂H0(X,L)isalinearsystem and X˜ (cid:15)(cid:15)π❉❉❉❉❉❉f❉❉❉"" X❴ ❴ ❴//P(V∗) istheinducedresolution,then f∗O(1)=π∗(L)⊗O(−E) whereE⊂X˜ istheexceptionaldivisor. In our case, one can show by a local calculation that the preimage of the base locus of the cameramaptoP˜3hasclass c1(O(−E))∩[P˜3]=2· X Pijk+ X Qi+ X Tij, 1≤i<j<k≤N 1≤i≤N 1≤i<j≤N sothatc (φ˜∗(O(1)))=c (π∗O(N))+c (O(−E))givesthestatedexpression. 1 1 1 7 Wecannowcomputethepushforwardφ˜∗c(P˜3)asanelementoftheChowringofP2N. Proposition4. ThepushforwardtoP2N ofc(P˜3)is N N N N φ˜ c(P˜3)= N3−(4+N) −N−2 [P3]+ 4N2−2 −6 −2N [P2] ∗ (cid:18) (cid:18)2(cid:19) (cid:18)3(cid:19)(cid:19) (cid:18) (cid:18)3(cid:19) (cid:18)2(cid:19) (cid:19) N N N + 6N+(N−4) [P1]+ 4+2N+2 +2 [P0] (cid:18) (cid:18)2(cid:19)(cid:19) (cid:18) (cid:18)3(cid:19) (cid:18)2(cid:19)(cid:19) Proof. Since we have already calculated c(P˜3) ∈ A•(P˜3) and φ˜∗(c1(OP2n(1))) ∈ A•(P˜3), the calculationofφ˜∗c(P˜3)isreducedtocalculatingthedegreesoftheintersections π∗(c (O (1)))k∩c(P˜3) 1 P2n for0 ≤ k ≤ 3. UsingtherelationsinA•(P˜3)thatwedescribedinsection3.2,theresultfollows byadirectcalculation. 4 Higher discriminants Higherdiscriminants,introducedin[9],provideaframeworkinwhichtostudythesingularities of a map. In particular, we will use them to understand how the Chern class of P˜3 computed abovepushesforwardalongφ˜. Wenowrecallthedefinitionsfrom[9],andphrasetheminaway thatwillbeeasiesttouseinourcontext. Definition1. Letf:Y Xbeamapofsmoothmanifolds. Thei-thhigherdiscriminantofthemapf isthelocusofpointsx ∈→Xsuchthatforeveryi−1dimensionalsubspaceV ⊂ TxX,thereexistsapoint y∈f−1(x)suchthat: hV,f∗TyYi6=TxX Wedenotethei-thhigherdiscriminantby∆i(f). Forexample,apointx∈Xisin∆1(f)ifandonlyifitisacriticalvalueoff. Indeed,according tothedefinitionthishappensexactlywhenthereisapointy∈f−1(x)whoseJacobian J(f) :T Y T X y y x → isnotsurjective. On the other extreme, x ∈ ∆dim(X)(f) if and only if for every codimension one subspace V ⊂TxX,thereexistsapointy∈f−1(x)thatsatisfies: f∗TyY ⊂V. It is instructive to consider the blow down map: f : Y = BlpP2 P2. For every point y ∈ Ep = f−1(p), f∗TyY is one dimensional. This means that p ∈ ∆1(→f). In addition, it is not hardtoseethatforeveryonedimensionalsubspaceV ⊂ TpP2,thereisapointy∈ Ep suchthat f∗TyY =V. Thisimpliesthatp∈∆2(f). Lemma2. [9,Rem.3]LetY Xbeapropermapofsmoothschemes.Thenallofthehigherdiscriminants offareclosed,andwehaveth→efollowingstratificationofX: ∆dim(X)(f)⊂··· ⊂∆2(f)⊂∆1(f)⊂X. Furthermore, codim(∆i(X))≥i. Thesignificanceofthehigherdiscriminantsisthattheytelluswhichstrataappearwhenwrit- ingf∗1Y inthebasisofEulerobstructionfunctionsonX. ForbackgroundonEulerobstructions werecommend[8]. 8 Proposition 5. [9, Cor. 3.3] Let f : Y X be a proper map of complex varieties. Let {∆i,α} be the codimensionicomponentsof∆i(f). Then,→ f∗1Y =Xηi,αEu∆i,α forsomeintegersηi,α. 4.1 Higher discriminantsof the resolution φ˜ Inthissectionwedescribethehigherdiscriminantsofthemap φ˜ :P˜3 MV ⊂P2N. N → Sincethedefinitionofhigherdiscriminantsassumesthatthesourceandtargetaresmooth,inthis sectionweconsiderφ˜ asamaptoP2N. LetX =∼ P1 ⊂MV denotetheimageofthepropertransformofthecameraplaneofthei-th i N camera. The restrictionof φ˜ to the complement of the preimageof the X ’s is anisomorphism, i whichmeansthatthesettheoreticsingularlocusofφ˜ iscontainedinthedisjointunion∐ X . i i Thefollowingpropositiondescribesthehigherdiscriminantsofφ˜. Proposition6. Thehigherdiscriminantsofφ˜ aregivenasfollows: • ∆2N−3(φ˜)=∆2N−2(φ˜)=MV N • ∆2N−1(φ˜)=∐ X i i • ∆2N(φ˜)=∅ To prove this proposition, we use the following lemma, which follows almost immediately fromthedefinitionofthehigherdiscriminants. Lemma3. Letf : Y Xbeamapofsmoothcomplexalgebraicvarieties. LetC ⊂ Xbeasmoothcurve. Supposethattherestr→ictionofftoChasnocriticalvalues. ThenC∩∆dim(X) =∅. Proof. Since f| has no critical values, for every point x ∈ C and every point y ∈ f−1(x) the C one dimensional space TxC ⊂ TxX is contained in f∗TyY. Therefore, if V ⊂ TxX is the or- thogonal complement to TxC, then f∗TyY is not contained in V. By definition, this implies that x∈/∆dim(X)(X). WeapplythislemmatoeachoftheP1’sX ⊂P2N.Letf:Y P1 =∼ X denotetherestriction i i of φ˜ toX . ThenY isisomorphic tothe blowup of P2 at1+ N→−1 points: q = q andp for i 2 i ijk j,k6=i. (cid:0) (cid:1) Themapfisobtainedasfollows. First,let g:BlqP2 P1 → betheresolutionoftheprojectionawayfromq. Then,let h:Blq,pijk(P2) Blp(P2) → betheblowupalongallofthepointsp forj,k6=i. ijk ∼ Finally,weclaimthatf=g◦h.Inparticular,fhasnocriticalvalues.Accordingtothelemma, thisprovesproposition6. 5 The Chern-Mather class of the multiview variety InthissectionwecomputetheChern-MatherclassofMV usingthetheoryofhigherdiscrimi- N nants. WethenusetheresulttodeterminetheEDdegreeofMV . N 9 5.1 The basicsetup Bypropositions5and6,thereexistsandintegerαsuchthat N φ˜∗(1P˜3)=EuMVN+α·XEuXi. (2) i=1 Atageneralpointx∈X ,χ(φ˜−1(x))=χ(P1)=2andEu (x)=1. Thisimpliesthat i Xi 2=Eu (x)+α α=2−Eu (x). MVN MVN ⇒ For the moment, suppose we knew the Euler obstruction Eu (x). Then, by taking the MVN Chern-Schwartz-MacPhersonclass(see[8])ofbothsidesofequation2andrecallingthatX =∼ P1 i weobtain φ˜∗(c(P˜3))=cM(MVN)+(2−EuMVN(x))cM(P1). (3) Sincewehavealreadycalculatedφ˜∗(c(P˜3))forallN,thiswouldgiveustheChern-Matherclass ofthemultiviewvarietyMV . N 5.2 Calculating Eu (x) MV N To compute Eu (x), first note that we can intersect MV with a general hypersurface H MVN N passingthroughx. Asaresult,weobtainasurfacesingularity: x∈S=MV ∩H. N ByawellknowntheoremaboutEulerobstructions(see[3,Sec.3]), Eu (x)=Eu (x). MVN S Now,supposewerestricttheresolutionφ˜ toS. Lemma4. φ˜| isaresolutionofSsuchthatthepreimageofxisarationalcurvenormalwithselfinter- S section−(N−1). Proof. LetEbethepreimageofx. NotethatEisthepropertransformofalineinthecameraplane of the i-th camera. To compute the self intersection of E in S˜ = φ˜−1(S) consider the following embeddings: E֒−i S˜ ֒−j P˜3. → → BytheWhitneysumformula,wehave (ji)∗(c(NE/S˜))=(ji)∗c(NE/P˜3)∩φ˜∗(OP2N(−1)). Aswehavealreadycomputedφ˜∗(OP2N(1)) ∈A•(P˜3),wejusthavetocalculate(ji)∗c(NE/P˜3). ByintersectingEwiththegeneratorsofA2(P˜3)wefind [E]=h2+Q2i +hXTij. j6=i UsingthisidentitytogetherwithourpresentationofA•(P˜3)gives (ji)∗c(NE/P˜3)=[E]−(N−1)h3. PluggingeverythingintotheWhitneysumformulashowsthatthedegreeofc(N )is−(N−1), E/S whichcompletestheproof. 10