Affine parts of abelian surfaces as complete intersection of three quartics

Ane parts of ab elian surfaes as omplete in tersetion of three quartis A. Lesfari Dep artment of Mathematis F aulty of Sien es University of Chouaïb Doukkali B.P. 20, El-Jadida, Mor o   o . E. mail : Lesfariahmedy aho o.fr, Lesfariud.a.ma Abstrat W e onsider an in tegrable system in v e unkno wns ha ving three quartis in v arian ts. W e sho w that the omplex ane v ariet y dened b y putting these in v arian ts equal to generi onstan ts, ompletes in to an ab elian surfae; the jaobian of a gen us t w o h yp erellipti urv e. This system is algebrai ompletely in tegrable and it an b e in tegrated in gen us t w o h yp erellipti funtions. Keywor ds . In tegrable systems, urv es, Kummer surfaes, Ab elian sur- faes. Mathematis Subje t Classi ation (2000) . 70H06, 37J35, 14H70, 14K20, 14H40. 1 In tro dution The problem of nding and in tegrating hamiltonian systems, has attrated a onsiderable amoun t of atten tion in reen t y ears. Beside the fat that man y in tegrable hamiltonian systems ha v e b een on the sub jet of p o w erful and b eautiful theories of mathematis, another motiv ation for its study is: the onepts of in tegrabilit y ha v e b een applied to an inreasing n um b er of ph ysial systems, biologial phenomena, p opulation dynamis,  hemial rate equations, to men tion only a few. Ho w ev er, it seems still hop eless to desrib e or ev en to reognize with an y failit y , those hamiltonian systems whi h are in tegrable, though they are quite exeptional. In this pap er, w e shall b e onerned with nite dimensional algebrai ompletely in tegrable systems. A dynamial system is algebrai ompletely in tegrable if it an b e linearized on a omplex algebrai torus C n /lattice (=ab elian v ariet y). The in v arian ts (often alled rst in tegrals or onstan ts) of the motion are p olynomials and the phase spae o ordinates (or some algebrai funtions of these) restrited 1 to a omplex in v arian t v ariet y dened b y putting these in v arian ts equals to generi onstan ts, are meromorphi funtions on an ab elian v ariet y . More- o v er, in the o ordinates of this ab elian v ariet y , the o ws (run with omplex time) generated b y the onstan ts of the motion are straigh t lines. Ho w ev er, b esides the fat that man y hamiltonian ompletely in tegrable systems p osses this struture, another motiv ation for its study whi h sounds more mo dern is: algebrai ompletely in tegrable systems ome up systematially whenev er y ou study the isosp etral deformation of some linear op erator on taining a rational indeterminate. Therefore there are hidden symmetries whi h ha v e a group theoretial foundation. The onept of algebrai omplete in tegra- bilit y is quite eetiv e in small dimensions and has the adv an tage to lead to global results, unlik e the existing riteria for real analyti in tegrabilit y , whi h, at this stage are p erturbation results. In fat, the o v erwhelming ma jorit y of dynamial systems, hamiltonian or not, are non-in tegrable and p ossess regimes of  haoti b eha vior in phase spae. In the presen t pap er, w e disuss an in teresting in teration b et w een omplex algebrai geometry and dynamial systems. w e presen t an in tegrable system in v e unkno wns ha ving three quartis in v arian ts. This system is algebrai ompletely in tegrable in C 5 , it an b e in tegrated in gen us 2 h yp erellipti funtions. W e sho w that the omplex ane v ariet y B(3) dened b y putting these in v arian ts equal to generi onstan ts, is a double o v er of a Kummer surfae and the system (1) an b e in tegrated in gen us 2 h yp erellipti funtions. W e mak e a are- ful study of the algebrai geometri asp et of the ane v ariet y B(3) of the system (1). W e nd via the P ainlev é analysis the prinipal balanes of the hamiltonian eld dened b y the hamiltonian. T o b e more preise, w e sho w that the system (1) p ossesses Lauren t series solutions in t , whi h dep end on 4 free parameters : α, β , γ and θ . These meromorphi solutions restrited to the surfae B (3) are parameterized b y t w o isomorphi smo oth h yp erellipti urv es H ε = ± i (8) of gen us 2 that in terset in only one p oin t at whi h they are tangen t to ea h other. The ane v ariet y B (3) is em b edded in to P 15 and ompletes in to an ab elian v ariet y e B (the jaobian of a gen us 2 urv e) b y adjoining a divisor D = H i + H − i . The latter has geometri gen us 5 and S = 2 D (v ery ample) has gen us 17. The o w (1) ev olv es on e B and is tan- gen t to ea h h yp erellipti urv e H ε at the p oin t of tangeny b et w een them. Consequen tly , the system (1) is algebrai in tegrable. Ab elian v arieties, v ery hea vily studied b y algebrai geometers, enjo y er- tain algebrai prop erties whi h an then b e translated in to dieren tial equa- tions and their Lauren t solutions. Among the results presen ted in this pap er, there is an expliit alulation of in v arian ts for a hamiltonian system whi h ut out an op en set in an ab elian v ariet y and v arious urv es related to this system are giv en expliitly . The in tegrable dynamial system presen ted here is in teresting, partiular to exp erts of ab elian v arieties who ma y w an t to see expliit examples of a orresp ondene for v arieties dened b y dieren t urv es. 2 2 A v e-dimensional in tegrable system Let us onsider the follo wing system of v e dieren tial equations in the unkno wns z 1 , . . . , z 5 : ˙ z 1 = 2 z 4 , ˙ z 2 = z 3 , ˙ z 3 = − 4 az 2 − 6 z 1 z 2 − 16 z 3 2 , (1) ˙ z 4 = − az 1 − z 2 1 − 8 z 1 z 2 2 + z 5 , ˙ z 5 = − 8 z 2 2 z 4 − 2 az 4 − 2 z 1 z 4 + 4 z 1 z 2 z 3 , where the dot denotes dieren tiation with resp et to the time t . Prop osition 2.1 The system (1) p ossesses thr e e quarti invariants and is  ompletely inte gr able in the sense of Liouvil le. The  omplex ane variety B(3) dene d by putting these invariants e qual to generi  onstants, is a double  over of a Kummer surfa e(4) and the system (1)  an b e inte gr ate d in genus 2 hyp er el lipti funtions. Pr o of . The follo wing three quartis are onstan ts of motion for this system F 1 = 1 2 z 5 + 2 z 1 z 2 2 + 1 2 z 2 3 + 1 2 az 1 + 2 az 2 2 + 1 4 z 2 1 + 4 z 4 2 , F 2 = az 1 z 2 + z 2 1 z 2 + 4 z 1 z 3 2 − z 2 z 5 + z 3 z 4 , (2) F 3 = z 1 z 5 − 2 z 2 1 z 2 2 − z 2 4 . The system (1) an b e written as a hamiltonian v etor eld ˙ z = J ∂ H ∂ z , z = ( z 1 , z 2 , z 3 , z 4 , z 5 ) ⊤ , where H = F 1 . The hamiltonian struture is dened b y the P oisson bra k et { F , H } =  ∂ F ∂ z , J ∂ H ∂ z  = 5 X k ,l =1 J k l ∂ F ∂ z k ∂ H ∂ z l , where ∂ H ∂ z = ( ∂ H ∂ z 1 , ∂ H ∂ z 2 , ∂ H ∂ z 3 , ∂ H ∂ z 4 , ∂ H ∂ z 5 ) ⊤ , and J =       0 0 0 2 z 1 4 z 4 0 0 1 0 0 0 − 1 0 0 − 4 z 1 z 2 − 2 z 1 0 0 0 2 z 5 − 8 z 1 z 2 2 − 4 z 4 0 4 z 1 z 2 − 2 z 5 + 8 z 1 z 2 2 0       , 3 is a sk ew-symmetri matrix for whi h the orresp onding P oisson bra k et satises the Jaobi iden tities. The seond o w omm uting with the rst is regulated b y the equations ˙ z = J ∂ F 2 ∂ z , z = ( z 1 , z 2 , z 3 , z 4 , z 5 ) ⊤ , and is written expliitly as ˙ z 1 = 2 z 1 z 3 − 4 z 2 z 4 , ˙ z 2 = z 4 , ˙ z 3 = z 5 − 8 z 1 z 2 2 − az 1 − z 2 1 , ˙ z 4 = − 2 az 1 z 2 − 4 z 2 1 z 2 − 2 z 2 z 5 , ˙ z 5 = − 4 az 2 z 4 − 4 z 1 z 2 z 4 − 16 z 3 2 z 4 − 2 z 3 z 5 + 8 z 1 z 2 2 z 3 . These v etor elds are in in v olution, i.e., { F 1 , F 2 } = h ∂ F 1 ∂ z , J ∂ F 2 ∂ z i = 0 , and the remaining one is asimir, i.e., J ∂ F 3 ∂ z = 0 . Let B b e the omplex ane v ariet y dened b y B = 2 \ k =1 { z : F k ( z ) = c k } ⊂ C 5 . (3) Sine B is the bre of a morphism from C 5 to C 3 o v er ( c 1 , c 2 , c 3 ) ∈ C 3 , for almost all c 1 , c 2 , c 3 , therefore B is a smo oth ane surfae. Note that σ : ( z 1 , z 2 , z 3 , z 4 , z 5 ) 7− → ( z 1 , z 2 , − z 3 , − z 4 , z 5 ) , is an in v olution on B . The quotien t B /σ is a Kummer surfae dened b y p ( z 1 , z 2 ) z 2 5 + q ( z 1 , z 2 ) z 5 + r ( z 1 , z 2 ) = 0 , (4) where p ( z 1 , z 2 ) = z 2 2 + z 1 , q ( z 1 , z 2 ) = 1 2 z 3 1 + 2 az 1 z 2 2 + az 2 1 − 2 c 1 z 1 + 2 c 2 z 2 − c 3 , r ( z 1 , z 2 ) = − 8 c 3 z 4 2 +  a 2 + 4 c 1  z 2 1 z 2 2 − 8 c 2 z 1 z 3 2 − 2 c 2 z 2 1 z 2 − 4 c 3 z 1 z 2 2 − 1 2 c 3 z 2 1 − 4 ac 3 z 2 2 − 2 ac 2 z 1 z 2 − ac 3 z 1 + c 2 2 + 2 c 1 c 3 . 4 Using F 1 = c 1 (2), w e ha v e z 5 = 2 c 1 − 4 z 1 z 2 2 − z 2 3 − az 1 − 4 az 2 2 − 1 2 z 2 1 − 8 z 4 2 , and substituting this in to F 2 = c 2 , F 3 = c 3 , (2) yields 2 az 1 z 2 + 3 2 z 2 1 z 2 + 8 z 1 z 3 2 − 2 c 1 z 2 + z 2 z 2 3 + 4 az 3 2 + 8 z 5 2 + z 3 z 4 = c 2 , 2 c 1 z 1 − 6 z 2 1 z 2 2 − z 1 z 2 3 − az 2 1 − 4 az 1 z 2 2 − 1 2 z 3 1 − 8 z 1 z 4 2 − z 2 4 = c 3 . (5) W e in tro due t w o o ordinates s 1 , s 2 as follo ws z 1 = − 4 s 1 s 2 , z 2 = s 1 + s 2 , z 3 = ˙ s 1 + ˙ s 2 , z 4 = − 2 ( ˙ s 1 s 2 + s 1 ˙ s 2 ) . Up on substituting this parametrization, (5) turns in to ( s 1 − s 2 )  ( ˙ s 1 ) 2 − ( ˙ s 2 ) 2  + 8 ( s 1 + s 2 )  s 4 1 + s 4 2 + s 2 1 s 2 2  +4 a ( s 1 + s 2 )  s 2 1 + s 2 2  − 2 c 1 ( s 1 + s 2 ) − c 2 = 0 , ( s 1 − s 2 )  s 2 ( ˙ s 1 ) 2 − s 1 ( ˙ s 2 ) 2  + 32 s 1 s 2  s 4 1 + s 4 2 + s 2 1 s 2 2  +32 s 2 1 s 2 2  s 2 1 + s 2 2  + 16 as 1 s 2  s 2 1 + s 2 2  + 16 as 2 1 s 2 2 − 8 c 1 s 1 s 2 − c 3 = 0 . These equations are solv ed linearly for ˙ s 2 1 and ˙ s 2 2 as ( ˙ s 1 ) 2 = − 32 s 6 1 − 16 as 4 1 + 8 c 1 s 2 1 + 4 c 2 s 1 − c 3 4 ( s 2 − s 1 ) 2 , (6) ( ˙ s 2 ) 2 = − 32 s 6 2 − 16 as 4 2 + 8 c 1 s 2 2 + 4 c 2 s 2 − c 3 4 ( s 2 − s 1 ) 2 , and an b e in tegrated b y means of the Ab el transformation H − → J ac ( H ) , where the h yp erellipti urv e H of gen us 2 is giv en b y an equation w 2 = − 32 s 6 − 16 as 4 + 8 c 1 s 2 + 4 c 2 s − c 3 . Consequen tly , the equations (1) are in tegrated in terms of gen us 2 h yp erel- lipti funtions. This establishes the prop osition. 3 Lauren t series solutions and algebrai urv es The in v arian t v ariet y B (3) is a smo oth ane surfae for generi v alues of c 1 , c 2 and c 3 . So, the question I address is ho w do es one nd the ompat- iation of B in to an ab elian surfae? F ollo wing the metho ds in A dler-v an 5 Mo erb ek e [1℄, the idea of the diret pro of is losely related to the geometri spirit of the (real) Arnold-Liouville theorem [8] . Namely , a ompat omplex n -dimensional v ariet y on whi h there exist n holomorphi omm uting v etor elds whi h are indep enden t at ev ery p oin t is analytially isomorphi to a n -dimensional omplex torus C n /Lattice and the omplex o ws generated b y the v etor elds are straigh t lines on this omplex torus. No w, the main problem will b e to omplete B (3) in to a non singular ompat omplex al- gebrai v ariet y e B = B ∪ D in su h a w a y that the v etor elds X F 1 and X F 2 generated resp etiv ely b y F 1 and F 2 , extend holomorphially along a divisor D and remain indep enden t there. If this is p ossible, e B is an algebrai omplex torus (an ab elian v ariet y) and the o ordinates z 1 , . . . , z 5 restrited to B are ab elian funtions. A naiv e guess w ould b e to tak e the natural ompatiation B of B b y pro jetivizing the equations: B = 3 \ k =1 { F k ( Z ) = c k Z 4 0 } ⊂ P 5 . Indeed, this an nev er w ork for a general reason : an ab elian v ariet y e B of dimension bigger or equal than t w o is nev er a omplete smo oth in tersetion, that is it an nev er b e desrib ed in some pro jetiv e spae P n b y n dim e B global p olynomial homogeneous equations. In other w ords, if B is to b e the ane part of an ab elian surfae, B m ust ha v e a singularit y somewhere along the lo us at innit y B ∩ { Z 0 = 0 } . In fat, w e shall sho w that the existene of meromorphi solutions to the dieren tial equations (1) dep ending on 4 free parameters an b e used to man ufature the tori, without ev er going through the deliate pro edure of blo wing up and do wn. Information ab out the tori an then b e gathered from the divisor. Prop osition 3.1 The system (1) p ossesses L aur ent series solutions whih dep end on 4 fr e e p ar ameters : α, β , γ and θ . These mer omorphi solutions r estrite d to the surfa e B(3) ar e p ar ameterize d by two isomorphi smo oth hyp er el lipti urves H ε = ± i (8) of genus 2. Pr o of . The rst fat to observ e is that if the system is to ha v e Lauren t solutions dep ending on 4 free parameters, the Lauren t deomp osition of su h asymptoti solutions m ust ha v e the follo wing form z 1 = 1 t ( z (0) 1 + z (1) 1 t + z (2) 1 t 2 + z (3) 1 t 3 + z (4) 1 t 4 + · · · ) , z 2 = 1 t ( z (0) 2 + z (1) 2 t + z (2) 2 t 2 + z (3) 2 t 3 + z (4) 2 t 4 + · · · ) , z 3 = 1 t 2 ( − z (0) 2 + z (2) 2 t 2 + 2 z (3) 2 t 3 + 3 z (4) 2 t 4 + · · · ) , z 4 = 1 2 t 2 ( − z (0) 1 + z (2) 1 t 2 + 2 z (3 t ) 1 t 3 + 3 z (4) 1 t 4 + · · · ) , z 5 = 1 t 3 ( z (0) 5 + z (1) 5 t + z (2) 5 t 2 + z (3) 5 t 3 + z (4) 5 t 4 + · · · ) . 6 Putting these expansions in to ¨ z 1 = − 2 az 1 − 2 z 2 1 − 16 z 1 z 2 2 + 2 z 5 , ¨ z 2 = − 4 az 2 − 6 z 1 z 2 − 16 z 3 2 , ˙ z 5 = − 8 z 2 2 z 4 − 2 az 4 − 2 z 1 z 4 + 4 z 1 z 2 z 3 , dedued from (1), solving indutiv ely for the z ( j ) k ( k = 1 , 2 , 5) , one nds at the 0 th step (resp. 2 th step) a free parameter α (resp. β ) and the t w o remaining ones γ , θ at the 4 th step. More preisely , w e ha v e z 1 = 1 t ( α − α 2 t + β t 2 + 1 6 α (3 β − 9 α 3 + 4 aα ) t 3 + γ t 4 + · · · ) , z 2 = ε √ 2 4 t (1 + αt + 1 3 ( − 3 α 2 + 2 a ) t 2 + 1 2 (3 β − α 3 ) t 3 − 2 ε √ 2 θ t 4 + · · · ) , z 3 = ε √ 2 4 t 2 ( − 1 + 1 3 ( − 3 α 2 + 2 a ) t 2 + (3 β − α 3 ) t 3 − 6 ε √ 2 θ t 4 + · · · ) , (7) z 4 = 1 2 t 2 ( − α + β t 2 + 1 3 α (3 β − 9 α 3 + 4 aα ) t 3 + 3 γ t 4 + · · · ) , z 5 = 1 t ( − 1 3 aα + α 3 − β + (3 α 4 − aα 2 − 3 αβ ) t +(4 ε √ 2 αθ + 2 γ + 8 3 aα 3 − 1 3 aβ − α 2 β − 3 α 5 − 4 9 a 2 α ) t 2 + · · · ) , with ε = ± i. Using the ma joran t metho d, w e an sho w that the formal Lauren t series solutions are on v ergen t. Substituting the solutions (7) in to F 1 = c 1 , F 2 = c 2 and F 3 = c 3 , and equating the t 0 -terms yields F 1 = 15 8 α 4 − 5 6 aα 2 − 5 4 αβ − 7 36 a 2 − 5 4 ε √ 2 θ = c 1 , F 2 = ε √ 2( 1 4 α 5 − γ + ε √ 2 2 αθ − 2 3 aα 3 + 1 3 aβ + 1 6 a 2 + 1 2 α 2 β ) = c 2 , F 3 = − 11 2 α 6 − β 2 + 4 αγ + 3 α 2 ε √ 2 θ + α 3 β − 1 3 a 2 α 2 + 10 3 aα 4 = c 3 . Eliminating γ and θ from these equations, leads to an equation onneting the t w o remaining parameters α and β : β 2 + 2 3 (3 α 2 − 2 a ) αβ − 3 α 6 + 8 3 aα 4 + 4 9 ( a 2 + 9 c 1 ) α 2 − 2 ε √ 2 c 2 α + c 3 = 0 , (8) A ording to Hurwitz' form ula, this denes t w o isomorphi smo oth h yp erel- lipti urv es H ε ( ε = ± i ) of gen us 2, whi h nishes the pro of of the prop o- sition. 4 Ane part of an ab elian surfae as the jaobian of a gen us t w o h yp erellipti urv e In order to em b ed H ε in to some pro jetiv e spae, one of the k ey underly- ing priniples used is the K o daira em b edding theorem, whi h states that a 7 smo oth omplex manifold an b e smo othly em b edded in to pro jetiv e spae P N with the set of funtions ha ving a p ole of order k along p ositiv e divisor on the manifold, pro vided k is large enough; fortunately , for ab elian v arieties, k need not b e larger than three aording to Lefshetz. These funtions are easily onstruted from the Lauren t solutions (7) b y lo oking for p olynomials in the phase v ariables whi h in the expansions ha v e at most a k-fold p ole. The nature of the expansions and some algebrai proprieties of ab elian v a- rieties pro vide a reip e for when to terminate our sear h for su h funtions, th us making the pro edure implemen table. Preisely , w e wish to nd a set of p olynomial funtions { f 0 , . . . , f N } , of inreasing degree in the original v ari- ables z 1 , . . . , z 5 , ha ving the prop ert y that the em b edding D of H i + H − i in to P N via those funtions satises the relation : geometri gen us (2 D ) ≡ g (2 D ) = N + 2 . A this p oin t, it ma y b e not so lear wh y the urv e D m ust really liv e on an ab elian surfae. Let us sa y , for the momen t, that the equations of the divisor D (i.e., the plae where the solutions blo w up), as a urv e traed on the ab elian surfae e B (to b e onstruted in prop osition 4.2), m ust b e understo o d as relations onneting the free parameters as they app ear rstly in the expansions (7). In the presen t situation, this means that (8) m ust b e understo o d as relations onneting α and β . Let L ( r ) =        p olynomials f = f ( z , . . . , z 5 ) of degre ≤ r , with at w orst a double p ole along H i + H − i and with z 1 , . . . , z 5 as in (7)        / [ F k = c k , k = 1 , 2 , 3] , and let ( f 0 , f 1 , . . . , f N r ) b e a basis of L ( r ) . W e lo ok for r su h that : g (2 D ( r ) ) = N r + 2 , 2 D ( r ) ⊂ P N r . W e shall sho w (prop osition 4.1) that it is unneessary to go b ey ond r=4. Lemma 4.1 The sp a es L ( r ) , neste d a  or ding to weighte d de gr e e, ar e gen- er ate d as fol lows L (1) = { f 0 , f 1 , f 2 , f 3 , f 4 , f 5 } , L (2) = L (1) ⊕ { f 6 , f 8 , f 9 , f 10 , f 11 , f 12 } , L (3) = L (2) , L (4) = L (3) ⊕ { f 13 , f 14 , f 15 } , (9) wher e f 0 = 1 , f 1 = z 1 = α t + . . . , 8 f 2 = z 2 = ε √ 2 t + . . . , f 3 = z 3 = − ε √ 2 4 t 2 + . . . , f 4 = z 4 = − α 2 t 2 + . . . , f 5 = z 5 = − η 3 t + . . . , f 6 = z 2 1 = α 2 t 2 + · · · , f 7 = z 2 2 = − 1 8 t 2 + · · · , f 8 = z 2 5 = η 2 9 t 2 + · · · , f 9 = z 1 z 2 = ε √ 2 α 4 t 2 + · · · , f 10 = z 1 z 5 = − αη 3 t 2 + · · · , f 11 = z 2 z 5 = − ε √ 2 η 12 t 2 + · · · , f 12 = [ z 1 , z 2 ] = − ε √ 2 α 2 2 t 2 + · · · , f 13 = [ z 1 , z 5 ] = 4 α 2 η 3 t 2 + · · · , f 14 = [ z 2 , z 5 ] = ε √ 2 αη 6 t 2 + · · · , f 15 = ( z 3 − 2 ε √ 2 z 2 2 ) 2 = − α 2 2 t 2 + · · · , with [ z j , z k ] = ˙ z j z k − z j ˙ z k , the wr onskien of z k and z j , and η ≡ 3 β − 3 α 3 + aα. Pr o of . The pro of of this lemma is straigh tforw ard and an b e done b y in- sp etion of the expansions (7). Note also that the funtions z 1 , z 2 , z 5 b eha v e as 1 /t and if w e onsider the deriv ativ es of the ratios z 1 /z 2 , z 1 /z 5 , z 2 /z 5 , the wronskiens [ z 1 , z 2 ] , [ z 1 , z 5 ] , [ z 2 , z 5 ] , m ust b eha v e as 1 /t 2 sine z 2 2 , z 2 5 b eha v e as 1 /t 2 . This nishes the pro of of the lemma. Note that dimL (1) = 6 , dimL (2) = dimL (3) = 13 , dimL (4) = 16 . Prop osition 4.1 L (4) pr ovides an emb e dding of D (4) into pr oje tive sp a e P 15 and D (4) (r esp. 2 D (4) ) has genus 5 (r esp. 17). Pr o of . It turns out that neither L (1) , nor L (2) , nor L (3) , yields a urv e of the righ t gen us; in fat g (2 D ( r ) ) 6 = dim L ( r ) + 1 , r = 1 , 2 , 3 . F or instane, the em b edding in to P 5 via L (1) do es not separate the sheets, so w e pro eed to L (2) and w e onsider the orresp onding em b edding in to P 12 . F or nite v alues of α and β , the urv es H i and H − i are disjoin t; dividing the v etor ( f 0 , . . . , f 12 ) b y f 7 and taking the limit t → 0 , to yield [0 : 0 : 0 : 2 ε √ 2 : 4 α : 0 : − 8 α 2 : 1 : − 8 9 η 2 : − 2 ε √ 2 α : 8 3 αη : 2 ε √ 2 3 η : 4 ε √ 2 α 2 ] . 9 The urv e (8) has t w o p oin ts o v ering α = ∞ , at whi h η ≡ 3 β − 3 α 3 + aα b eha v es as follo ws : η = − 6 α 3 + 3 aα ± 3 q 4 α 6 − 4 aα 4 − 4 c 1 α 2 + 2 ε √ 2 c 2 α − c 3 , = ( − 3( a 2 +4 c 1 ) 4 α + lo w er order terms , pi king the + sign , − 12 α 3 + O ( α ) , pi king the - sign . Then b y pi king the - sign and b y dividing the v etor ( f 0 , . . . , f 12 ) b y f 8 , the orresp onding p oin t is mapp ed in to the p oin t [0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 1 : 0 : 0 : 0 : 0] , in P 12 whi h is indep enden t of ε, whereas pi king the + sign leads to t w o dieren t p oin ts, aording to the sign of ε. Th us, adding at least 2 to the gen us of ea h urv e, so that g (2 D (2) ) − 2 > 12 , 2 D (2) ⊂ P 12 6 = P g − 2 , whi h on tradits the fat that N r = g (2 D (2) ) − 2 . The em b edding via L (2) (or L (3) ) is unaeptable as w ell. Consider no w the em b edding 2 D (4) in to P 15 using the 16 funtions f 0 , . . . , f 15 of L (4) (9). It is easily seen that these funtions separate all p oin ts of the urv e (exept p erhaps for the p oin ts at ∞ ) : The urv es H i and H − i are disjoin t for nite v alues of α and β ; dividing the v etor ( f 0 , . . . , f 15 ) b y f 7 and taking the limit t → 0 , to yield [0 : 0 : 0 : 2 ε √ 2 : 4 α : 0 : − 8 α 2 : 1 : − 8 9 η 2 : − 2 ε √ 2 α : 8 3 αη : 2 ε √ 2 3 η : 4 ε √ 2 α 2 : − 32 3 α 2 η : − 4 ε √ 2 3 αη : 4 α 2 ] . Ab out the p oin t α = ∞ , it is appropriate to divide b y g 8 ; then b y pi king the sign - in η ab o v e, the orresp onding p oin t is mapp ed in to the p oin t [0 : 0 : 0 : 0 : 0 : 0 : 0 : 0 : 1 : 0 : 0 : 0 : 0 : 0 : 0 : 0] , in P 15 whi h is indep enden t of ε, whereas pi king the + sign leads to t w o dieren t p oin ts, aording to the sign of ε. Hene, the divisor D (4) obtained in this w a y has gen us 5 and th us g (2 D (4) ) has gen us 17 and 2 D (4) ⊂ P 15 = P g − 2 , as desired. This ends the pro of of the prop osition. 10 Let L = L (4) , D = D (4) , and S = 2 D (4) ⊂ P 15 . Next w e wish to onstrut a surfae strip around S whi h will supp ort the omm uting v etor elds. In fat, S has a go o d  hane to b e v ery ample divisor on an ab elian surfae, still to b e onstruted. Prop osition 4.2 The variety B (3) generi al ly is the ane p art of an ab elian surfa e e B , mor e pr e isely the ja obian of a genus 2 urve. The r e du e d divisor at innity e B \ B = H i + H − i ,  onsists of two smo oth isomorphi genus 2 urves H ε (8). The system of dif- fer ential e quations (1) is algebr ai  omplete inte gr able and the  orr esp onding ows evolve on e B . Pr o of . W e need to atta hes the ane part of the in tersetion of the three in v arian ts (2) so as to obtain a smo oth ompat onneted surfae in P 15 . T o b e preise, the orbits of the v etor eld (1) running through S form a smo oth surfae Σ near S su h that Σ \ B ⊆ e B , and the v ariet y e B = B ∪ Σ , is smo oth, ompat and onneted. Indeed, let ψ ( t, p ) = { z ( t ) = ( z 1 ( t ) , . . . , z 5 ( t )) : t ∈ C , 0 < | t | < ε } , b e the orbit of the v etor eld (1) going through the p oin t p ∈ S . Let Σ p ⊂ P 15 b e the surfae elemen t formed b y the divisor S and the orbits going through p . Consider the urv e S ′ = H ∩ Σ , where H ⊂ P 15 is a h yp erplane transv ersal to the diretion of the o w and Σ ≡ ∪ p ∈S Σ p . If S ′ is smo oth, then using the impliit funtion theorem the surfae Σ is smo oth. But if S ′ is singular at 0 , then Σ w ould b e singular along the tra jetory ( t − axis) whi h go immediately in to the ane part B. Hene, B w ould b e singular whi h is a on tradition b eause B is the bre of a morphism from C 5 to C 2 and so smo oth for almost all the three onstan ts of the motion c k . Next, let B b e the pro jetiv e losure of B in to P 5 , let 11 Z = [ Z 0 : Z 1 : . . . : Z 5 ] ∈ P 5 and let I = B ∩ { Z 0 = 0 } b e the lo us at innit y . Consider the map B ⊆ P 5 − → P 15 , Z 7− → f ( Z ) , where f = ( f 0 , f 1 , ..., f 15 ) ∈ L ( S ) (9) and let e B = f ( B ) . In a neigh b ourho o d V ( p ) ⊆ P 15 of p , w e ha v e Σ p = e B and Σ p \S ⊆ B . Otherwise there w ould exist an elemen t of surfae Σ ′ p ⊆ e B su h that Σ p ∩ Σ ′ p = ( t − axis ) , orbit ψ ( t, p ) = ( t − axis ) \ p ⊆ B , and hene B w ould b e singular along the t − axis whi h is imp ossible. Sine the v ariet y B ∩ { Z 0 6 = 0 } is irreduible and sine the generi h yp erplane setion H g en. of B is also irreduible, all h yp erplane setions are onneted and hene I is also onneted. No w, onsider the graph Γ f ⊆ P 5 × P 15 of the map f , whi h is irreduible together with B . It follo ws from the irreduibilit y of I that a generi h yp erplane setion Γ f ∩ { H g en. × P 15 } is irreduible, hene the sp eial h yp erplane setion Γ f ∩ {{ Z 0 = 0 } × P 15 } is onneted and therefore the pro jetion map pr oj P 15 { Γ f ∩ {{ Z 0 = 0 } × P 15 }} = f ( I ) ≡ S , is onneted. Hene, the v ariet y B ∪ Σ = e B is ompat, onneted and em b eds smo othly in to P 15 via f . W e wish to sho w that e B is an ab elian surfae equipp ed with t w o ev erywhere indep enden t omm uting v etor elds. F or doing that, let φ τ 1 and φ τ 2 b e the o ws orresp onding to v etor elds X F 1 and X F 2 . The latter are generated resp etiv ely b y F 1 and F 2 . F or p ∈ S and for small ε > 0 , φ τ 1 ( p ) , ∀ τ 1 , 0 < | τ 1 | < ε, is w ell dened and φ τ 1 ( p ) ∈ B . Then w e ma y dene φ τ 2 on B b y φ τ 2 ( q ) = φ − τ 1 φ τ 2 φ τ 1 ( q ) , q ∈ U ( p ) = φ − τ 1 ( U ( φ τ 1 ( p ))) , where U ( p ) is a neigh b ourho o d of p . By omm utativit y one an see that φ τ 2 is indep enden t of τ 1 ; φ − τ 1 − ε 1 φ τ 2 φ τ 1 + ε 1 ( q ) = φ − τ 1 φ − ε 1 φ τ 2 φ τ 1 φ ε 1 , = φ − τ 1 φ τ 2 φ τ 1 ( q ) . W e arm that φ τ 2 ( q ) is holomorphi a w a y from S . This b eause φ τ 2 φ τ 1 ( q ) is holomorphi a w a y from S and that φ τ 1 is holomorphi in U ( p ) and maps bi-holomorphially U ( p ) on to U ( φ τ 1 ( p )) . No w, sine the o ws φ τ 1 and φ τ 2 are holomorphi and indep enden t on S , w e an sho w along the same lines as in the Arnold-Liouville theorem [1,6℄ that e B is a omplex torus C 2 /lattice 12 and so in partiular e B is a Kähler v ariet y . And that will done, b y onsidering the lo al dieomorphism C 2 − → e B , ( τ 1 , τ 2 ) 7− → φ τ 1 φ τ 2 ( p ) , for a xed origin p ∈ B . The additiv e subgroup { ( τ 1 , τ 2 ) ∈ C 2 : φ τ 1 φ τ 2 ( p ) = p } , is a lattie of C 2 , hene C 2 /lattice − → e B , is a biholomorphi dieomorphism and e B is a Kähler v ariet y with Kähler metri giv en b y dτ 1 ⊗ dτ 1 + dτ 2 ⊗ dτ 2 . No w, a ompat omplex Kähler v ariet y ha ving the required n um b er as (its dimension) of indep enden t meromorphi funtions is a pro jetiv e v ariet y [11℄. In fat, here w e ha v e e B ⊆ P 15 . Th us e B is b oth a pro jetiv e v ariet y and a omplex torus C 2 /lattice and hene an ab elian surfae as a onsequene of Cho w theorem. By the lassiation theory of ample line bundles on ab elian v arieties, e B ≃ C 2 /L Ω with p erio d lattie giv en b y the olumns of the matrix  δ 1 0 a c 0 δ 2 c b  , I m  a c c b  > 0 , and δ 1 δ 2 = g ( H ε ) − 1 = 1 , implying δ 1 = δ 2 = 1 . Th us e B is prinipally p olarized and it is the jaobian of the h yp erellipti urv e H ε . This ompletes the pro of of the prop osition. Remark 4.1 W e have se en that the r ee tion σ on the ane variety B amounts to the ip σ : ( z 1 , z 2 , z 3 , z 4 , z 5 ) 7− → ( z 1 , z 2 , − z 3 , − z 4 , z 5 ) , hanging the dir e tion of the  ommuting ve tor elds. It  an b e extende d to the (-Id)-involution ab out the origin of C 2 to the time ip ( t 1 , t 2 ) 7→ ( − t 1 , − t 2 ) on e B , wher e t 1 and t 2 ar e the time  o or dinates of e ah of the ows X F 1 and X F 2 . The involution σ ats on the p ar ameters of the L aur ent solution (7) as fol lows σ : ( t, α, β , γ , θ , ε ) 7− → ( − t, − α, − β , − γ , − θ , − ε ) , inter hanges the urves H ε = ± i (8) and the line ar sp a e L  an b e split into a dir e t sum of even and o dd funtions. Ge ometri al ly, this involution inter- hanges H i and H − i , i.e., H − i = σ H i . 13 Remark 4.2 Consider on e B the holomorphi 1-forms dt 1 and dt 2 dene d by dt i ( X F j ) = δ ij , wher e X F 1 and X F 2 ar e the ve tor elds gener ate d r esp e tively by F 1 and F 2 . T aking the dier entials of ζ = 1 /z 1 and ξ = z 1 /z 2 viewe d as funtions of t 1 and t 2 , using the ve tor elds and the L aur ent series (7) and solving line arly for dt 1 and dt 2 , we obtain as exp e te d the hyp er el lipti holomorphi dier entials ω 1 = dt 1 | H ε , = 1 △ ( ∂ ξ ∂ t 2 dζ − ∂ ζ ∂ t 2 dξ ) | H ε , = αdα p P ( α ) , ω 2 = dt 2 | H ε , = 1 △ ( − ∂ ξ ∂ t 1 dζ − ∂ ζ ∂ t 1 dξ ) | H ε , = √ 2 dα 2 p P ( α ) , with P ( α ) ≡ 4 α 6 − 4 aα 4 − 4 c 1 α 2 + 2 ε √ 2 c 2 α − c 3 , and ∆ ≡ ∂ ζ ∂ t 1 ∂ ξ ∂ t 2 − ∂ ζ ∂ t 2 ∂ ξ ∂ t 1 . The zer o es of ω 2 pr ovide the p oints of tangeny of the ve tor eld X F 1 to H ε . W e have ω 1 ω 2 = − ε √ 2 α, and X F 1 is (doubly) tangent to H ε at the p oint  overing α = ∞ , i.e., wher e b oth the urves touh. Referenes [1℄ M. A d ler, P. van Mo erb eke. The omplex geometry of the K o w alewski- P ainlev é analysis. In v en t. Math. 97 (1989) 3-51. [2℄ A.I. Belokolos, V.Z. Bob enko, V.Z. Enol'skii, A.R. Its, V.B. Matve ev. Algebro-Geometri approa h to nonlinear in tegrable equa- tions. Springer-V erlag 1994. [3℄ P.A. Griths, J. Harris. Priniples of algebrai geometry . Wiley- In tersiene 1978. [4℄ L. Haine. Geo desi o w on S O (4) and Ab elian surfaes. Math. Ann. 263 (1983) 435-472. [5℄ A. L esfari. Ab elian surfaes and K o w alewski's top. Ann. Sien t. Éole Norm. Sup. P aris sér. 4, 21 (1988) 193-223. 14 [6℄ A. L esfari. Completely in tegrable systems : Jaobi's heritage. J. Geom. Ph ys. 31 (1999) 265-286. [7℄ A. L esfari. Le théorème d'Arnold-Liouville et ses onséquenes. Elem. Math. 58 (2003) 6-20. [8℄ A. L esfari. 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