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schwarz maps for hypergeometric systems in single and several variables are studied by several authors ( cf . @xcite ) for more than hundred years . these systems treated were irreducible , maybe because specialists believed that reducible systems would not give interesting schwarz maps . we study in this paper appell s hypergeometric system @xmath0 of rank four when its parameters satisfy @xmath2 or @xmath3 . in this case , the system @xmath0 is reducible , and has a @xmath4-dimensional subsystem isomorphic to appell s @xmath5 ( proposition [ prop : s2 ] ) . if @xmath6 then @xmath0 has two such subsystems . by proposition [ prop : s2 g ] , the intersection of these subsystems is equal to the gauss hypergeometric equation . as a consequence , we have inclusions on @xmath0 , two @xmath5 s and @xmath7 ( theorem [ matome ] ) . we give the monodromy representation of the system @xmath0 which can be specialized to the case @xmath6 in theorem [ th : monod - rep ] . as for explicit circuit matrices with respect to a basis @xmath8 , see corollary [ cor : monod - matrix ] . we further specialize the parameters of the system @xmath0 as @xmath9 in  [ schmap ] . in this case , the restriction of its monodromy group to the invariant subspace is arithmetic and isomorphic to the triangle group of type @xmath10 $ ] . we show that its schwarz map admits geometric interpretations : the map can be considered as the universal abel - jacobi map of a 1-dimensional family of curves of genus 2 in theorem [ th : gen - schwarz ] . the system @xmath0 is equivalent to the restriction of a hypergeometric system @xmath11 to a two dimensional stratum in the configuration space @xmath12 of six lines in the projective plane . in appendix [ 3-dim - s ] , we study a system of hypergeometric differential equations in three variables , which is obtained by restricting @xmath11 to the three dimensional strata corresponding to configurations only with one triple point . the methods to prove proposition [ prop : s2 ] are also applicable to this system under a reducibility condition . in appendix [ genus2 ] , we classify families of genus @xmath13 branched coverings of the projective line , whose period maps yield triangle groups .    in a forthcoming paper @xcite , we study this schwarz map using period domains for mixed hodge structures . moreover , we explicitly give its inverse in terms of theta functions . gauss hypergeometric series @xmath14 where @xmath15 , admits an integral representation : @xmath16 the function @xmath17 is a solution of the hypergeometric equation @xmath18 where @xmath19 the collection of solutions is denoted by @xmath20 . appell s hypergeometric series @xmath21 admits an integral representation : @xmath22 the function @xmath23 is a solution of the hypergeometric system @xmath24   ( d'(c-1+d+d')-y(a+d+d')(b'+d'))z=0 ,   \end{array } \right.\ ] ] where @xmath25 , which can be written as @xmath26 where @xmath27 & q_1(a , b , b',c;x , y)=y(1-y)\partial_{yy}+x(1-y)\partial_{yx }      + ( c-(a+b'+1)y)\partial_y - b'x\partial_x - ab ' , & \\[2 mm ] & r_1(a , b , b',c;x , y)=(x - y)\partial_{xy}-b'\partial_x+b\partial_y , \end{aligned}\ ] ] and @xmath28 , etc . the last equation @xmath29 is derived from the integrability condition of the first two equations . the collection of solutions is denoted by @xmath30 . appell s hypergeometric series @xmath31 admits an integral representation : @xmath32@xmath33 the function @xmath34 satisfies the system @xmath35 where @xmath36 & & q_2(a , b , b',c , c';x , y)=d'(c'-1+d')-y(a+d+d')(b'+d ) . \end{aligned}\ ] ] the collection of solutions is denoted by @xmath37 .      as for the reducibility of the systems @xmath0 and @xmath5 , the following is known :    [ redf2]@xmath38@xcite@xmath39 appell s system @xmath40 is reducible if and only if at least one of @xmath41 is an integer . [ redf1]@xmath38@xcite@xmath39 appell s system @xmath42 is reducible if and only if at least one of @xmath43 is an integer . the system @xmath40 is reducible when @xmath44 , fact [ redf2 ] . in fact , we see that the system @xmath45 is a subsystem of @xmath46 ; precisely , we have    [ prop : s2 ] @xmath47    we give three `` proof ' 's : one using power series , subsection [ subsec : power ] , one using integral representations , subsection [ subsec : integ ] , and one manipulating differential equations , subsection [ subsec : equat ] . the former two are valid only under some non - integral conditions on parameters , which we do not give explicitly . though the last one is valid for any parameters , it would be not easy to get a geometric meaning . the following fact explains the inclusion in proposition [ prop : s2 ] . [ bailey1 ] @xmath48      we consider the integral @xmath49 which is a solution of the system @xmath50 . we change the coordinate @xmath51 into @xmath52 as @xmath53 which sends @xmath54 the inverse map is @xmath55 since @xmath56 we have @xmath57 this implies , if @xmath58 , then the double integral above becomes the product of the beta integral @xmath59 and the integral @xmath60 which is an element of the space @xmath61 . this shows @xmath62 which is equivalent to @xmath63    the bi - rational coordinate change @xmath64 is so made that the lines defining the integrand of the integral @xmath65 may become the union of vertical lines and horizontal lines in the @xmath66-space . actual blow - up and down process is as follows ( see figure [ st ] ) . name the six lines in the @xmath67-projective plane as : @xmath68 blow up at the 4 points ( shown by circles ) @xmath69 and blow - down along the proper transforms of the line @xmath70 and two lines : @xmath71 these three lines are dotted . this takes the @xmath67-projective plane to @xmath72 . in the figure , lines labeled @xmath73 stand for @xmath74 , and the lines labeled @xmath75 on the right are the blow - ups of the intersection points @xmath76 , respectively . the line obtained by blowing up the point @xmath77 is the line defined by @xmath78 , which should be labeled by @xmath79 .    , width=491 ]      a proof of the inclusion in proposition [ prop : s2 ] that is valid for any parameters is done as follows . let @xmath80 be a solution of the system @xmath81 . then , the system @xmath82 yields @xmath83-linear expressions of @xmath84 and @xmath85 in terms of @xmath86 and @xmath80 . substitute these expressions into the system @xmath87 . then , we get two linear forms in @xmath86 and @xmath80 . we now have only to see their coefficients vanish for the given parameters after a change of coordinates and a change of the unknown by multiplying a simple factor . we do not here present the actual computation , because if we put @xmath88 in the proof of proposition [ x3fd ] in subsection [ secondproof ] , manipulating differential equations , it gives essentially a proof of proposition [ prop : s2 ] . when @xmath90 , applying proposition 1.3 also for @xmath91 , we see that the system @xmath92 has two subsystems isomorphic to @xmath5 . the intersection of the two @xmath5 s would be the gauss hypergeometric equation . in fact , we have the following proposition .    [ prop : s2 g ] @xmath93    similar to the argument of the previous subsection , we can give three `` proof ' 's : one using power series , one using integral representations , and one manipulating differential equations . we give a sketch of them in the following . the following identity explains the inclusion above . [ bailey2 ] @xmath94 we continue the argument in [ subsec : integ ] . in the integral @xmath95 in  [ subsec : integ ] above , change the coordinate from @xmath96 to @xmath97 as @xmath98 sending @xmath99 since @xmath100 we have @xmath101 this implies , if @xmath91 , then @xmath102 this shows @xmath103 which of course implies the inclusion relation in proposition [ prop : s2 g ] by combination with that in proposition [ prop : s2 ] .      put @xmath104 we have @xmath105 and so on . assume that @xmath106 . the equation @xmath107 gives a linear expression of @xmath108 in @xmath109 and @xmath110 . substitute these expressions in @xmath111 and we get the product of @xmath112 and a @xmath83-linear combination of @xmath110 and @xmath109 . the coefficients vanish if @xmath91 . if we do the same for @xmath113 , then we find that it vanishes when @xmath58 . we show that some indefinite integrals solve the system @xmath92 . we begin with some well - known facts . [ wellknown1 ] @xmath114 where @xmath115    note that @xmath116 this implies @xmath117 since @xmath118 we have @xmath119    & = \dsp b x(d_s+1)\frac{1-s}{x - s}\phi \\[4 mm ]    & = \dsp b x\frac{\partial}{\partial s}\left(\frac{s(1-s)}{x - s}\phi\right ) . \end{array}\ ] ]    [ wellknown2 ] the indefinite integral @xmath120 solves @xmath121 . in particular , @xmath122 .    since @xmath123 , we have @xmath124 lemma [ wellknown1 ] leads to @xmath125 let @xmath126 and @xmath127 be the operators generating the system @xmath121 : @xmath128 q_1(a,0,b , c;s , t ) &   = p(a , b , c;t)/t+s(1-t)\partial_{st}-bs\partial_s,\\[2 mm ] r_1(a,0,b , c;s , t)&=(s - t)\partial_{st}-b\partial_s;\end{array}\ ] ] refer to section 1 . note that @xmath129 . by using the above identities , we have @xmath130 and @xmath131 @xmath132 furthermore , for @xmath133 , the forms of operators above imply that @xmath80 lies in @xmath134 . we now use the following fact :    @xcite . [ bailey3 ] @xmath135    from this fact we get , when @xmath136 , @xmath137 if we put @xmath138 then @xmath139 thus we have @xmath140 & = ( 1-y)^{-b'}(1-x)^{-b}s_1\left(b,0,b',a;\dsp\frac{-x}{1-x},\frac{xy}{(1-x)(1-y)}\right)\\[3 mm ] & \supset ( 1-y)^{-b'}(1-x)^{-b } s\left(b , b',a;\dsp\frac{xy}{(1-x)(1-y)}\right ) . \end{array}\ ] ] this agrees with the inclusion in proposition [ prop : s2 g ] . in particular , by the inclusion @xmath141 and lemma [ wellknown2 ] we get solution of @xmath142 represented by the indefinite integral : @xmath143 starting point of the path of integration can be any point @xmath144 , so we choose @xmath145 , just for simplicity . by exchanging the role of @xmath146 and @xmath147 , we get an inclusion @xmath148 and another solution of @xmath142 represented by the indefinite integral : @xmath149 after the change @xmath150 , it can be also expressed as @xmath151 thus , we have :    [ matome ] we have the following inclusions of the spaces of solutions : @xmath152    moreover , the collection of solutions @xmath153 is spanned by @xmath154 and @xmath155    this will play a key role to understand the schwarz map of a system @xmath0 with specific parameters , which will be introduced in  [ e36 ] . in this section , we study the monodromy representation of @xmath156 . though it is assumed in @xcite that the parameters satisfy the irreducibility condition in fact 1.1 : @xmath157 in this section , we only assume the weaker condition @xmath158 we modify theorem 7.1 in @xcite so that the statements remain valid for these parameters . we will apply the result of this section in [ special ] . let @xmath159 be the complement of the singular locus of @xmath0 . for each @xmath160 , we consider a multi - valued function @xmath161 on @xmath162    as in  1.6 and  1.7 of chapter 2 in @xcite , we define the twisted homology group @xmath163 associated with @xmath164 and locally finite one @xmath165 . under some genericity condition , the integral of @xmath164 over a twisted cycle gives a solution of @xmath0 . if @xmath166 then the natural map @xmath167 is bijective , and the inverse map @xmath168 is called the regularization . in general , the map @xmath169 is neither injective nor surjective , however we still have the isomorphism @xmath170    under condition ( 3.1 ) , thanks to the vanishing theorem of cohomology groups in @xcite , rank of @xmath165 and @xmath163 are equal to the euler number @xmath171 of @xmath172 , and the bilinear form  _ the intersection form _ @xmath173 is non - degenerate . let @xmath174 be a small simply connected domain in @xmath175 . we can identify the local solution space to @xmath0 on @xmath174 with the trivial vector bundle @xmath176 via the euler type integral representation of solutions to @xmath0 . the monodromy representation of @xmath0 is equivalent to that of the local system @xmath177 over @xmath175 . we also consider a local system @xmath178 over @xmath175 . we fix a small positive real number @xmath179 , and let @xmath180 a base point in @xmath175 . denote the germs at this point of the local systems @xmath181 and @xmath182 by @xmath183 respectively . let @xmath184 be the monodromy representations of @xmath181 and @xmath182 with respect to @xmath180 . @xmath185 and @xmath186 are called the circuit transformations along @xmath187 . [ prop : inv - subspace ]    * the image @xmath188 of the natural map @xmath189 is invariant under the monodromy representation @xmath190 . * the kernel @xmath191 of the natural map @xmath192 is invariant under the monodromy representation @xmath193 . it is clear that @xmath194 and @xmath195 are invariant under @xmath190 and @xmath193 , respectively . we have only to note that the natural maps @xmath196 and @xmath197 commute with @xmath190 and @xmath193 . we will see that if @xmath2 or @xmath3 , then both of @xmath188 and @xmath191 are proper subspaces . thus monodromy representations @xmath190 and @xmath193 are reducible in this case . [ lem : duality ]    1 . let @xmath198 and @xmath199 be elements of @xmath200 and @xmath195 , respectively . then we have @xmath201 2 . suppose that @xmath202 is decomposed into the direct sum of the eigenspaces @xmath203 of @xmath204 of eigenvalues @xmath205 . then @xmath206 is decomposed into the direct sum of the eigenspaces @xmath207 of @xmath208 of eigenvalues @xmath209 . the eigenspace @xmath210 is characterized as @xmath211    \(1 ) the intersection number @xmath212 is stable under small deformations of @xmath198 and @xmath199 . \(2 ) since @xmath204 belongs to the general linear group , we have @xmath213 . let @xmath214 be any element of @xmath215 @xmath216 and let @xmath217 be any element of @xmath218 . since we have @xmath219 for @xmath220 , @xmath221 belongs to the space @xmath222 . thus @xmath208 induces a linear transformation of @xmath222 . let @xmath223 be an eigenvalue of the restriction of @xmath208 to @xmath222 , and let @xmath224 be an eigenvector of @xmath223 . since the intersection from @xmath225 is non degenerate , there exists @xmath226 such that @xmath227 . note that @xmath228 hence we have @xmath229 , i.e. , @xmath230 and @xmath231 . similarly , we have @xmath232 for @xmath233 . to show @xmath234 , we consider the restriction of @xmath208 to @xmath235 , where @xmath236 take its eigenvector @xmath237 and repeat the above argument . in this way , we have @xmath238 since @xmath239 we have @xmath240 for @xmath241 . let @xmath242 @xmath243 be locally finite chains shown in figure [ fig : basis ] . we specify a branch of @xmath164 on each chain by the assignment of @xmath244 on it as in table [ tab : arg ] , where @xmath245 and load @xmath164 to get the locally finite twisted cycles @xmath246 . -cycles , width=377 ]        .list of @xmath244 on @xmath247 [ cols="^,^,^,^,^,^",options="header " , ]     it will be shown that @xmath248 form a basis of @xmath200 in corollary [ cor : basis ] . we choose elements in @xmath195 as @xmath249 @xmath250 where @xmath251 and @xmath252 note that in terms of @xmath253 s the irreducible condition in fact 1.1 is @xmath254 the condition ( 3.1 ) is @xmath255 and ( see figure [ p2fig ] ) @xmath256 and that @xmath257 satisfies ( 3.1 ) and @xmath258 for any @xmath259 . explicitly , @xmath260 and @xmath261 can be written as : @xmath262- \frac{\circlearrowleft_{0}^-}{\mu_1^{-1}-1}\big)\times   \big(\frac{\circlearrowleft_0^+}{\mu_3^{-1}-1 } + [ \delta,1-\delta]- \frac{\circlearrowleft_{1}^-}{\mu_4^{-1}-1}\big)\big]^{\psi^{-1}},\ ] ] @xmath263- \frac{\circlearrowleft_{1}^-}{\mu_2^{-1}-1}\big ) \times   \big(\frac{\circlearrowleft_\infty^+}{\mu_{345}-1 } + [ \frac{-1}{\delta},-\delta]- \frac{\circlearrowleft_{0}^-}{\mu_3^{-1}-1}\big)\big]^{\psi^{-1}},\ ] ] where @xmath264 is a small positive real number , @xmath265 $ ] and @xmath266 $ ] are closed intervals , @xmath267 is the negatively oriented circle of which radius , center and terminal are @xmath268 , @xmath269 and @xmath270 , @xmath271 @xmath272 is the positively oriented circle of which radius , center and terminal are @xmath264 , @xmath273 and @xmath274 . notice that the definition of the twisted cycles @xmath275 @xmath276 make sense even in the case @xmath2 or @xmath3 . indeed , this specialization gives no harm to @xmath277 @xmath278 , and thanks to the above expression , when @xmath279 and @xmath280 , we have    @xmath281- \frac{\circlearrowleft_{1}^-}{\mu_4^{-1}-1}\big)\big]^{\psi^{-1 } } , \\ { { \widetilde{\delta}}_4}&= & \big [ \big(\frac{\circlearrowleft_0^+}{\mu_1^{-1}-1 } + [ \delta,1-\delta]- \frac{\circlearrowleft_{1}^-}{\mu_2^{-1}-1}\big ) \times   \circlearrowleft_\infty^+\big]^{\psi^{-1}},\end{aligned}\ ] ]    respectively . [ rem : kernel ] suppose that @xmath282 . then the twisted cycles @xmath283 and @xmath284 are homologous to @xmath269 in @xmath285 , since they are the boundary of locally finite @xmath4-chains given by the replacement @xmath286 in their expressions , where @xmath287 is the annulus @xmath288 . they belong to @xmath191 . by proposition [ prop : int - mat ] , it turns out that @xmath188 is spanned by @xmath289 and @xmath290 . [ prop : int - mat ] the intersection matrix @xmath291 for @xmath292,@xmath293,@xmath294 and @xmath295,@xmath293,@xmath296 is given by @xmath297 its determinant is @xmath298 which does not vanish under the assumption @xmath299 . follow  3 of chapter viii in @xcite for the computation of the intersection numbers . by a straightforward calculation , we have its determinant .    proposition [ prop : int - mat ] yields the following corollary . [ cor : basis ] the twisted cycles @xmath292,@xmath293,@xmath300 and @xmath295,@xmath293,@xmath301 form a basis of @xmath200 and that of @xmath195 , respectively . we express the twisted cycles @xmath302 and @xmath303 as linear combinations of @xmath304 . [ lem : squares ] we have @xmath305    set @xmath306 and compute the intersection numbers @xmath307 . then we have @xmath308 which yields the expression of @xmath302 .    since @xmath309 we have the expression of @xmath303 .    in @xcite , we take a basis @xmath292 , @xmath310 , @xmath302 and @xmath303 of @xmath200 . if @xmath6 then each of @xmath302 and @xmath303 is a scalar multiple of @xmath292 by lemma [ lem : squares ] . similar to lemma [ lem : squares ] , we have the following .    [ lem : dual - squares ] we have @xmath311        we give generators of the fundamental group @xmath312 . let @xmath313 ( and @xmath314 , @xmath315 ) be a loop in @xmath316 starting from @xmath317 , approaching to the point @xmath318,(and @xmath319 , @xmath320 ) with @xmath321 , turning once around the point positively , and tracing back to @xmath180 . let @xmath322 ( and @xmath323 ) be a loop in @xmath324 starting from @xmath317 , approaching to the point @xmath325 , ( and @xmath326 ) with @xmath327 , turning once around the point positively , and tracing back to @xmath180 . it is known that the loops @xmath328 generate @xmath312 . the circuit matrices of @xmath190 and @xmath329 along @xmath330 will be denoted as @xmath331      [ th : monod - rep ] under the assumption ( 3.1 ) , the circuit transformations @xmath332 @xmath333 are given as @xmath334 @xmath335 where @xmath336 is any element of @xmath200 , @xmath337 @xmath338 is the submatrix of @xmath339 consisting of the @xmath340 , @xmath341 , @xmath342 and @xmath343 entries of @xmath339 , and @xmath344 @xmath345    under the condition @xmath346 , the linear transformation @xmath347 satisfies the assumption of lemma [ lem : duality ] ( 2 ) . in fact , a fundamental system of solutions can be given by the hypergeometric series @xmath348 multiplied by the power functions @xmath349 thus the eigenvalues of @xmath347 are @xmath1 and @xmath350 , and each of the eigenspaces is two dimensional . it is easy to see that the locally finite chains @xmath351 and @xmath352 are invariant under the deformation along @xmath313 . hence the twisted cycles @xmath289 and @xmath353 span the eigenspace of @xmath347 of eigenvalue @xmath1 . similarly we can show that the twisted cycles @xmath354 and @xmath355 span the eigenspace of @xmath356 of eigenvalue @xmath1 . by lemma [ lem : duality ] ( 2 ) , the eigenspace of @xmath347 of eigenvalue @xmath357 is @xmath358 it is easy to see that the right hand side @xmath359 of ( [ eq : monod1 ] ) satisfies @xmath360 by proposition [ prop : int - mat ] , @xmath8 form a basis even in the case @xmath361 or @xmath362 . thus the representation matrix @xmath363 of @xmath347 with respect to this basis is continuous on the parameters @xmath364 . on the other hand , the expression @xmath365 is also continuous since the factor @xmath366 in the denominator of @xmath367 \dfrac{\mu_4 - 1}{\mu_{34}(\mu_5 - 1 ) } & \dfrac{\mu_{34}-1}{\mu_{34}(\mu_5 - 1 ) } \end{pmatrix}.\ ] ] cancels by @xmath368 . similarly we have the expression of @xmath369 . to study @xmath370 , we work temporarily under the condition @xmath371 . we decompose @xmath315 into @xmath372 , where @xmath373 is the approach to @xmath374 and @xmath375 is the turning path . we trace the deformation of the triangle @xmath376 made by @xmath377 and @xmath378 along @xmath373 . after the deformation , this becomes a small triangle near the point @xmath379 . we see the argument of @xmath380 on this triangle . since @xmath381 and @xmath382 in @xmath373 , @xmath383 in this triangle , and @xmath384 on the line @xmath385 , @xmath380 varies negative to positive via the upper - half space , i.e , @xmath386 decreases by @xmath387 along @xmath373 . note that this change is compatible with our assignment of @xmath386 on @xmath377 and @xmath378 in table [ tab : arg ] . thus the twisted cycle @xmath388 plays the role of a vanishing cycle as the line @xmath389 approaches the point @xmath390 . since @xmath375 corresponds to the move of @xmath391 turning around the point @xmath392 the cycle @xmath393 is an eigenvector of @xmath370 of eigenvalue @xmath394 . we can similarly show that @xmath395 is an eigenvector of @xmath396 of eigenvalue @xmath397 . on the other hand , we can find three chambers not affected by the move of the line @xmath389 along @xmath375 . for example , @xmath398 , @xmath399 and @xmath400 . hence @xmath370 has three dimensional eigenspace of eigenvalue @xmath1 . lemma [ lem : duality ] ( 2 ) yields that this eigenspace is expressed as @xmath401 so @xmath370 has desired eigenvalues and eigenspaces . since the factor @xmath402 is continuous on @xmath394 at @xmath1 , the expression of @xmath370 is valid even in the case @xmath403 . similarly we have the expressions of @xmath404 and @xmath405 . [ cor : monod - rep - dual ] under the assumption ( 3.1 ) , the circuit transformations @xmath406 @xmath333 are given as @xmath407 @xmath408 where @xmath199 is any element of @xmath195 . let @xmath409 and @xmath410 @xmath333 be the circuit matrices along the loop @xmath330 with respect to the basis @xmath411 of @xmath200 , and to @xmath412 of @xmath195 , respectively . that is , we have transformations @xmath413 by the continuation along @xmath330 . [ cor : monod - matrix ] the circuit matrices are expressed as @xmath414 @xmath415 where @xmath416 @xmath417 @xmath418 @xmath419 their explicit forms are @xmath420 @xmath421    @xmath422    @xmath423    they satisfy @xmath424    we identify @xmath425 with the row and column vectors @xmath426 respectively . note that @xmath427 theorem [ th : monod - rep ] yields these expressions . we have @xmath428 by lemma [ lem : duality ] ( 1 ) .    as a result , @xmath409 @xmath333 coincides with the circuit matrix with respect to the basis @xmath8 by theorem 7.1 in @xcite . in this section we introduce a system @xmath430 , and describe its schwarz map , which is the main result of this paper . we introduce in this subsection a system @xmath430 , which is a system @xmath0 with specific parameters , and mention a reason why this system is of special interest . let @xmath12 be the configuration space of six lines @xmath432 in general position in the projective plane @xmath433 . we identify the space @xmath12 with @xmath434 where @xmath70 is the line at infinity in the @xmath435-plane given by @xmath436 . the system @xmath437 is generated by the linear differential equations which annihilate functions on @xmath12 defined by the integral @xmath438 the schwarz map of the system @xmath439 is studied ( cf . @xcite , @xcite ) in two cases @xmath440 and @xmath441 . we have been interested in the case @xmath442 . on the other hand , let @xmath443 be the 2-dimensional stratum defined by @xmath444 , which is the space of six lines such that the three lines @xmath445 meet at a point , the three lines @xmath446 meet at another point , and nothing further special occurs . it is known ( @xcite ) that the restriction of @xmath439 onto @xmath443 is the appell s hypergeometric system @xmath447 , which is projectively equivalent ( multiplying a function to the unknown ) to @xmath448 setting @xmath449 , we define @xmath450 we believe that the first step of understanding @xmath451 is the study of the system @xmath429 .    the schwarz map of a system is defined by the ratio of linearly independent solutions . the main objective of this paper is the schwarz map of the hypergeometric system @xmath452 the system @xmath429 admits solutions stated in proposition [ matome ] . the next subsection gives a geometric background of understanding these solutions . consider a family of curves of genus 2 given as triple covers of @xmath453 : @xmath454 branching at four points @xmath455 . we choose two linearly independent holomorphic 1-forms : @xmath456 and put @xmath457    for a fixed @xmath51 , the _ abel - jacobi map _ for the curve @xmath458 is a multi - valued map @xmath459 it is a single - valued map to its jacobian @xmath460 , where @xmath461 is a lattice generated by its periods : integrals over possible loops with base @xmath462 : @xmath463      proposition [ matome ] for @xmath464 implies that after the coordinate change @xmath465 and the change of unknown : @xmath466 , two linearly independent solutions of @xmath467 and the two indefinite integrals @xmath468 form a set of fundamental solutions of @xmath429 .    on the other hand , the integral representation of the gauss hypergeometric equation given in section 1 asserts that the integral above along any _ closed _ path gives a solution of @xmath469 . thus we find that the schwarz map of @xmath429 is the totality of the abel - jacobi map of the family @xmath470 after a slight modification ( multiplying @xmath471 to the second coordinate ) . thus we get    [ th : gen - schwarz ] if we change the coordinates @xmath472 of @xmath473 as @xmath474 the schwarz map of the system @xmath429 is equivalent to the projectivization of the family of the abel - jacobi map of the family @xmath470 of curves of genus 2 , explicitly given as @xmath475 the latter two @xmath476 and @xmath477 are @xmath478 and @xmath479 in section 2 . the map by means of the former two @xmath480 is the schwarz map of the hypergeometric equation @xmath481 . its image is a disc , and the inverse map of @xmath482 is single - valued automorphic function on the disc with respect to the triangle group of type @xmath10 $ ] ; in other words , the disc is tessellated by schwarz triangles of type @xmath10 $ ] . the image surface under @xmath483 can be regarded as lying in a fiber bundle with the @xmath482-image disc as its base and the jacobian variety of @xmath458 as the fiber on the image point @xmath484 . a triangle of type @xmath485 $ ] is a hyperbolic triangle with angles @xmath486 and @xmath487 ; the above triangle has angles @xmath488 and @xmath269 . the triangle group of type @xmath485 $ ] is the group consisting of the even products of the reflections with the sides of the triangle of type @xmath485 $ ] as axes . it is known that the triangle group of type @xmath10 $ ] is conjugate to the congruence subgroup @xmath489 for arithmetic triangle groups , see @xcite . other than this family of curves , there are two families of curves of genus 2 branching at four points in @xmath453 ; see appendix 2 .      from monodromy side , theorem [ th : gen - schwarz ] can be understood as follows . define @xmath490 and @xmath491 @xmath333 by substituting @xmath492 into @xmath409 and @xmath410 defined in  [ circuit ] , respectively . they are the circuit matrices for @xmath429 with respect to @xmath493 and those for @xmath494 with respect to @xmath495 where @xmath496{t_1(1-t_1)t_2(1-t_2)(1-t_1x - t_2y ) } } .\ ] ]    [ cor : monodromy ] we have @xmath497 @xmath498 they satisfy @xmath499    by proposition [ prop : inv - subspace ] and remark [ rem : kernel ] , the subspace spanned by solutions @xmath500 and @xmath501 is invariant under the monodromy representation . in fact , the top - left @xmath502 block matrices @xmath503 of @xmath490 @xmath333 act on this space . note that @xmath504 , @xmath505 . let @xmath506 be the group generated by @xmath507 , @xmath508 and @xmath509 . the group @xmath506 is isomorphic to the triangle group @xmath10 $ ] , and is contained in the unitary group @xmath510)\mid g h ' \;^t \overline { g}=h'= \begin{pmatrix } -1 & -\omega \\ \omega+1 & 0 \end{pmatrix } \big\}.\ ] ] by a matrix @xmath511 the hermite matrix @xmath512 and circuit matrices @xmath513 @xmath514 are transformed as @xmath515 @xmath516 hence the projectivization of @xmath506 is isomorphic to the congruence subgroup @xmath517 of @xmath518 and the ratio @xmath519 can be regarded as the map @xmath482 in theorem [ th : gen - schwarz ] and as an element of the upper - half space . let @xmath520 denote the stratum defined by @xmath521 and let us restrict the system @xmath439 to this stratum , which is the space of six lines such that the three lines @xmath445 meet at a point , and nothing further special occurs . this system is denoted by @xmath522 or @xmath523 . little is known about this system .    before stating the proposition in this section , we briefly recall the appell - lauricella s system @xmath524 @xmath525 where @xmath526 are variables , and @xmath527 and @xmath528 . this is a 3-variable version of the appell s @xmath5 . it admits solutions given by a power series @xmath529 where @xmath530 , and by an integral @xmath531the collection of solutions is denoted by @xmath532    in this section , we prove the following proposition .    [ x3fd ] if @xmath533 , then the system @xmath522 is reducible and has a subsystem isomorphic to the appell - lauricella s system @xmath534 in 3 variables with 4 free parameters . more precisely , the collection of the solutions of @xmath522 includes @xmath535 note @xmath536 . if we apply the proposition under the further restriction @xmath88 , we find a subsystem isomorphic to @xmath5 in @xmath0 , which is equivalent to proposition 1.1 . we give three `` proof ' 's : one using power series , one using integral representations , and one manipulating differential equations . it is known that the system @xmath537 has a solution given by the series @xmath538 where @xmath539 ; refer to @xcite . a computation shows that the identity @xmath540 holds if and only if @xmath541      we manipulate the integral @xmath542 the three lines @xmath543 meet at @xmath544 . introduce new coordinate @xmath545 @xmath546 which send @xmath547 to @xmath548 . since @xmath549 @xmath550 we have @xmath551 @xmath552 where @xmath553 . if @xmath554 then the double integral above becomes the product of the beta integral @xmath555 and the integral @xmath556 which can be written as @xmath557 on the other hand , the integral @xmath558 solves @xmath559 . by solving the system @xmath560 we complete the proof of proposition [ x3fd ]      we manipulate the system @xmath523 ( given in [ msy , p.24 ] ) : @xmath561 where @xmath562 , @xmath563 , @xmath564 , and @xmath565 , and the appell - lauricella system @xmath566 . we show that the system @xmath567 is a subsystem of @xmath523 when @xmath533 . change the unknown @xmath568 of @xmath523 into @xmath110 by @xmath569 and the variables @xmath570 into @xmath571 as @xmath572 then @xmath523 can be written as @xmath573 where @xmath574 where @xmath575 write the system @xmath567 as @xmath576 , where @xmath577 eliminating the second derivatives in @xmath578 by using @xmath579 , we see that @xmath578 are linear combination , over @xmath580 , of the @xmath579 s if and only if @xmath581 and @xmath582 this completes the proof of proposition [ x3fd ] .    * remark : * actually we have , under the condition @xmath583 , @xmath584@xmath585 we encountered a family of curves @xmath458 of genus 2 given as triple covers of @xmath453 . this is the case 3 in the following proposition . indeed , since the @xmath587 fold cyclic cover @xmath588 of @xmath453 branching at four points with indices @xmath589 has euler characteristic @xmath590 if we assume the genus of @xmath588 is two ( euler characteristic of @xmath588 is @xmath591 ) , we have @xmath592 it is easy to see that only three cases above are possible .    the three cases can be realized by the following families of curves : @xmath593 { \rm case\ 6:}&c_t^{(6 ) } : & s^6=s^2(1-s)^4(t - s)^3,&\quad t:{\rm \ parameter},\\[2 mm ] { \rm case\ 4:}&c_t^{(4 ) } : & s^4=s^2(1-s)^2(s - t),&\quad t:{\rm \ parameter}. \end{array}\ ] ] note that the double cover of the base space of case 6 branching at the two points of index 2 is equivalent to case 3 . aomoto k. and kita m. , translated by k. iohara , _ theory of hypergeometric functions _ , springer verlag , now york , 2011 . w. n. bailey , _ generalized hypergeometric series _ , cambridge , 1935 e. bod , algebraicity of the appell - lauricella and horn hypergeometric functions , j. diff . equations * 252 * ( 2012 ) , 541566 . k. cho , a generalization of kita and noumi s vanishing theorems of cohomology groups of local system , nagoya math . j. , * 147 * ( 1997 ) , 6369 . k. mimachi and t. sasaki , irreducibility and reducibility of lauricella s system of differential equations @xmath594 and the jordan - pochhammer differential equation @xmath595 , kyushu j. math . * 66*(2012 ) , 6187 . k. matsumoto , t. sasaki and m. yoshida , the monodromy of the period map of a @xmath596-parameter family of k3 surfaces and the hypergeometric function of type @xmath597 , intern . j. math . * 3*(1992 ) , 1164 . k. matsumoto , t. sasaki , n. takayama and m. yoshida , monodromy of the hypergeometric differential equation of type @xmath597 , ii the unitary reflection group of order @xmath598 , annali della scuola normale superiore di pisa ( 4 ) * 20 * ( 1993 ) , 617631 . k. matsumoto and t. terasoma , period maps of reducible hypergeometric equations and mixed hodge structures , in preparation . k. matsumoto and m. yoshida , monodromy of lauricella s hypergeometric @xmath599-system , ann . ( 5 ) , * 13 * ( 2014 ) , 551577 . takeuchi , k , commensurability classes of arithmetic triangle groups , j. fac . tokyo sect . ia math . 24(1977 ) , 201212 . m. yoshida , _ hypergeometric functions , my love _ , vieweg , 1997 .
we study an appell hypergeometric system @xmath0 of rank four which is reducible and show that its schwarz map admits geometric interpretations : the map can be considered as the universal abel - jacobi map of a @xmath1-dimensional family of curves of genus 2 .