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let @xmath2 be the cubic surface defined by @xmath3 then @xmath4 is a singular del pezzo surface with a unique singularity @xmath5 of type @xmath1 and three lines , each of which is defined over @xmath6 .     on the @xmath1 cubic surface.,width=491 ]    let @xmath7 be the zariski open subset formed by deleting the lines from @xmath4 . our principal object of study in this paper is the cardinality @xmath8 for any @xmath9 . here @xmath10 is the usual height on @xmath0 , in which @xmath11 is defined as @xmath12 , provided that the point @xmath13 is represented by integral coordinates @xmath14 that are relatively coprime . in figure [ fig : d5 ] we have plotted an affine model of @xmath4 , together with all of the rational points of low height that it contains . the following is our principal result . we have @xmath15 where the leading constant is @xmath16 with @xmath17    it is straightforward to check that the surface @xmath4 is neither toric nor an equivariant compactification of @xmath18 . thus this result does not follow from the work of tschinkel and his collaborators @xcite . our theorem confirms the conjecture of manin @xcite since the picard group of the minimal desingularisation @xmath19 of the split del pezzo surface @xmath4 has rank @xmath20 . furthermore , the leading constant @xmath21 coincides with peyre s prediction @xcite . to check this we begin by observing that @xmath22 by ( * ? ? ? * theorem  4 ) and ( * ? ? ? * theorem  1.3 ) , where @xmath23 is a split smooth cubic surface and @xmath24 is the order of the weyl group of the root system @xmath1 . next one easily verifies that the constant @xmath25 in the theorem is the real density , which is computed by writing @xmath26 as a function of @xmath27 and using the leray form @xmath28 . finally , it is straightforward to compute the @xmath29-adic densities as being equal to @xmath30 . our work is the latest in a sequence of attacks upon the manin conjecture for del pezzo surfaces , a comprehensive survey of which can be found in @xcite . a number of authors have established the conjecture for the surface @xmath31 which has singularity type @xmath32 . the sharpest unconditional result available is due to la bretche @xcite . furthermore , in joint work with la bretche @xcite , the authors have recently resolved the conjecture for the surface @xmath33 which has singularity type @xmath34 . our main result signifies only the third example of a cubic surface for which the manin conjecture has been resolved . the proof of the theorem draws upon the expanding store of technical machinery that has been developed to study the growth rate of rational points on singular del pezzo surfaces . in particular , we will take advantage of the estimates involving exponential sums that featured in @xcite . in the latter setting these tools were required to get an asymptotic formula for the relevant counting function with error term of the shape @xmath35 . however , in their present form , they are not even enough to establish an asymptotic formula in the @xmath1 setting . instead we will need to revisit the proofs of these results in order to sharpen the estimates to an extent that they can be used to establish the theorem . in addition to these refined estimates , we will often be in a position to abbreviate our argument by taking advantage of @xcite , where several useful auxiliary results are framed in a more general context .    in keeping with current thinking on the arithmetic of split del pezzo surfaces , the proof of our theorem relies on passing to a universal torsor , which in the present setting is an open subset of the hypersurface @xmath36 embedded in @xmath37 $ ] . furthermore , as with most proofs of the manin conjecture for singular del pezzo surfaces of low degree , the shape of the cone of effective divisors of the corresponding minimal desingularisation plays an important role in our work . for the surfaces treated in @xcite , @xcite , @xcite , the fact that the effective cone is simplicial streamlines the proofs considerably . for the surface studied in @xcite , this was not the case , but it was nonetheless possible to exploit the fact that the dual of the effective cone is the difference of two simplicial cones . for the cubic surface , the dual of the effective cone is again the difference of two simplicial cones . however , we choose to ignore this fact and rely on a more general strategy instead .    while working on this paper the first author was supported by epsrc grant number ` ep / e053262/1 ` . the second author was partially supported by a feodor lynen research fellowship of the alexander von humboldt foundation . the authors are grateful to the referee for a number of useful comments that have improved the exposition of this paper . define the multiplicative arithmetic functions @xmath38 for any @xmath39 , where @xmath40 denotes the number of distinct prime factors of @xmath41 . these functions will feature quite heavily in our work and we will need to know the average order of the latter .    [ lem : sum_h_k ] for any @xmath42 we have @xmath43    let @xmath42 be given and let @xmath44 . then we have @xmath45}2^{\omega(u)}\\ & \ll_{\varepsilon}q\log q \sum_{d_1,\ldots , d_k=1}^{\infty}\frac{(d_1\cdots d_k)^{{\varepsilon}-1/2}}{[d_1,\ldots , d_k]},\end{aligned}\ ] ] where @xmath46 $ ] denotes the least common multiple of @xmath47 . we easily check that the final sum is absolutely convergent by considering the corresponding euler product , which has local factors of the shape @xmath48 . given integers @xmath49 , with @xmath50 , we will be led to consider the quadratic exponential sum @xmath51 our study of this should be compared with the corresponding sum studied in ( * ? ? ? * eq . ( 3.1 ) ) , involving instead a cubic phase @xmath52 . in ( * ? ? * lemma  4 ) an upper bound of the shape @xmath53 is established for the cubic sum . the following result shows that we can do better in the quadratic setting . [ lem : lemma4 ] for any @xmath54 with @xmath55 , we have @xmath56    writing @xmath57 in the second step , we find that @xmath58 the inner sum is @xmath41 if @xmath59 and @xmath60 otherwise . let @xmath61 and write @xmath62 , @xmath63 with @xmath64 . then @xmath65 and the result follows . our next results concern the function @xmath66 , where @xmath67 is the fractional part of @xmath68 . the following estimate improves upon ( * ? ? ? * lemma  5 ) . [ lem : sum_mod_q ] for any @xmath69 , @xmath70 , @xmath71 with @xmath72 , we have @xmath73    let @xmath74 denote the sum that is to be estimated . by mbius inversion it follows that @xmath75 we claim that @xmath76 for any @xmath69 , @xmath70 and @xmath71 with @xmath72 . under this assumption , it therefore follows that @xmath77 this is satisfactory for the lemma , since @xmath78 .    to establish we follow the proof of ( * lemma  4 ) , finding that @xmath79 where @xmath80 rather than applying weyl s inequality as in ( * ? ? ? * lemma  4 ) , we simply break into @xmath81 residue classes modulo @xmath82 and apply lemma [ lem : lemma4 ] to deduce that @xmath83    now @xmath84 for any @xmath85 . hence @xmath86 which thereby concludes the proof of .    for positive integers @xmath87 , we define the function @xmath88 we combine lemma  [ lem : sum_mod_q ] with the proof of ( * ? ? ? * lemma  1 ) to obtain the following result . [ lem : sum_mod_q_2 ] let @xmath89 and @xmath90 . we have @xmath91    in the proof of ( * ? ? ? * lemma  1 ) , @xmath92 is estimated as @xmath93 , for given @xmath94 coprime to @xmath41 . using ( * ? ? ? * lemma  7 ) , we make this precise as @xmath95 where @xmath96 is chosen such that @xmath97 . our task is to compute @xmath98    for the main term , we may extend the summation over @xmath94 to all positive integers , since @xmath99 as in ( * ? ? ? * lemma  1 ) , we see that the sum over @xmath100 is @xmath101 , with @xmath102 summing this over @xmath103 , we get @xmath104 . it is easy to see that @xmath105 agrees with the leading constant in the statement of the lemma .    for the error term , we exchange the summations over @xmath94 and @xmath103 . applying lemma  [ lem : sum_mod_q ] , we obtain the contribution @xmath106 with @xmath107 . this completes the proof of the lemma . given @xmath108 such that @xmath50 and a real - valued function @xmath109 defined on an interval @xmath110 , let @xmath111 it is interesting to compare this sum with the sort of sums that featured in our corresponding investigation of the @xmath34 cubic surface . the sole difference between ( * ? ? ? * eq . ( 4.1 ) ) and @xmath112 is that the argument involves @xmath113 , rather than @xmath114 .    we will be interested in studying @xmath112 when @xmath115 . here , if @xmath116 $ ] and @xmath117 , then @xmath118 is defined to be the set of real - valued differentiable functions @xmath109 , such that @xmath119 is monotonic and of constant sign on @xmath120 , with @xmath121 . it will be convenient to define @xmath122 we will need a version of ( * ? ? ? * lemma  10 ) , in which the factor @xmath123 is made more explicit . this is achieved in the following result . [ lem : lemma10 ] let @xmath124 . assume @xmath125 and @xmath115 . for any @xmath44 , we have @xmath126 where @xmath127 is the divisor function .    in comparing this with ( * ? ? ? * lemma  10 ) , one sees that the first and third term in both results share the same approximate order of magnitude . however , the middle term is improved from @xmath128 to @xmath129 . this saving is crucial in our work . it arises from the fact that the current set - up leads us to estimate the quadratic exponential sums with @xmath130 , rather than the corresponding cubic sums with phase @xmath52 and @xmath131 . in the former case we are dealing with linear exponential sums , for which we have very good control , and in the latter case we only have the bound @xmath132 available . let @xmath133 . replacing the bound @xmath134 by @xmath135 in the application of vaaler s trigonometric formula in the proof of ( * ? ? ? * lemma  10 ) , we obtain @xmath136 for any @xmath137 , where @xmath138 as in ( * ? ? ? * lemma  10 ) , we rewrite this as @xmath139 with @xmath140 and @xmath141 where @xmath142 is the multiplicative inverse of @xmath143 modulo @xmath41 . since @xmath144 , we have ( with @xmath145 ) @xmath146    write each @xmath147 modulo @xmath41 uniquely as @xmath148 with @xmath149 and @xmath150 . then @xmath151 with @xmath152 , just as in ( * ? ? ? * lemma  10 ) . therefore , @xmath153    for the contribution from the case @xmath154 , note that @xmath155 . we have @xmath156 trivially , and @xmath157 the inner sum is @xmath158 if @xmath159 ( which is possible only in the case @xmath160 since @xmath161 ) and @xmath60 otherwise . thus @xmath162 , whence the total contribution to @xmath163 from the case @xmath154 is @xmath164 where @xmath165 is the sum of divisors function .    for the total contribution to @xmath163 from the case @xmath166 , we note that @xmath167 by ( * ? ? ? * lemma  5 ) for @xmath115 . also @xmath168 where @xmath169 . by lemma  [ lem : lemma4 ] , @xmath170 the contribution from the case @xmath171 is therefore @xmath172    plugging the contribution from @xmath173 and @xmath166 to @xmath163 into @xmath112 , we deduce , for any @xmath137 , that @xmath112 is @xmath174 observing that @xmath175 we therefore deduce that @xmath176    let @xmath177 if @xmath137 , we may use this @xmath10 in the estimate above , together with @xmath178 , in order to obtain the lemma . if @xmath179 , so that @xmath180 , we deduce from the trivial estimate @xmath181 that the lemma holds in this case too . let @xmath4 be the @xmath1 cubic surface , let @xmath182 be the open subset formed by deleting the lines from @xmath4 and let @xmath19 be the minimal desingularisation of @xmath4 . in this section we will establish an explicit bijection between @xmath183 and the integral points on the universal torsor above @xmath19 , subject to a number of coprimality conditions . for this we will follow the strategy explained in @xcite .    to establish the bijection we will introduce new variables @xmath184 and @xmath185 . it will be convenient to henceforth write @xmath186 and @xmath187 for any @xmath188 . let us recall some information concerning the geometry of @xmath4 from ( * ? ? ? * section  8) . blowing up the singularity @xmath5 on @xmath4 results in the exceptional divisors @xmath189 in a @xmath1-configuration on the minimal desingularisation @xmath190 . let @xmath191 resp . @xmath192 on @xmath19 be the strict transforms under @xmath193 of the three lines @xmath194 , @xmath195 , @xmath196 resp .  the curves @xmath197 and @xmath198 on @xmath4 . the extended dynkin diagram in figure  [ fig : dynkin ] is the dual graph of the configuration of the curves @xmath199 on @xmath19 . @xmath200 \ar@{-}[dr ] \ar@{-}[dd ] & & e_6   \ar@{-}[rr ] & & { * + < 10pt>[o][f]{e_2 } } \ar@{-}[dr]\\        & e_8 \ar@{-}[r ] & e_7 \ar@{-}[r ] & { * + < 10pt>[o][f]{e_5 } } \ar@{-}[r ] & { * + < 10pt>[o][f]{e_4 } } \ar@{-}[r ] & { * + < 10pt>[o][f]{e_1}}\\        a_1 \ar@{-}[rrrr ] \ar@{-}[ur ] & & & & { * + < 10pt>[o][f]{e_3 } } \ar@{-}[ur]}\ ] ]    by ( * ? ? ? * section  8) , non - zero global sections @xmath201 corresponding to @xmath199 form a generating set of the cox ring of @xmath19 . the ideal of relations in @xmath202 is generated by @xmath203 . we express the sections @xmath204 , for @xmath205 , of the anticanonical class @xmath206 in terms of the generators of @xmath202 as follows : @xmath207    the general strategy of @xcite suggests that @xmath183 should be parametrised by certain integral points on the variety @xmath208 . this is confirmed in the the following result . [ lem : bijection ] we have @xmath209 where @xmath210 is the set of @xmath211 such that holds , with @xmath212 and @xmath213    the coprimality conditions in are achieved by taking @xmath214 and @xmath215 to be coprime if and only if the divisors @xmath216 and @xmath217 are not adjacent in the diagram . the reader is invited to consider the correspondence between    * the variables of the parametrisation and the generators of @xmath202 , * the torsor equation  ( [ eq : torsor ] ) and the relation in @xmath202 , * the height conditions  ( [ eq : height ] ) and the expressions of @xmath204 in terms of the generators of @xmath202 , * the coprimality conditions  ( [ eq : cpa2])  ( [ eq : cpe ] ) and the configuration of the curves associated to the generators of @xmath202 encoded in figure  [ fig : dynkin ] . the proof of lemma  [ lem : bijection ] is elementary , but modelled according to the geometry of @xmath4 . the following additional geometric information is relevant . contracting @xmath218 in this order leads to a map @xmath219 that is the blow - up of six points in the projective plane . we may choose @xmath220 as the coordinate lines in @xmath221 . then @xmath222 is the quadric @xmath223 . the morphisms @xmath224 and the projection @xmath225{cccc }      \phi_2 : & s & { \dashrightarrow } & { { { \mathbb{p}}^2}},\\      & { \mathbf{x } } & \mapsto & ( x_0:x_1:x_2 )    \end{array}\ ] ] from the singularity @xmath5 , form a commutative diagram of rational maps between @xmath226 and @xmath227 . the inverse map of @xmath228 is @xmath225{cccc }      \phi_3 : & { { { \mathbb{p}}^2}}&{\dashrightarrow}&s,\\      & ( { \eta}_4':{\alpha}_1':{\eta}_8')&\mapsto&({\eta}_4'^3:{\eta}_4'^2{\alpha}_1':{\eta}_4'^2{\eta}_8':{\eta}_8'{\alpha}_2 ' )    \end{array}\ ] ] where @xmath229 . the maps @xmath230 give a bijection between the complement @xmath7 of the lines on @xmath4 and @xmath231 , and furthermore , induces a bijection between @xmath183 and the integral points @xmath232    motivated by the way the curves @xmath233 occur in @xmath234 as the blow - ups of intersection points of @xmath235 , one introduces the following further variables @xmath225{lll }      { \eta}_5=\gcd({\eta}_4,{\eta}_8),&{\eta}_7=\gcd({\eta}_5,{\eta}_8 ) , &       { \eta}_3=\gcd({\eta}_4 , { \alpha}_1 , { \alpha}_2),\\      { \eta}_1=\gcd({\eta}_3,{\eta}_4,{\alpha}_2),&{\eta}_2=\gcd({\eta}_1,{\alpha}_2)&{\eta}_6=\gcd({\eta}_2,{\alpha}_2 ) . \end{array}\ ] ] although we omit the details here , it is now straightforward to derive the bijection described in the statement of lemma [ lem : bijection ] using elementary number theory .    in analysing the height conditions apparent in we will meet a number of real - valued functions , whose size it will be crucial to understand . we begin with the observation that is equivalent to @xmath236 , where @xmath237 in what follows we will need to work with the regions @xmath238 in keeping with the philosophy of @xcite , the definitions of these regions is dictated by the polytope whose volume is defined to be the constant @xmath239 , as computed using an alternative method in the introduction . in fact one has @xmath240 to which @xmath241 is closely related . perhaps a few more words are in order concerning the role of the cone of effective divisors in our work . the parametrisation of @xmath183 in lemma [ lem : bijection ] suggests that @xmath242 should be comparable to the volume of @xmath243 . on the other hand , the factors @xmath239 and @xmath25 of the conjectured leading constant in our theorem suggest the appearance of @xmath244 instead . the latter is constructed from @xmath241 , which comes from the dual of the effective cone , and from @xmath245 , which is obtained from the region whose volume is @xmath25 . at some point we will therefore need to make a transition from @xmath243 to @xmath244 . rather than distributing this procedure over the entire proof , as in our previous investigation @xcite , we will save this transition until lemma  [ lem : final_step ] , where it signifies the final step in our argument . we are now ready to record the various integrals that will feature in our work , together with some basic estimates for them . all of the bounds are simple enough to deduce in themselves , but readily follow from applications of ( * ? ? ? * lemma  5.1 ) . bearing this in mind , we have @xmath246 and @xmath247 and finally @xmath248 we now have everything in place to start the proof of the theorem . for fixed @xmath249 , let @xmath250 be the number of @xmath251 that contribute to @xmath242 . let @xmath252 be the set of @xmath253 satisfying @xmath254 . by definition , @xmath255 . we would like to begin by applying ( * ? ? ? * proposition  2.4 ) , which is concerned with a much more general setting . in order to facilitate our use of this result , table  [ t : dict ] presents a dictionary between the notation adopted in @xcite and the special case considered here . .dictionary for applying ( * ? ? ? * proposition  2.4 ) [ cols=">,<,>,<",options="header " , ]     we may now apply ( * ? ? ? * proposition  2.4 ) to deduce that @xmath256 where @xmath257 and the error term @xmath258 is the sum of terms of the form @xmath259 with @xmath260 one for each of the intervals that form @xmath261 , with start and end points @xmath262 and @xmath263 . here , @xmath264 denotes the multiplicative inverse of an integer @xmath265 . our first task is to show that the overall contribution from @xmath266 makes a satisfactory contribution to @xmath242 . [ lem : d5_first_sum ] we have @xmath267    we must show that once summed over @xmath268 such that ( [ eq : cpe8 ] ) , ( [ eq : cpe7 ] ) and ( [ eq : cpe ] ) hold , the term @xmath258 contributes @xmath269 . let @xmath270 . we remove ( [ eq : cpe8 ] ) by a mbius inversion . this leads us to estimate @xmath271 where @xmath272 is defined to be @xmath273 with @xmath274 where @xmath275 is the allowed interval for @xmath276 and @xmath277 as above depend on @xmath278 and @xmath279 . we may split the summation over @xmath280 into subintervals @xmath281 where we have @xmath282 as functions of @xmath276 . in view of the bounds for @xmath283 and @xmath284 that follow from the inequalities in the definition of @xmath285 , it follows that @xmath286    since @xmath287 , we may restrict the summation over @xmath288 to @xmath289 such that @xmath290 . then @xmath291 and @xmath292 , so that we can apply lemma  [ lem : lemma10 ] to obtain @xmath293    note that @xmath294 for any @xmath42 . writing , temporarily , @xmath295 we deduce that the total contribution from the first term is @xmath296 by lemma  [ lem : sum_h_k ] . the total contribution from the second term is @xmath297 finally , the total contribution from the third term is @xmath298 this therefore completes the proof of the lemma . let @xmath299 be the number of @xmath300 subject to @xmath301 , and let @xmath302 be the remaining number of elements of @xmath210 . lemma  [ lem : d5_first_sum ] can be modified in an obvious way to give estimates for @xmath299 and @xmath302 . for @xmath299 , we sum over @xmath303 first and over @xmath304 afterwards , and for @xmath302 , we do the reverse . we rewrite the result of lemma  [ lem : d5_first_sum ] as follows . removing ( [ eq : cpe8 ] ) by a mbius inversion , and adding @xmath306 to prevent that @xmath307 , we arrive at the formula @xmath308 where @xmath309    [ lem:8 ] we have @xmath310 where @xmath311 let @xmath312 and @xmath313 where @xmath314 \mid    { { { \eta}_4{\eta}_7k_8{\eta}_8 ' } \equiv { -{\varrho}^2{\eta}_3}\ ( \mathrm{mod}\ { q})}\}.\ ] ]    as in ( * ? ? ? * section  8.3 ) , we have @xmath315 where @xmath316 is the unique integer modulo @xmath41 with @xmath317 clearly @xmath318 is equivalent to @xmath319 for any such @xmath316 . using lemma  [ lem : sum_mod_q ] , we deduce that @xmath320 is @xmath321 a straightforward application of partial summation therefore reveals the total error as being @xmath322 here , in the second step , we have used @xmath323 and @xmath324 and the bound ( [ eq : estimate_v1 ] ) for @xmath325 . the final step uses lemma [ lem : sum_h_k ] . we rewrite the result of lemma  [ lem : d5_first_sum ] . recall the definition of the function @xmath327 for positive integers @xmath87 . noting that we may replace ( [ eq : cpe7 ] ) by @xmath328 , it follows that @xmath329 where @xmath330 here we automatically have @xmath331 . thus the congruence involving @xmath103 in @xmath332 determines @xmath303 uniquely modulo @xmath333 . [ lem:9 ] we have @xmath334 where @xmath335    let @xmath312 , @xmath336 and @xmath337 it follows from lemma  [ lem : sum_mod_q_2 ] that @xmath320 is @xmath338    a little thought reveals that the main term here is @xmath339 . using partial summation , we estimate the total error as @xmath340 using @xmath341 and ( [ eq : estimate_v1 ] ) in the second step and lemma [ lem : sum_mod_q ] in the final step . throughout the remainder of the paper we set @xmath342 for the total error term that appears in our main result . in this section and the next we will need to compute the average order of certain complicated multi - variable arithmetic functions , sometimes weighted by piecewise continuous functions . as previously , we will place ourselves in the more general investigation carried out in @xcite . here , given @xmath343 and @xmath344 , a number of rather general sets of functions are introduced : @xmath345 ( * ? ? ? * definition  3.8 ) , @xmath346 ( * ? ? ? * definition  4.2 ) , @xmath347 ( * ? ? ? * definition  7.7 ) and @xmath348 ( * ? ? ? * definition  7.8 ) . we will not redefine these sets here , but content ourselves with recording the inclusions @xmath349 ( * ? ? ? * corollary  7.9 ) .    in the notation of ( * definition  7.7 ) , our manipulations will involve the function @xmath350 for any @xmath351 , where @xmath352 and @xmath353      [ lem : d5_third_sum_b ] we have @xmath354 where @xmath355 is given by . our proof of the lemma is based on combining ( * ? ? ? * proposition  3.9 ) with lemma [ lem:8 ] . we will apply the former to @xmath356 summed over @xmath357 , with @xmath358 and @xmath359    there are a number of preliminary hypotheses that need to be checked in using ( * ? ? ? * proposition  3.9 ) . local factors of @xmath360 are given by @xmath361 , equal to @xmath362 we see that @xmath363 , for an appropriate @xmath364 .    for @xmath365 , we observe that ( [ eq : estimate_v2b ] ) implies @xmath366 and that @xmath367 unless @xmath323 .    thus everything is in place for an application of ( * ? ? ? * proposition  3.9 ) , giving @xmath368 where we check @xmath369 by ( * ? ? ? * corollary  7.10 ) . [ lem : d5_third_sum_a ] we have @xmath370 with @xmath355 given by . this time our argument is based on combining ( * ? ? ? * proposition  3.10 ) with lemma [ lem:9 ] , the former being applied with @xmath371 . as previously , there are a number of preliminary hypotheses that need to be checked in order to use this result . for the first of these , we define @xmath372 as in the proof of lemma  [ lem : d5_third_sum_b ] , we have @xmath373 , for some @xmath364 .    next , ( [ eq : estimate_v2a ] ) implies that @xmath374 an application of ( * ? ? ? * proposition  3.10 ) now gives the expected main term , together with a total error term @xmath375 . we put back together our estimates for @xmath302 and @xmath299 that were obtained in lemmas  [ lem : d5_third_sum_b ] and [ lem : d5_third_sum_a ] , respectively . this yields the following result . since @xmath379 , there is a @xmath364 such that @xmath380 . this and the bound ( [ eq : estimate_v3 ] ) for @xmath381 show that we are able to apply ( * ? ? ? * proposition  4.3 ) with @xmath382 to conclude that @xmath383 here , @xmath384 and @xmath385 is the `` average '' of @xmath386 over @xmath387 , which is computed as @xmath388 using ( * ? ? ? * corollary  7.10 ) . it turns out that in applying ( * ? ? ? * lemma  5.1 ) to obtain , only the inequality @xmath396 is used in the definition of @xmath243 . hence the same bounds hold if we replace @xmath243 by @xmath244 in the definitions of @xmath397 . finally , for @xmath415 we use ( [ eq : estimate_v2b ] ) for the integration over @xmath416 . then , integrating over @xmath417 , @xmath399 and @xmath418 we obtain @xmath419 this completes the proof of the lemma .    substituting @xmath420 into @xmath25 , for fixed @xmath421 , we obtain @xmath422 finally , by substituting @xmath423 for @xmath424 into ( [ eq : alpha_volume ] ) , written as an integral , we deduce that @xmath425 this completes the proof of the theorem . t.  d. browning , an overview of manin s conjecture for del pezzo surfaces . _ analytic number theory  a tribute to gauss and dirichlet ( gttingen , 20th june 24th june , 2005 ) _ , 3956 , clay mathematics proceedings * 7 * , ams , 2007 . u. derenthal and yu . tschinkel , universal torsors over del pezzo surfaces and rational points . _ equidistribution in number theory , an introduction _ , 169196 , nato sci . phys . chem . * 237 * , springer , 2006 .
the manin conjecture is established for a split singular cubic surface in @xmath0 , with singularity type @xmath1 .