Manin’s Conjecture For Del Pezzo Surfaces

Manin’s conjecture is the interface of analytic number theory and Diophantine geometry.It mainly studies the distribution of rational points on Fano varieties.When rational points on Fano varieties satisfy some conditions,one predicts that an asymptotic formula on the number of rational points can be established.S_ince del Pezzo surfaces belong to Fano varieties,Manin’s conjecture for del Pezzo surfaces has important theoretical significance.Let S（?）C P^d be a del Pezzo surface defined over Q,we define a height function H（x）:=max{|x₀|,...,|x_d|｝for x=（x₀,...,xn）∈Zd+1 with gcd（x₀,...,x_d）=1.Then,for an open subset U（?）S,Manin’s conjecture states that （?） holds,as B→∞,where p is the rank of the Picard group of a minimal desingularization of S and c is the constant predicted by Peyre.We define the split singular del Pezzo surfaces S_i （?）P_di with 1<i<3 for d₁=5 and d₂=d₃=4 over Q by the following equations:S1:x₀x₂-x₁x₅=x₀x₂-x³x₄=x₀x³+x₁²x₁x₄=x₀x₅+x₁x₄+x₄²=x³x₅+x₁x₂+x₂x₄=0,S2:x₀+x₀x³+x₂x₄=x₁x³-x₂²=0,S3:x₀x₁-x₂x³=x₀x₄+x₁x₂+x₃²=0.In this paper,we use a unified and different method from the previous to deal with the above three split singular del Pezzo surfaces,prove the corresponding Manin’s conjecture and improve the error terms.Let N_Ui,H（B）be defined as N_U,H（B）,we have （?）,where c_i is the Peyre constant.Proofs of Manin’s conjecture for a split del Pezzo surface S which is defined as above,consist of three main steps.·One constructs an explic_it bijection between rational points of bounded height on S and integral points in a region on a universal torsor which is related to S.In this way,we can transform rational points into integral points,which is called "universal torsors" method.·Using methods of analytic number theory（partial summation,partial integral,the theory on congruence equation,the summation on divisor function）,one establishes an asymptotic formula on the number of integral points in this region.·One shows that the volume of this region grows asymptotically as predicted by Peyre.