A strong form of the Manin–Peyre conjecture with a power‐saving error term is proved for a certain cubic fourfold.

This paper is organised as follows. In the first three sections we give a very general version of the determinant method in [He 2 ] and construct the auxiliary hypersurfaces mentioned above. In section 4 we use these auxiliary hypersurfaces to count points on space curves. In section 5 we collect some results on the geometry of algebraic surfaces in P 3 . These are used in section 6 in combination with the estimates for curves in sections 4 to prove theorem 0.4 for affine surfaces by means of the determinant method. In section 7 we collect some results on the geometry of varieties with many linear subspaces , which we then apply in section 8 in the proof of theorem 0.4 for affine hypersurfaces of dimension >2. Finally, in section 9 we deduce theorem 0.1 from theorem 0.4.We shall in this paper use the convention that ε may change between occurrences. We may thus, for example, first write f(B)=O ε (B 2ε ) and then f(B)=O ε (B ε ). Let us also point out that all intersections with projective linear subspaces are scheme-theoretical. The hyperplane section of an integral variety may thus be a reducible or non-reduced scheme.

For any N ≥ 2, let Z ⊂ P N be a geometrically integral algebraic variety of degree d. This paper is concerned with the number N Z (B) of Q-rational points on Z which have height at most B. For any ε > 0 we establish the estimateprovided that d ≥ 6. As indicated, the implied constant depends at most upon d, ε and N .

We give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces. Mathematics Subject Classification (2000) 14G05 · 11G35√ d+ε ) for surfaces of degree d ≥ 4. Then in [17] we showed that N (X ; B) d,ε min(B 4/3+16/9d+ε + B 14/9+ε , B 1+5/2 √ d+ε ). The aim of this paper is to prove the following stronger result.Theorem 0.1 Let X ⊂ P n be a geometrically integral projective surface over Q of degree d. Let X be the complement of the union of all lines on X . Then,This exponent is best possible for d ≥ 21, since N (X ; B) >> B whenever there is a conic on X . For quartic surfaces, we get N (X ; B) = O ε (B 13/8+ε ) compared to N (X ; B) = P. Salberger (B) 123 806 P. Salberger O ε (B 16/9+ε ) in [17]. From Theorem 0.1 we obtain a bound for the number of non-trivial equal sums of two powers. Corollary 0.2 Let n d (B) be the number of positive integer solutions to the equation x dThis estimate is superior to the estimates n d (B) = O d,ε B 5/3+ε of Hooley [14] and n d (B) = O d,ε B 3/2+1/(d−1)+ε of Skinner and Wooley [20]. It also improves upon the recent estimate n d (B) = O d,ε B 3/2+1/(2d−2)+ε of Browning and Heath-Brown [3, Theorem 2]. But for d ≥ 9, then 0.2 is superseeded by the bound n d (B) = O d,ε B 3/

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