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52 T.D. BROWNING
But now (18) implies that Y14 ¿ B1/2/(Y1/2Y1/2Y24Y1/2), and (20) and (21) together imply that Y03 ¿ Y33Y34/Y04. We therefore deduce that
1/2 3/4 3/4 1/2 1/2 1/4 1/4 03 04 23 24 33 34
Y1,Yi3,Yi4 (20) holds
Y03,Y04,Y33 Y1,Y23,Y24,Y34
¿ B1/2 Y1/2Y1/2Y33Y34.
Y1,Y04,Y33 Y23,Y24,Y34
Finally it follows from (17) and (21) that Y33 ¿ B1/2/(Y1/2Y1/2Y34), whence
N ¿ B 1 ¿ B(logB)5,
Y1,Yi3,Yi4 Y04,Y13,Y14,Y23,Y34 (20) holds
which is satisfactory for the theorem.
Next we suppose that (22) holds, so that (23) also holds. In this case it follows from (19), together with the inequality Y1Y13Y14 ¿ Y03Y04, that
1/2 1/2 1/4 3/4 1/2 1/4 04 14 24 34 03 04 03 04 24 34
1/2 1/2 1/2 1/2 1/4 03 23 33 1 23 33
On combining this with the inequality Y14 ¿ B1/2/(Y1/2Y1/2Y24Y1/2), that follows
from (18), we may therefore deduce from (25) that X X
N ¿ Y1Y13Y14Y23Y24Y33Y34
Y1,Yi3,Yi4 (22) holds
Y1,Yi3,Yi4 (22) holds
¿
1/2 1/4 3/4 1/2 3/2 3/4 5/4 1 03 04 14 23 24 33 34
Y1,Y03,Y04,Y33 Y14,Y23,Y24,Y34
¿ B1/2
1/4 1/4 1/2 1/2 3/4 3/4 03 04 23 24 33 34
Y1,Y03,Y04 Y23,Y24,Y33,Y34
Now it follows from (23) that Y33 ¿ Y03Y04/Y34. We may therefore combine this
with the first inequality in (17) to conclude that
X N ¿ B1/2 X Y03Y04Y1/2Y1/2 ¿ B(logB)5,
Y1,Yi3,Yi4 (22) holds
Y1,Y03,Y04 Y23,Y24,Y34
which is also satisfactory for the theorem.
Finally we suppose that (24) holds. On combining (19) with the fact that Y33Y34 ¿ Y03Y04, we obtain
Y04Y14Y24Y34 Y03Y04 Y04Y14Y24 Y03Y13Y23 Y34 Y13Y23
Summing (25) over Y33 first, with min{Y03Y04,Y33Y34} 6 Y1/2Y1/2Y1/2Y 1/2, we therefore obtain
1/2 1/2 3/2 1/2 3/2 1/2 1 03 04 13 14 23 24 34
Y1,Yi3,Yi4 (24) holds
Y1,Y03,Y04,Y13 Y14,Y23,Y24,Y34
AN OVERVIEW OF MANIN’S CONJECTURE FOR DEL PEZZO SURFACES 53
But then we may sum over Y03,Y13 satisfying the inequalities in (17), and then Y1 satisfying the second inequality in (18), in order to conclude that
X 1/4 X 1/2 1/2 3/2 1/4 5/4 1/2 04 13 14 23 24 34
Y1,Yi3,Yi4 (24) holds
Y1,Y04,Y13 Y14,Y23,Y24,Y34
¿ B1/2 Y1/2Y1/2Y14Y24Y1/2 ¿ B(logB)5.
Y1,Y04,Y14 Y23,Y24,Y34
This too is satisfactory for Theorem 3, and thereby completes its proof.
4. Open problems
We close this survey article with a list of five open problems relating to Manin’s conjecture for del Pezzo surfaces. In order to encourage activity we have deliberately selected an array of very concrete problems.
(i) Establish (3) for a non-singular del Pezzo surface of degree 4.
The surface x0x1 −x2x3 = x0 + x1 + x2 −x3 −2x4 = 0 has Picard group of rank 5.
(ii) Establish (3) for more singular cubic surfaces.
Can one establish the Manin conjecture for a split singular cubic surface whose universal torsor has more than one equation? The Cayley cubic surface (8) is such a surface.
(iii) Break the 4/3-barrier for a non-singular cubic surface.
We have yet to prove an upper bound of the shape NU,H(B) = OS(Bθ), with θ < 4/3, for a single non-singular cubic surface S ⊂ P3. This seems
to be hardest when the surface doesn’t have a conic bundle structure over Q. The surface x0x1(x0 + x1) = x2x3(x2 + x3) admits such a structure; can one break the 4/3-barrier for this example?
(iv) Establish the lower bound NU,H(B) À B(logB)3 for the Fermat cubic.
The Fermat cubic x0 + x1 = x2 + x3 has Picard group of rank 4. (v) Better bounds for del Pezzo surfaces of degree 2.
Non-singular del Pezzo surfaces of degree 2 take the shape t2 = F(x0,x1,x2),
for a non-singular quartic form F. Let N(F;B) denote the number of integers t,x0,x1,x2 such that t2 = F(x) and |x| 6 B. Can one prove that we always have N(F;B) = Oε,F (B2+ε)? Such an estimate would be essentially best possible, as consideration of the form F0(x) = x0 +x1 −x2 shows. The best result in this direction is due to Broberg [Bro03a], who has established the weaker bound N(F;B) = Oε,F (B9/4+ε). For certain quartic forms, such as F1(x) = x0 +x1 +x2, the Manin conjecture implies that one ought to be able to replace the exponent 2+ε by 1+ε. Can one prove that N(F1;B) = O(Bθ) for some θ < 2?
Acknowledgements. The author is extremely grateful to Professors de la Bret`eche and Salberger, who have both made several useful comments about an earlier version of this paper. It is also a pleasure to thank the anonymous referee for his careful reading of the manuscript.
54 T.D. BROWNING
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School of Mathematics, University of Bristol, Bristol BS8 1TW, UK E-mail address: t.d.browning@bristol.ac.uk
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