Rational points of bounded height on Del Pezzo surfaces of degree six

Marcello Robbiani
Let K be a number field. Dénote by V3 a split Del Pezzo surface of degree six over K and by o) its canonical divisor. Dénote by W3 the open complément of the exceptional Unes in K3. Let NWs(-w, X) be the number of ^-rational points on W3 whose anticanonical height H.a is bounded by X. Manin has conjectured that asymptotically NW3(co, X) tends to cX(\o% X)3, where c is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following
more » ... For each number field K there exists a constant cK such that NW3(co, X) < cfCX(\ogX)3 + 2r, where r is the rank of the group of units of OK. The constant cK is far from being optimal. However, if AT is a purely imaginary quadratic field, this proves an upper bound with a correct power of log X. The proof of Manin's conjecture for arbitrary number fields and a précise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the case K Q. Up to now the best resuit for arbitrary K goes back, as far as we know, to [Manin-Tschinkel], who gives an upper bound N^ia), X) < cXl +\ The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support.
doi:10.5169/seals-53005 fatcat:wa6muzhqcng4fjomi7hjjgdily