Diagonal quartic surfaces with a Brauer–Manin obstruction

In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer–Manin obstruction to Hasse principle. We are able find an asymptotic formula for such ordered by height. The proof uses generalization method Heath-Brown on sums over linked variables. also show that there exists no uniform generic generator in family.