Counting rational points on quartic del Pezzo surfaces with a rational   conic
  Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$. 
T. D. Browning, E. Sofos
math.NT math.AG