On the factor alpha in Peyre's constant
  For an arbitrary del Pezzo surface S, we compute alpha(S), which is the volume of a certain polytope in the dual of the effective cone of S, using Magma and Polymake. The constant alpha(S) appears in Peyre's conjecture for the leading term in the asymptotic formula for the number of rational points of bounded height on S over number fields. 
Ulrich Derenthal, Andreas-Stephan Elsenhans, J\"org Jahnel
math.AG math.NT