Weak Approximation For Cubic Hypersurfaces And Degree 4 Del Pezzo Surfaces

In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic hypersurfaces of dimension at least two defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residue field has at least eleven elements. (2) For del Pezzo surfaces of degree 4 defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residue field has at least thirteen elements. (3) Weak approximation holds for cubic hypersurfaces of dimension at least ten defined over a global function field of characteristic not equal to 2, 3, 5 or a purely imaginary number field.