On the Hasse principle for complete intersection varieties

In 1961, Birch used a general version of the Hardy-Littlewood circle method to verify the Hasse principle for systems of forms of all degrees, provided that these forms had sufficiently many variables (this depends both on the degree, and the number of forms). There have been many developments in this area since then which have led to a significant reduction in the number of variables required to verify the Hasse principle in specific cases, however none of these results have been applicable to systems of two cubic forms. We aim to improve on Birch's result for two cubic forms which states that the Hasse principle is true provided that the forms are in at least 50+phi variables, where phi is the dimension of the singular locus of the intersection variety of the two forms.
We firstly aim to develop a two-dimensional version of the averaged Van-der Corput differencing method, and then use this to find a better bound for the minor arcs by taking advantage of the extra saving gained by averaging over both integrals. We will then perform Weyl differencing to get an explicit bound for the exponential sums which appear in the minor arcs. This will enable us to save 6 or 7 variables over Birch's method.
After this, we will adapt the Van-der Corput differencing step further to get partial Kloosterman refinement in the a sum. In order to take advantage of this, we will then use Poisson summation instead of Weyl differencing. We will need to adapt and improve upon current state of the art techniques in order to prove that square-root cancellation occurs in the exponential sums which arise in the minor arcs. This will hopefully enable us to save an additional 3-5 variables.
Finally, we aim to incorporate a version of the circle method which uses larger intervals as building blocks for the minor arcs, in order to get further saving over the a sum and potentially save another variable.