On the Hasse principle for complete intersections

TL;DR

This paper proves the Hasse principle for certain smooth projective varieties defined by two cubic forms over the rationals, using a new Kloosterman refinement technique applicable when the dimension is at least 39.

Contribution

It introduces a novel Kloosterman refinement method for systems of equations over  and establishes the Hasse principle for varieties defined by two cubic forms when the dimension is at least 39.

Findings

Hasse principle holds for smooth projective varieties defined by two cubic forms with dimension   39 or more.

Development of a new Kloosterman refinement technique for systems over .

Proof of the Hasse principle in this setting extends previous results to higher dimensions.

Abstract

We prove the Hasse principle for a smooth projective variety $X\subset \PP^{n-1}_\Q$ defined by a system of two cubic forms $F,G$ as long as $n\geq 39$. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over $\Q$.