The circle method applies to rational points on Perazzo-type cubic fourfolds and exceeds the square-root barrier.
Browning, Rational points on cubic hypersurfaces that split off a form.Compositio Math.146 (2010), 853–885
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Beyond the square-root barrier: cubic forms of Perazzo type
The circle method applies to rational points on Perazzo-type cubic fourfolds and exceeds the square-root barrier.