The circle method applies to rational points on Perazzo-type cubic fourfolds and exceeds the square-root barrier.
Perazzo, Sulle variet` a cubiche la cui hessiana svanisce identicamente.Giornale di Matematiche (Battaglini)38(1900), 337–354
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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.