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In this paper, we establish a strategy for the calculation of the proportionof everywhere locally soluble diagonal hypersurfaces of $\mathbb{P}^{n}$ offixed degree. Our strategy is based on the product formula established byBright, Browning and Loughran. Their formula reduces the problem into thecalculation of the proportions of $\mathbb{Q}_{v}$-soluble diagonalhypersurfaces for all places $v$. As worked examples, we carry out our strategyin the cases of quadratic and cubic hypersurfaces. As a consequence, we provethat around $99.99\%$ of diagonal cubic $4$-folds have $\mathbb{Q}$-rationalpoints under a hypothesis on the Brauer-Manin obstruction.

How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces