Norm hypergraphs

We introduce a high uniformity generalization of the so-called (projective) norm graphs of Alon, Koll'ar, R'onyai, and Szab'o, and use it to show that $${\mathrm{ex}}_d(n,K_{s_1,\dots ,s_d}^{(d)})=\mathrm{\Theta }\left(n^{d-\frac{1}{s_1\dots s_{d-1}}}\right)$$ holds for all integers $s_1,\dots ,s_d\ge 2$ such that $s_{d} \geq \left((d-1)(s_{1}\ldots s_{d-1}-1)\right)!+1$. This improves upon a recent result of Ma, Yuan and Zhang, and thus settles (many) new cases of a conjecture of Mubayi.