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In our previous work we conjectured - inspired by an algebro-geometric resultof Fujita - that the height of an arithmetic Fano variety X of relativedimension n is maximal when X is the projective space$\mathbb{P}^n_{\mathbb{Z}}$ over the integers, endowed with the Fubini-Studymetric, if the the corresponding complex Fano variety is K-semistable. In thiswork the conjecture is settled for diagonal hypersurfaces in$\mathbb{P}^{n+1}_{\mathbb{Z}}$. The proof is based on a logarithmic extensionof our previous conjecture, of independent interest, which is established fortoric log Fano varieties of relative dimension at most three, hyperplanearrangements on $\mathbb{P}^n_{\mathbb{Z}}$, as well as for general logarithmicarithmetic Fano surfaces.

Sharp bounds on the height of K-semistable toric Fano varieties II, the log case