The del Pezzo surface of degree 5

The 10 del Pezzo surfaces are precisely the smooth Fano surfaces, and their geometry is classically elegant (involving root systems, iconic graphs, and line counting). For instance, the smooth cubic surface and its 27 lines occur as the degree 3 del Pezzo surface. There is some motivation in finding ‘nice’ effective representatives of the anticanonical divisor class on such surfaces since it allows one to express these surfaces in the framework of log Calabi-Yau pairs, which are central to the GHK approach to mirror symmetry via cluster varieties. Thus motivated (and after being prompted by a good friend) I decided to work out one of these examples in detail: the del Pezzo surface of degree 5. It seems like there a few appendices that could be added to this story – there is a tropical angle to it, and a good reason for the appearance of the Petersen graph is not apparent to me – but the intention is worked through!

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