Revista Matemática Iberoamericana

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Volume 35, Issue 1, 2019, pp. 125–172
DOI: 10.4171/rmi/1051

Published online: 2019-01-08

Smooth models of singular $K3$-surfaces

Alex Degtyarev[1]

(1) Bilkent University, Ankara, Turkey

We show that the classical Fermat quartic has exactly three smooth spatial models. As a generalization, we give a classification of smooth spatial (as well as some other) models of singular $K3$-surfaces of small discriminant. As a by-product, we observe a correlation (up to a certain limit) between the discriminant of a singular $K3$-surface and the number of lines in its models. We also construct a $K3$-quartic surface with 52 lines and singular points, as well as a few other examples with many lines or models.

Keywords: $K3$-surface, smooth quartic surface, Fermat quartic, sextic curve, sextic model, octic model, Niemeier lattice, Mukai group

Degtyarev Alex: Smooth models of singular $K3$-surfaces. Rev. Mat. Iberoam. 35 (2019), 125-172. doi: 10.4171/rmi/1051