Let $X$ be a supersingular K3 surface over an algebraically closed field $k$ of positive characteristic $\!p$. Artin proved in the paper https://eudml.org/doc/81948 that the determinant $\mathrm{disc}(X)$ of a matrix of the intersection pairing on the Néron-Severi group $NS(X)$ is equal to $-p^{2\sigma}$, where $1 \leqslant \sigma \leqslant 10$.

If X has a quasi-elliptic fibration, then $\sigma$ is computable according to http://arxiv.org/pdf/math/0311057.pdf (16 page). What if X has only an elliptic fibration?

  • $\begingroup$ All supersingular K3 surfaces admit an elliptic or quasi-elliptic fibration. In fact, all $K3$ surfaces with the picard number at least $6$ are (quasi) elliptic. So, probably your question has no easy answer. $\endgroup$ – Edward Teach Sep 11 '16 at 17:08

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