## The non-symplectic index of supersingular $$K3$$ surfaces.(English)Zbl 1471.14078

This coincise and carefully written paper studies the non-symplectic index of supersingular $$K3$$ surfaces over algebraically closed fields of odd characteristic.
Given a $$K3$$ surface $$X$$ over an algebraically closed field $$k$$, the image of the representation of its automorphism group $\mathrm{Aut}(X)\to GL(H^{0}(X,\Omega_{X/k}^{2}))$ is a finite cyclic group. Its order $$N_{X}$$ is called the non-symplectic order of $$X$$.
If $$k=\mathbb C$$ is the field of complex numbers, it is known that the non-symplectic index can be any integer $$N$$ with $$1\leq N\leq 66$$ and $$N\not= 60$$.
Recall that if $$k$$ is of positive characteristic, a $$K3$$ surface over $$k$$ is said to be supersingular when it has maximal Picard number, i.e. when its Neron-Severi lattice $$NS(X)$$ has rank 22. Recall also that the Artin invariant of $$X$$ is half the dimension of the discriminant group of $$NS(X)$$ as an $$\mathbb{F}_{p}$$-vector space.
Let $$k$$ be of characteristic $$p>2$$. The result in the paper under revision shows that for a supersingular $$K3$$ surface $$X$$ over $$k$$ of Artin invariant $$\sigma$$, the non-symplectic index is $$p^{m}+1$$ where $$m=0$$ or $$\sigma/m$$ is an odd integer. Dimensions of the families they form are also computed in terms of $$m$$. The main ingredient of the proof is the crystalline Torelli theorem. Furthermore, a supersingular $$K3$$ surface with maximal non-symplectic index, i.e. $$N_{X}=p^{\sigma}+1$$, is unique up to isomorphism. Examples of these are provided.

### MSC:

 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties 14J28 $$K3$$ surfaces and Enriques surfaces 14G17 Positive characteristic ground fields in algebraic geometry
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### References:

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