Existence of supersingular reduction for families of \(K3\) surfaces with large Picard number in positive characteristic (Q1652781)

The paper under review proves two main results about the existence of \(K3\) surfaces over function fields in positive characteristic. The first is that in large enough characteristic, if the height \(h(X)\) of the formal Brauer group of a \(K3\) surface \(X\) over the function field of a curve \(C\) over an algebraically closed field \(k\) of finite characteristic is between \(3\) and \(10\), then there is some closed point \(v\) of \(C\) over which the height of \(X\) increases after some finite extension of the function field \(k(C)\). Combined with a result of Artin, Igusa, and Mazur, this means that in many cases, \(X\) has potentially supersingular reduction at some place \(v\). The other result is that there are non-isotrivial \(K3\) surfaces \(X\) over \(k(C)\) with prescribed height between \(2\) and \(10\), provided that \(p\geq 5\). If \(p=3\), the author also constructs a non-isotrivial surface with height \(10\).