Dynkin diagrams of rank 20 on supersingular \(K3\) surfaces
A \textit{Dynkin type} is a finite formal sum of \(A_l,\, l\geq 1,\, D_m, \, m\geq 4,\, E_6,\, E_7,\) and \(E_8\) with non-negative integer coefficients corresponding to an even negative-definite lattice \(L(R)\) of discriminant denoted by \(\mathrm{disc} (R)\). Let \(R\) be a Dynkin type of rank \(20\). A prime integer \(p\) is called a \textit{\(R\)-supersingular \(K3\) prime} if \(p\) is odd not dividing \(\mathrm{disc}(R)\), and there exists a normal \(K3\) surface over an algebraically-closed field \(k\) of characterstic \(p\) whose singularity is of type \(R\). The minimal resolution of such a normal \(K3\) surface is the supersingular \(K3\) surface of Artin invariant \(1\) over \(k\). The question asked in the paper under review is whether or not there exist \(R\)-supersingular \(K3\) primes for a given Dynkin type \(R\) of rank \(20\). In other words, for a given \(R\), the existence of normal \(K3\) surfaces with singularity being of type \(R\) is the problem. This question in pure geometry is reduced to be an existence problem of certain lattices in Theorem 2.1. Denote by \(D_L\) the discriminant group, and \(q_L\) the discriminant form of a lattice \(L\). The theorem asserts that that an odd prime integer \(p\) being a \(R\)-supersingular \(K3\) prime when it does not divide \(\mathrm{disc} (R)\) is equivalent to that there existing an overlattice \(\tilde{L}\) of \(L(R)\) whose set of roots is the same as that of \(L(R)\), and a lattice \(T'\) of rank \(2\) and signature \((1,1)\) such that \((D_{T'}, pq_{T'})\simeq (D_{\tilde{L}}, q_{\tilde{L}})\). Besides, a tuple \(\tau(p):=(\tau_l(p))_l\) defined in sections 3 and 4, where \(l\) runs all odd prime divisors of \(\mathrm{disc}(R)\), characterizes whether or not this criterion holds. In conclusion, a list of all \(R\)-supersingular \(K3\) primes is given as a subset of the multiplicative group of units of a cyclic group.
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