Supersingular quartic surfaces
The aim of the paper under review is to study supersingular \(K3\) surfaces over an algebraically-closed field \(k\) of odd characteristic which is not \(5\). The main result is to give a projective model of them by showing that any of these \(K3\) surfaces is isomorphic to a smooth quartic surface in \(\mathbb{P}^3_k\). In the proof, key lemmas allow it to be sufficient to prove the statement for a supersingular \(K3\) surface of Artin invariant \(1\), which uniquely exists. The main body of the proof is to show that a \(K3\) surface \(X_\lambda\) defined by a polynomial \(f_{\lambda}:=x^4+y^4+z^4+w^4+\lambda xyzw\) with \(\lambda^4\not=256\), which is either ordinary or supersingular, is supersingular of Artin invariant \(1\). By a criterion for \(X_\lambda\) not to be ordinary in terms of the Frobenius morphism, the condition in the main theorem is finally given. The author also makes a remark about the projective model of the \(K3\) surfaces defined over a finite field. The present result is resembled to a well-known fact that all complex \(K3\) surfaces are diffeomorphic in particular to a smooth quartic surface in \(\mathbb{P}^3_{\mathbb{C}}\).
scientific article; zbMATH DE number 7076014
scientific article