Canonical coverings of Enriques surfaces in characteristic 2 (Q2165752)

Consider a normal canonical covering \(Y\to X\) of an Enriques surface \(X\) over an algebraically closed field of characteristic \(2\). The aim of the paper under review is to study such coverings of an Enriques surface. More precisely, the first part of the main result is to give a classification of types of Enriques surface (i.e., classical or supersingular) in accordance with the types of singularities in \(Y\), and the second part is devoted to determine possible types of singularities on \(Y\) with the covering. A key technique is a correspondence between regular derivation of \(\mathcal{O}_Y\) and the quotient \(Y^D\). Indeed, if a regular derivation \(D\) of \(\mathcal{O}_Y\) is fixed-point free, then, the quotient \(Y^D\) admits the normal canonical covering \(Y\to Y^D\). Moreover, it is known that of type classical or supersingular of \(Y^D\) is determined by the type of \(D\) being multipricative or additive. The proof of the first part depends on case-by-case analysis of the types of singularities of \(Y\). The second result is achieved by giving an explicit example of a pair of a projective model of \(Y\) and an appropriate derivation \(D\) for each of the candidate singularities.