Smooth quotients of bi-elliptic surfaces

We consider the quotient X of bi-elliptic surface by a finite automorphism group. If X is smooth, then it is a bi-elliptic surface or ruled surface with irregularity one. As a corollary any bi-elliptic surface cannot be Galois covering of projective plane, hence does not have any Galois embedding

Galois embedding of K3 surface --abelian case--

We study Glois embeddings of K3 surfaces in the case where the Galois groups are abelian. We show several properties of K3 surfaces concerning the Galois embeddings. In particular, if the Galois group G is abelian, then G is isomorphic to Z/4Z, Z/6Z or Z/2Z\oplusZ/2Z\oplusZ/2Z and S is a smooth complete intersection of hypersurfaces. Further, we state the detailed structure of such surfaces

Sextic variety as Galois closure variety of smooth cubic

Let V be a nonsingular projective algebraic variety of dimension n. Suppose there exists a very ample divisor D such that D^n=6 and dim H^0(V, O(D))=n+3. Then, (V, D) defines a D_6-Galois embedding if and only if it is a Galois closure variety of a smooth cubic in P^{n+1} with respect to a suitable projection center such that the pull back of hyperplane of P^n is linearly equivalent to D