** OPEN QUESTIONS**

**Galois points for plane curves and Galois embeddings
of algebraic
varieties
**(for the details and problems, please click HERE..

We make questions public on the internet. Before stating them we introduce

briefly the notion of Galois point and its genelarization.

Let k be an algebraically closed field of characteristic p 0. We assume it to

be a ground field of our discussions. Let C be an irreducible projective plane

curve of degree d ( 3) and k(C) the function field. Let P be a point in

consider the projection

to C, we get a dominant rational map f = f

extension of fields f

point respectively. Note that [k(C) : f

mult

separable, we take the Galois closure L

extension is inseparable, the point is called a strange center.) Let

G

The nonsingular projective model C

C at P. We denote by g(P) the genus of C

These notions can be extended naturally to projective space curves or to higher

dimensional projective varieties cc. In such cases we call Galois lines,

Galois subspaces c, respectively.

For smooth algebraic varieties not in projective spaces, we extend the notion

above as follows: Let X be a smooth algebraic variety with a very ample divisor D.

Let =

dim L = N+1 ( 3). Put V = f (X)

subvariety of

center W;

map f = f

extension is Galois, then W is called a Galois subspace and the pair (X, D) is said

to defines a

V W is empty and V W is not empty.

These correspond to outer and inner Galois point respectively. If there exists such

a divisor, then X is said to have a Galois embedding. On the other hand, if the

extension is not Galois but separable, we take the Galois closure L

extension. Let G

group at W. The normal projective variety V

called the Galois closure variety for V at W.

Concerning the framework above we have obtained several results. However, we

have lots of problems remain unsolved. We ask the questions in this webpage.

The results differ a great deal according to the characteristic of the ground field,

so we arrange the questions dividing in two cases: p = 0 and p > 0.

The open questions are written by H. Yoshihara in characteristic zero case and

S. Fukasawa in positive characteristic case.

For the details, please click HERE