OPEN QUESTIONS

Galois points for plane curves and Galois embeddings
of algebraic varieties
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 P2 and
consider the projection  πP : P
2 …-> P1  with the center P. Restricting πP
to C, we get a dominant rational map f = fP : C …-> P1, which induces a finite
extension of fields f* : k( P1) ⊂ k(C). If the extension is Galois, we call P a
Galois point for C. If P \notin C and P ∈ C, we call it outer and inner Galois
point respectively.  Note that  [k(C) : f*k( P1)] = d - m, where m is 0 and
multP(C) if P is outer and inner respectively. If the extension is not Galois but
separable, we take the Galois closure LP of the extension. (In case the
extension is inseparable, the point is called a strange center.)  Let
GP = Gal(LP/ f*k(P1)) be the Galois group and we call it the
Galois group at P.
The nonsingular projective model CP of LP is called the
Galois closure curve for
C at P
. We denote by g(P) the genus of CP, which is called the genus at P.
These notions can be extended naturally to projective space curves or to higher
dimensional projective varieties ……. In such cases we call
Galois lines,
Galois subspaces …, 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 ψ = ψL be the morphism associated with L = H0(X, Ο(D)), where
dim L = N+1 (≧ 3).  Put  V = f (X) ⊂ PN　 and n = dim V.　Let W be a linear
subvariety of PN of dim W = N-n-1 (≧ 0). Consider the projection πW with the
center W; πW : PN …-> Pn.　Restricting πW to V, we get a dominant rational
map  f = fW : V …-> Pn and a finite extension of fields k(V)/ f*k(Pn). If this
extension is Galois, then W is called a Galois subspace and the pair (X, D) is said
to defines a
Galois embedding. Note that there are two cases:
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 LW of the
extension. Let GW = Gal(LW/ f*k(Pn) be the Galois group, which is called the Galois
group at W. The normal projective variety VW, which is the LW-normalizatio of V is
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,