This page is a copy of a webpage written by Hisao Yoshihara.
Latest version of "List of Problems": List of Problems (Latest version)
Changes of "List of Problems" (version: 6 September, 2021) after September 2021 are the responsibility of the second author.
We make questions public on the internet. Before stating them we introduce briefly the notion of Galois point and its generalization.
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 P^2 and consider the projection \pi_P : P^2 \cdots P^1 with the center P.
Restricting \pi_P, we get a dominant rational map f = f_P : C \dashrightarrow P^1,
which induces a finite extension of fields f* : k(P^1) \subset k(C).
If the extension is Galois, we call P a Galois point for C.
If P \notin C and P \in C, we call it outer and inner Galois point respectively.
Note that [k(C) : f*k(P^1)] = d - m, where m is 0 and mult_P(C) if P is outer and inner respectively.
If the extension is not Galois but separable, we take the Galois closure L_P of the extension.
(In case the extension is inseparable, the point is called a strange center.)
Let G_P = Gal(L_P/f*k(P^1)) be the Galois group and we call it the Galois group at P.
The nonsingular projective model C_P of L_P is called the Galois closure curve for C at P.
We denote by g(P) the genus of C_P, 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 \psi = \psi_L be the morphism associated with L = H^0(X, O(D)), where dim L = N+1 (≧ 3).
Put V = \psi(X) \subset P^N and n = dim V.
Let W be a linear subvariety of P^N of dim W = N-n-1 (≧ 0).
Consider the projection \pi_W with the center W; \pi_W : P^N \dashrightarrow P^n.
Restricting \pi_W to V, we get a dominant rational map f = f_W : V \dashrightarrpw P^n and a finite extension of fields k(V)/ f*k(P^n).
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 \cap W is empty and V \cap 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_W of the extension.
Let G_W = Gal(L_W/f*k(P^n) be the Galois group, which is called the Galois group at W.
The normal projective variety V_W, which is the L_W-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.
See for the details: List of Problems (Latest version)