Degree of irrationalities of algebraic surfaces with Kodaira dimension zero

1.Let Si(I=1，2)be smooth projective surfaces and f：S-D 21-D 2→S-D 22-D 2 be the surjective morphism.There has been a problem that whether the inequality dr(S-D 21-D 2)[greater than or equal]dr(S-D 22-D 2)hold true，where dr is the degree of irrationality.We have found examples which do not satisfy the inequality in the class of hyperelliptic surfaces.2。Let S be a hyperelliptic surface.We have proved that dr(S)=2，3or4，and that dr(S)=2if and only if2K I D2s I D2is trivial，where K I D2s ii D2 is the canonical bundle of S.3。Let C be a plane curve of degree d and K=k(C)be the rational function field of C.We consider a projection™p of C from P∈C to a line l=I D4~I D 4 P I D 11 D 1.The projection induces the extension of fieldsπI D2 P ii D1*I D1：k(L)*K.We have studied the structure of this extension from geometrical viewpoint.If the extension is Galois，we call P Galois point.We have determined all the Galois points.If P is not a Galois point，we cosider the minimal splitting field Lp of the extension K/k(L)and the Galois group Gal(Lp/k(L))。Our study have been done in this latter case only for d=4，the general case must be studied in the furture.

1. Let Si (I = 1,2) be smooth projective surfaces and f : SィイD21ィエD2 → SィイD22ィエD2 be the surjective morphism. There has been a problem that whether the inequality dr(SィイD21ィエD2) 【greater than or equal】 dr(SィイD22ィエD2) hold true, where dr is the degree of irrationality. We have found examples which do not satisfy the inequality in the class of hyperelliptic surfaces.2. Let S be a hyperelliptic surface. We have proved that dr(S) = 2, 3 or 4, and that dr(S) = 2 if and only if 2KィイD2sィエD2 is trivial, where KィイD2sィエD2 is the canonical bundle of S.3. Let C be a plane curve of degree d and K = k(C) be the rational function field of C. We consider a projection пp of C from P ∈ C to a line l =ィイD4〜ィエD4 PィイD11ィエD1. The projection induces the extension of fields πィイD2PィエD2ィイD1*ィエD1 : k(l) * K. We have studied the structure of this extension from geometrical viewpoint. If the extension is Galois, we call P Galois point. We have determined all the Galois points. If P is not a Galois point, we cosider the minimal splitting field Lp of the extension K/k(l) and the Galois group Gal(Lp/k(l)). Our study have been done in this latter case only for d = 4, the general case must be studied in the furture.