Degree of irrationalities of algebraic surfaces with Kodaira dimension zero

Let Si (I = 1,2) be smooth projective surfaces and f：S - D21 - D2→S - D22 - D2 be the surjective morphism. There has been a problem that whether theinequality dr（si D21 D2）【greater than or equal】dr(si D22 D2) hold true，在那里博士是irrationality的学位，我们已经找到了一个例子，那就是不满意inequalityin the class of hyperelliptic surfaces.2. Let S be a hyperelliptic surface. We have proved thatdr(S) = 2, 3或4，and that dr(S) = 2 if and only if 2kwhere K - d - 2s - D2 is the canonical bundle of S.3. Let C be a plane curve of degree d and K = K (C) bethe rational function field of C. We consider a projection sinp of C from p∈C to a line l = D4 ~ D4P - D11 - D1. The projection induces The extension of fields π D2P - D2 - D1* - D1：k(l) * k . We have studied the structure of this extension from geometrical viewpoint. If theextension is Galois，我们call P Galois point.我们已确定所有Galois points. If P is not a Galois point，我们cosider the minimal splitting field Lp of the extension K/ K (l) and the Galois group Gal（Lp/ K (l)）。Our study 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.