Hisao Yoshihara

Field theory for the function field of the quintic fermat curve

We study the structure of the function field K of the quintic Fermat curve F(5) in the following way: Let K m be a g-maximal rational subfield of K. Then the field extension K/K m is obtained by the projection from F(5) to a line with a center p, ∈ F(5). By using this fact, we consider the field extension K/K m from several point of view.

Rational curve with one cusp

In this paper we shall give new examples of plane curves C satisfying C {P}A for some point P E C by using automorphisms of open surfaces. 1. In this paper we assume the ground field is an algebraically closed field of characteristic zero. Let C be an irreducible algebraic curve in P2. Then let us call C a curve of type I if C {P} -A1 for some point P E C, and a curve of type II if C\L _ A1 for some line L. Of course a curve of type II is of type I. Curves of type II have been studied by Abhyankar and Moh [1] and others. On the other hand, if the logarithmic Kodaira dimension of P2 C is -o0 or the automorphism group of p2 C is infinite dimensional, then C is of type I [3,6]. So the study of type I seems to be important, but only a few examples are known that are of type I and not of type II. Moreover the definitions of those examples are not so plain [4,5]. Here we shall give new examples by using automorphisms of open surfaces. 2. First we recall the properties of type I. Let (e,,...,ee) be the sequence of the multiplicities of the infinitely near singular points of P in case C is of type I with degree d 3. Then put R d22 e2 e + 1. I~ I Since a curve of type II is an image of a line by some automorphism of A2 [1], we see that R 2 2 for this curve by the same argument as below. Denoting by G the automorphism group of p2 C, we have G D Am for all m > 1 if R > 0, and G is a finite group if R e + e2, where L is the tangent at P. Thus we have d = Ip(C, L), which implies c n L = (P1, i.e., C is type II. Received by the editors November 16, 1982. 1980 Mathematics Subject Classification. Primary 14H45. ?1983 American Mathematical Society 0002-9939/83/0000-1455/

Galois Lines for Normal Elliptic Space Curves, II

01.75

Plane curves whose singular points are cusps and triple coverings of ℙ2

Let C be a curve, and l and l0 be lines in the projective three space ℙ3. Consider a projection πl: ℙ3 ⋯ → l0 with center l, where l ⋂ l0= ∅. Restricting πl to C, we obtain a morphism πl|C : C → l0 and an extension of fields (πl|C)* : k(l0) ↪ k(C). If this extension is Galois, then l is said to be a Galois line. We study the defining equations, automorphisms and the Galois lines for quartic curves, and give some applications to the theory of plane quartic curves.

Galois Lines for Space Curves

We study a plane curve C whose singular points are cusps and the surface which is a triple covering of ℙ2 branched along C. As a result, especially we obtain an inequality for the sum of the Milnor numbers at the singularities of C and new surfaces of general type.

Degree of irrationality of a product of two elliptic curves

Let C be a curve, and l, l0 be lines in the projective three space ℙ3. Consider a projection πl: ℙ3 ⋯ → l0 with center l, where l ∩ l0 = ∅. Restricting πl to C, we get a morphism πl|C: C → l0 and an extension of fields (πl|C)* : k(l0) ↪ k(C). We study the algebraic structure of the extension and the geometric structure of C. In particular, we study the structure of the Galois group and the number of Galois lines for some special cases.

Double coverings of rational surface

We consider the degree of irrationality dr(S) of some algebraic surface S. Firstly we give an estimate of dr(S) for a surface S with a structure of a fiber space. Secondly we prove the existence of a nonsingular curve of genus 3 on E × E for a certain elliptic curve E with complex multiplications. As a corollary, we obtain that dr(E × E) = 3.

A Note on the Existence of Some Curves

Let K be a purely trancendental extension of a field k with two variables. We want to classify fields {L}, which are quadratic extension of K. First we prove the existence of a normal form of the defining equation of L. Then we give a rough classification and several results. Lastly we mention of an application to the theory of plane curves.

Galois lines for space elliptic curve with \(j=12^3\)

Publisher Summary There is a classical problem whether there exist curves in the complex projective plane P 2 with assigned numerical characters satisfying the genus formula of Clebsch. More than half a century ago, Lefschetz and Zariski studied such a problem for Pluckerian characters. This chapter discusses the case where (d, e) [resp. (d, e)] is a pair of positive integers, then it is said to be effective if there is a curve C in P 2 with C – { P } = A 1 [resp. G m ] for some point P , where d = deg C and e = mult p C . Then, there is the following conjecture: if a pair (d, e) [resp. (d, e)] is effective, then d d ≤ 3e]. In the former case, Tsunoda showed that d ≤ 3e + 2 by using the logarithmic version of Miyaokas inequality. The chapter proves the following theorem by considering the double covering of P 2 branched along C.

Smooth quotients of bi-elliptic surfaces

The