Hypersurfaces in the three dimensional projective spaces and its complement

Let k be an algebraically closed field of characteristic zero. We denote by k a purelytranscendental extension of k, and by L a quadratic extension of k，If dim_k k = 1 or equivalently If k = k (x)， then L可以expressed as k (y)，where the element y satisfies the equation y^2 = (x-a_1)…with distinct a_ie_k . themodel of L is a rational，an elliptic or a hyperelliptic curve. In this study we consider the similar subject In the case whendim_kk = 2. Suppose K = K (x，让我们成为一个非单一模型L. Then we study the structure of S from the birationalviewpoint. First we obtain the following result:THEOREM 1. If L/K是一个quadratic extension，then the field L can be written as K(ROO<f(x, y)> where f(x，y) is a reduced polynomial of even degree such that the curve C：f = 0 in P2 has at most ordinary singularities.From the above we conclude the following:THEOREM 2.A belian and hyperelliptic surfaces cannot be birationally equivalent to double coverings of P^2，Except those, every class in the classification can appear。

Let k be an algebraically closed field of characteristic zero. We denote by K a purely transcendental extension of k, and by L a quadratic extension of K, If dim_k K = 1 or equivalently if K = K(x), then L can be expressed as K(y), where the element y satisfies the equation y^2 = (x-a_1)・・・(x-a_<2n+1>) with distinct a_iE_k. The model of L is a rational, an elliptic or a hyperelliptic curve. In this study we consider the similar subject in the case when dim_kK = 2. Suppose K = k(x, y) and let S be a nonsingular model of L. Then we study the structure of S from the birational viewpoint. First we obtain the following result :THEOREM 1. If L/K is a quadratic extension, then the field L can be written as K(ROO<f(x, y)> where f(x, y) is a reduced polynomial of even degree such that the curve C : f = 0 in P2 has at most ordinary singularities.From the above we conclude the following :THEOREM 2. A belian and hyperelliptic surfaces cannot be birationally equivalent to double coverings of P^2, Except those, every class in the classification can appear.