Galois points for plane curves and Cremona transformations
Let \(C\) be an irreducible curve of degree \(d \geq 3\) over a fixed algebraically closed field \(k\). Previously [J. Algebra 226, No. 1, 283--294 (2000; Zbl 0983.11067)], the author and \textit{H. Yoshihara} introduced the concept of Galois point for \(C\). Let \(\pi_P\) be the projection from \(C\) to a line \(l\) with the center \(P \in \mathbb{P}^2\). It induces a field extension \(\pi^*_P : k(\mathbb{P}^1) \hookrightarrow k(C)\). The point \(P\) is called Galois point if \(k(C)\) is a Galois extension of the field \(K_P = \pi^*_P(k(\mathbb{P}^1))\). Let \(L_P\) be the Galois closure of \(k(C)/k_P)\), and \(G_P = \mathrm{Gal}\,(L_P/K_P)\). Denote by \(\varphi\) the standard quadratic transformation \[ \mathbb{P}^2_X \ni (X_0:X_1:X_2) \mapsto \left( \frac{1}{X_0}: \frac{1}{X_1}: \frac{1}{X_2} \right) = (Y_0:Y_1: Y_2) \in \mathbb{P}^2_Y \] of projective planes. The author consider the points \(p_0, = (0:0:1), p_1 = (0:1:0), p_2 = (0:0:1)\) in \(\mathbb{P}^2_X\) and the same points \(p'_0,p'_1,p'_2 \in \mathbb{P}^2_Y\), and proves that for any plane curve \(C\) and any \(i =0,1,2\) there is an isomorphism between \(\pi^*_{P_i} : k(\mathbb{P}^1) \hookrightarrow k(C)\) and \(\pi^*_P : k(\mathbb{P}^1) \hookrightarrow k(\varphi[C])\). So, \(L_{p_i} \cong L_{p'_i}\) and \(G_{p_i} \cong G_{p'_i}\). This is the main result of the paper. Also, the author introduces the following minimality condition for a Galois point \(P\) on a plane curve \(C\) of degree \(d\). Let \(m_P\) be the multiplicity of \(C\) at \(P\), and \(m_1 \geq m_2 \geq \ldots \geq m_l\) be the multiplicities of \(C\) at its singular points (except \(P\)); in case where is no singular points, one puts \(m_1=m_2=1\). The pair \((C,P)\) is said to be \(GN\)-minimal if \(\mathrm{deg}\, C \geq m_P+m_1+m_2\). In the case where a Galois points \(P\) satisfies \(G_P \subset PGL(2,k)\), the author determines defining equations of \(GN\)-minimal curves of degree \(d \geq 3\). The paper is concluded with a list of four problems on Galois points and birational transformations.
scientific article; zbMATH DE number 5354919