arXiv subjects

Satoru Fukasawa

Publications and source records attributed to Satoru Fukasawa.

38 records · Page 3Linked to original sources

arXiv2011

Complete determination of the number of Galois points for a smooth plane curve

Let $C$ be a smooth plane curve. A point $P$ in the projective plane is said to be Galois with respect to $C$ if the function field extension induced from the point projection from $P$ is Galois. We denote by $δ(C)$ (resp. $δ'(C)$) the number of Galois points contained in $C$ (resp. in $\mathbb P^2 \setminus C$). In this article, we determine the numbers $δ(C)$ and $δ'(C)$ in any remaining open cases. Summarizing results obtained by now, we will have a complete classification theorem of smooth plane curves by the number $δ(C)$ or $δ'(C)$. In particular, we give new characterizations of Fermat curve and Klein quartic curve by the number $δ'(C)$.

math.AG↗

arXiv2009

Galois points for a normal hypersurface

We study Galois points for a hypersurface $X$ with $\dim {\rm Sing}(X) \le \dim X-2$. The purpose of this article is to determine the set $Δ(X)$ of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number of Galois points in terms of $\dim X$ and $\dim {\rm Sing}(X)$ if $Δ(X)$ is a finite set, and prove that $X$ is a cone if $Δ(X)$ is infinite. To achieve our purpose, we need a certain hyperplane section theorem on Galois point. We prove this theorem in arbitrary characteristic. On the other hand, the hyperplane section theorem has other important applications: For example, we can classify the Galois group induced from a Galois point in arbitrary characteristic and determine the distribution of Galois points for a Fermat hypersurface of degree $p^e+1$ in characteristic $p>0$.

math.AG↗