Projective Surface with canonical hyperplane section
In this post, I will try to give an Introduction and some basic properties of canonical hyperplane section of a projective surface. The definitions, theorems or properties related to this kind of surface are taken from the thesis of Epema. This is also a part of my master 2 thesis studies. I. Introduction: First we come to the definition of surface with canonical hyperplane sections: Definition 1.1 : A curve $C$ with genus $g \geq 2$ is called canonically embedded if there exists an embedding $i_{|K_C|}: C \hookrightarrow \mathbb{P}^{g-1}$. Here this curve must be smooth since we assume that $deg K_C = 2g - 2$. Definition 1.2 : $X$ is called a surface with canonical hyperplane sections if there exists an embedding $i : X \hookrightarrow \mathbb{P}^g$, $g \geq 3$, such that a general hyperplane sections $C$ of $i(X)$, i.e., the intersection of $X$ with a hyperplane of degree $1$ is a canonically embedded curve with genus $g$. Remark 1 : Here, $C$ is the general curve of th...