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 the linear system $|L|$ which defines the embedding $i$. Indeed, we see that there is a global section $s$ of $O_X(L)$ corresponds to a global section $x_i$ of $O_{\mathbb{P}^g}$, and $H = Z(x_i)$, which states that $C$ must be a zero locus of $s$ and hence $C \sim L$. Since $|K_C|$ is complete, then $|L|$ must also be complete. Also $C$ is required to be a smooth curve by the definition 1. Then by the definition of closed immersion defined by a complete linear system, we must have that $O_X (L) = O_X (C) \cong O_X (1) \cong i^{*} O_{\mathbb{P}^g} (1)$ and $O_X (1) \otimes O_C \cong O_X(C) \otimes O_C \cong O_C(K_C)$. By adjunction formula, that is $C.(C+K_X)=K_C$, we have that $C.K_X = 0$. We can also see that since $C$ is a smooth curve, then $X$ is a surface with at most isolated singularities, because by the Jacobian's criterion of singular locus of projective surface embedded in projective space, we have that $Sing(X) \cap H = Sing( X \cap H)$ with $H$ a general hyperplane. Since $X \cap H$ is $C$ which is smooth, so $Sing(X) \cap H = \varnothing $, which means $Sing (X)$ must be points or empty.
Now let $\pi: X' \rightarrow X$ be the minimal resolution of $X$, i.e., the resolution of singularites in which fibers of $\pi$ on $X'$ do not contain any contractible curves (curves which have negative self-intersection). Let $C' = \pi^{*} C$, since $C$ is smooth in $X$ then the neighborhoods of $C$ and $C'$ are isomorphic to each other, which means $C'$ is also an irreducible curve and $(K_X'+\sum_{i=1}^{n} E_i ).C' = K_X.C = 0$ (by the properties monoidal transformation), but $C'.E_i = 0$ also, hence $C'.K_{X'} = 0$ which implies $C'^{2} = K_C'$, i.e., $O_{X'}(C') \otimes O_{C'} = O_{C'} (K_{C'})$. Here my version is a little bit different from Epema, since Epema said the isomorphic of neighborhoods of $C'$ and $C$ implies immediately that $O_{X'}(C') \otimes O_{C'} = O_{C'} (K_{C'})$, and then get the other fact by using adjunction. It's still a little bit unclear for me about the argument of Epema, since the isomorphic of $C'$ and $C$ just only reserve the data of these curves on the smooth locus, but not the canonical divisor.
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 the linear system $|L|$ which defines the embedding $i$. Indeed, we see that there is a global section $s$ of $O_X(L)$ corresponds to a global section $x_i$ of $O_{\mathbb{P}^g}$, and $H = Z(x_i)$, which states that $C$ must be a zero locus of $s$ and hence $C \sim L$. Since $|K_C|$ is complete, then $|L|$ must also be complete. Also $C$ is required to be a smooth curve by the definition 1. Then by the definition of closed immersion defined by a complete linear system, we must have that $O_X (L) = O_X (C) \cong O_X (1) \cong i^{*} O_{\mathbb{P}^g} (1)$ and $O_X (1) \otimes O_C \cong O_X(C) \otimes O_C \cong O_C(K_C)$. By adjunction formula, that is $C.(C+K_X)=K_C$, we have that $C.K_X = 0$. We can also see that since $C$ is a smooth curve, then $X$ is a surface with at most isolated singularities, because by the Jacobian's criterion of singular locus of projective surface embedded in projective space, we have that $Sing(X) \cap H = Sing( X \cap H)$ with $H$ a general hyperplane. Since $X \cap H$ is $C$ which is smooth, so $Sing(X) \cap H = \varnothing $, which means $Sing (X)$ must be points or empty.
Now let $\pi: X' \rightarrow X$ be the minimal resolution of $X$, i.e., the resolution of singularites in which fibers of $\pi$ on $X'$ do not contain any contractible curves (curves which have negative self-intersection). Let $C' = \pi^{*} C$, since $C$ is smooth in $X$ then the neighborhoods of $C$ and $C'$ are isomorphic to each other, which means $C'$ is also an irreducible curve and $(K_X'+\sum_{i=1}^{n} E_i ).C' = K_X.C = 0$ (by the properties monoidal transformation), but $C'.E_i = 0$ also, hence $C'.K_{X'} = 0$ which implies $C'^{2} = K_C'$, i.e., $O_{X'}(C') \otimes O_{C'} = O_{C'} (K_{C'})$. Here my version is a little bit different from Epema, since Epema said the isomorphic of neighborhoods of $C'$ and $C$ implies immediately that $O_{X'}(C') \otimes O_{C'} = O_{C'} (K_{C'})$, and then get the other fact by using adjunction. It's still a little bit unclear for me about the argument of Epema, since the isomorphic of $C'$ and $C$ just only reserve the data of these curves on the smooth locus, but not the canonical divisor.
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