In the following context: Let $S$ be a connected smooth projective surface over $\mathbb{C}$, and let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$. Let $d=\dim(\Sigma)$ and $\phi_{\Sigma}: S\hookrightarrow \mathbb{P}^{d} $ the closed embedding of $S$ on $\mathbb{P}^{d}$, induced by $\Sigma$.
Let $\mathbb{P}^{d*}$ be the dual projective space of $\mathbb{P}^{d}$ parametrizing hyperplanes in $\mathbb{P}^{d}$ and let $\overline{\eta}$ be the geometric generic point of $\mathbb{P}^{d*}$. Recall that by definition $\Sigma=\mathbb{P}^{d*}$.
For any closed point $t\in\Sigma=\mathbb{P}^{d*}$, let $H_t$ be the hyperplane in $\mathbb{P}^{d}$ corresponding to $t$, $C_t=H_t\cap S$ the corresponding hyperplane section (which are curves over $\mathbb{C}$) of $S$, and
\begin{equation*} r_t:C_t\hookrightarrow S \end{equation*} the closed embedding of the curve $C_t$ into $S$.
My questions are:
For the geometric generic point $\overline{\eta}\in\Sigma=\mathbb{P}^{d*}$, can I also consider $H_{\overline{\eta}}$ be the hyperplane in $\mathbb{P}^{d}$ corresponding to $\overline{\eta}$ and $C_{\overline{\eta}}=H_{\overline{\eta}}\cap S$ be the corresponding hyperplane section of $S$ (which is a curve over $\overline{\eta}$), and
\begin{equation*} r_{\overline{\eta}}:C_{\overline{\eta}} \hookrightarrow S \end{equation*} the closed embedding of the curve $C_{\overline{\eta}}$ into $S$ ?.In the paper: https://arxiv.org/abs/1405.6430v2 the autors work in a more general setting: instead of $S$ they work with a variety $X$ of dimension $2n$, and the geometric generic hyperplane section is denoted by $Y_{\overline{\eta}}$. They work over an uncountable algebraically closed field of characteristic zero and they work with the etale cohomoly.
Since I am working over the complex numbers I think I can use the usual cohomology (for example with integers coeficients) instead of the etale cohomology, but when I want to prove some facts about the geometric generic curve $C_{\overline{\eta}}$ (which is a curve over $\overline{\eta}$ ) it seems weird to use the cohomology with integer coefficients (i.e. $H^{1}(C_{\overline{\eta}},\mathbb{Z})$ ), so I am wondering if I should use etale cohomology to proof properties about the geoemetric generic curve $C_{\overline{\eta}}$ of the family parametrized by $\Sigma$?
Thank you very much for your help!
