Let $R$ be compact a Riemann surface of genus $g$ and $ J (R) $ be its Jacobian. For a subvariety $X$ of $J(R)$ of dimension $d$, denote the set of non-singular points of $X$ by $X_{reg}$. Then the Gauss map of $X$ can be written as $$ \begin{array}{llll} G:&X_{reg}&\longrightarrow&(\mathbb{P}^{g-1})^* \\ &x&\longrightarrow& \mathbb P[T_x(X_{reg})]\\ \end{array} $$ here $(\mathbb{P}^{g-1})^*$ is the dual projective of $\mathbb{P}^{g-1}$, that is, $(\mathbb{P}^{g-1})^*$ identified with the set of hyperplanes in $\mathbb{P}^{g-1}$.

Let's say I want to calculate the degree of the Gauss map above. I'm following Principles of Algebraic Geometry, GRIFFITHS and HARRIS and also Advances in Moduli Theory, Shimizu and Ueno.

It follows from facts proven in the cited references that $G(X_{reg})$ is a set of points of $(\mathbb{P}^{g-1})^*$ corresponding to hyperplanes of $\mathbb{P}^{g-1} $ with certain properties $P_1$, and in addition also $\overline{G(X_{reg})}= (\mathbb{P}^{g-1})^*$, to Zariski closure of $G(X_{reg})$ . As I said, I would like to calculate the degree of the Gauss map $G$, so the references calculate $\#G^{-1}(H)$, at where $H \in G(X_{reg}) \subset (\mathbb{P}^{g-1})^*$. I imagined that first, we would need to verify that $ H $ is not a branch point of $G$, but this is not done in the references. What they actually do is take $H \in A \subset G(X_{reg})$, where $A$ is a dense open subset of $(\mathbb{P}^{g-1})^*$ that has certain properties $P_2 \subset P_1$.Thus, taken $H \in A$ is then calculated $\#G^{-1}(H)$. Only in future steps do the references verify that $H \in A$ is not a branch point of $G$.

Why is this correct? That is, why do not you need, in this case, to verify that $H \in A$ is not a branch point of $G$, before calculating $\#G^{-1}(H)$? It would be because $A$ is a dense open subset of $(\mathbb{P}^{g-1})^*$??

Thanks!

Let $R$ be compact a Riemann surface of genus $g$ and $ J (R) $ be its Jacobian. For a subvariety $X$ of $J(R)$ of dimension $d$, denote the set of non-singular points of $X$ by $X_{reg}$. Then the Gauss map of $X$ can be written as $$ \begin{array}{llll} G:&X_{reg}&\longrightarrow&(\mathbb{P}^{g-1})^* \\ &x&\longrightarrow& \mathbb P[T_x(X_{reg})]\\ \end{array} $$ here $(\mathbb{P}^{g-1})^*$ is the dual projective of $\mathbb{P}^{g-1}$, that is, $(\mathbb{P}^{g-1})^*$ identified with the set of hyperplanes in $\mathbb{P}^{g-1}$.

Let's say I want to calculate the degree of the Gauss map above. I'm following Principles of Algebraic Geometry, Griffiths and Harris and also Advances in Moduli Theory, Shimizu and Ueno.

It follows from facts proven in the cited references that $G(X_{reg})$ is a set of points of $(\mathbb{P}^{g-1})^*$ corresponding to hyperplanes of $\mathbb{P}^{g-1} $ with certain properties $P_1$, and in addition also $\overline{G(X_{reg})}= (\mathbb{P}^{g-1})^*$, to Zariski closure of $G(X_{reg})$ . As I said, I would like to calculate the degree of the Gauss map $G$, so the references calculate $\#G^{-1}(H)$, at where $H \in G(X_{reg}) \subset (\mathbb{P}^{g-1})^*$. I imagined that first, we would need to verify that $ H $ is not a branch point of $G$, but this is not done in the references. What they actually do is take $H \in A \subset G(X_{reg})$, where $A$ is a dense open subset of $(\mathbb{P}^{g-1})^*$ that has certain properties $P_2 \subset P_1$.Thus, taken $H \in A$ is then calculated $\#G^{-1}(H)$. Only in future steps do the references verify that $H \in A$ is not a branch point of $G$.

Why is this correct? That is, why do not you need, in this case, to verify that $H \in A$ is not a branch point of $G$, before calculating $\#G^{-1}(H)$? It would be because $A$ is a dense open subset of $(\mathbb{P}^{g-1})^*$??

Thanks!