Assume we have a smooth algebraic curve $C$ over complex numbers. Consider then all possible embeddings of $C$ into a smooth algebraic surface $X$.

Here are two questions:

1. How many are there different such embeddings for fixed $C$ and $X$? (in moduli-theoretic sense)

(I understand that this must heavily depend on $X$: type, algebraic structure etc. as well as on $C$).

The second question is easily answered below by Sasha but I leave it here:

2. Is there a way to characterize the divisor classes on $C$ corresponding to normal bundles of $C$ for various such embeddings? (Here $X$ is still fixed, although it makes sense to ask a generalized version of it for $X$ varying in the moduli space of the type to which $X$ belongs).

Assume we have a smooth algebraic curve $C$ over complex numbers. Consider then all possible embeddings of $C$ into a smooth algebraic surface $X$.

Here are two questions:

1. How many are there different such embeddings for fixed $C$ and $X$? (in moduli-theoretic sense)

(I understand that this must heavily depend on $X$: type, algebraic structure etc. as well as on $C$).

2. Is there a way to characterize the divisor classes on $C$ corresponding to normal bundles of $C$ for various such embeddings?

Here $X$ is still fixed. A generalized version of this question for $C$ fixed and $X$ varying in the moduli space of the type to which $X$ belongs is partially answered by Sasha below, where he chose $X$ to vary in the class of geometrically ruled surfaces.

Assume we have a smooth algebraic curve $C$ over complex numbers. Consider then all possible embeddings of $C$ into a smooth algebraic surface $X$.

Here are two questions:

1. How many are there different such embeddings for fixed $C$ and $X$? (in moduli-theoretic sense)

(I understand that this must heavily depend on $X$: type, algebraic structure etc. as well as on $C$).

2. Is there a way to characterize the divisor classes on $C$ corresponding to normal bundles of $C$ for various such embeddings? (Here $X$ is still fixed, although it makes sense to ask a generalized version of it for $X$ varying in the moduli space of the type to which $X$ belongs).

Assume we have a smooth algebraic curve $C$ over complex numbers. Consider then all possible embeddings of $C$ into a smooth algebraic surface $X$.

Here are two questions:

1. How many are there different such embeddings for fixed $C$ and $X$? (in moduli-theoretic sense)

(I understand that this must heavily depend on $X$: type, algebraic structure etc. as well as on $C$).

The second question is easily answered below by Sasha but I leave it here:

2. Is there a way to characterize the divisor classes on $C$ corresponding to normal bundles of $C$ for various such embeddings? (Here $X$ is still fixed, although it makes sense to ask a generalized version of it for $X$ varying in the moduli space of the type to which $X$ belongs).