These are some questions concerning Mumford's "Lectures on curves on an algebraic surface".

We concern ourselves with questions of the Picard variety $P$, and its dimension, of a complete nonsingular surface $F$ over an algebraically closed ground field $k$ of arbitrary characteristic. Let $\mathfrak{o}$ be the sheaf of local rings on $F$, $\{C_\alpha: \alpha \in S\}$ a family of curves on $F$ such that $C_0 = C$ is nonsingular and $N$ the sheaf of sections of the normal bundle to $C$ on $F$. Then the equality (A): $\dim P = h^1(\mathfrak{o})$ is equivalent to (B): $\{\text{Tangent space to }S \text{ at }\alpha = 0\} \overset{\rho}{\to} H^0(N)$ is surjective for suitable $\{C_\alpha : \alpha \in S\}$.

I was wondering if someone could give me their explanation/intuitions behind the following, as I am finding the book to be quite terse.

1. The algebraic solution of problem (B) for characteristic $0$, following Kodaira-Spencer.
2. The algebraic solution of problem (A) for characteristic $0$, following Grothendieck, using a theorem of Cartier on an algebraic group scheme.
3. Problem (A) in characteristic $p$, i.e. $\dim P$ may not be given in general by $h^1(\mathfrak{o})$ but that the tangent space to $P$ (at a point) corresponds to the subspace of $H^1(\mathfrak{o})$ annihilated by the Bockstein operators.

Thanks in advance!

These are some questions concerning Mumford's "Lectures on curves on an algebraic surface".

We concern ourselves with questions of the Picard variety $P$, and its dimension, of a complete nonsingular surface $F$ over an algebraically closed ground field $k$ of arbitrary characteristic. Let $\mathfrak{o}$ be the sheaf of local rings on $F$, $\{C_\alpha: \alpha \in S\}$ a family of curves on $F$ such that $C_0 = C$ is nonsingular and $N$ the sheaf of sections of the normal bundle to $C$ on $F$. Then the equality (A): $\dim P = h^1(\mathfrak{o})$ is equivalent to (B): $\{\text{Tangent space to }S \text{ at }\alpha = 0\} \overset{\rho}{\to} H^0(N)$ is surjective for suitable $\{C_\alpha : \alpha \in S\}$.

I was wondering if someone could give me their explanation/intuitions behind the following, as I am finding the book to be quite terse.

1. The algebraic solution of problem (A) for characteristic $0$, following Grothendieck, using a theorem of Cartier on an algebraic group scheme.
2. Problem (A) in characteristic $p$, i.e. $\dim P$ may not be given in general by $h^1(\mathfrak{o})$ but that the tangent space to $P$ (at a point) corresponds to the subspace of $H^1(\mathfrak{o})$ annihilated by the Bockstein operators.

Thanks in advance!