I am trying to understand the construction of the Jacobian of a curve following the notes of J. S. Milne
The question is going to be about a particular step in the proof of Proposition 4.2b in Chapter III, but I will first briefly recall the setup.
Let $X$ be a scheme flat over $T$, a divisor $D$ on $X$ is called relative effective divisor on $X/T$ if it is effective and flat over $T$ as a subscheme of $X$ (definition 3.4). There is a one-to-one correspondence benween relative effective divisors and sheaves $\mathcal L$ over $X$ with a global section $s$ such that $\mathcal{L}/s\mathcal{O}_X$ is flat over $T$.
We are working over a field. Let $C$ be a non-singular curve of genus $\geq 2$.
We are trying to construct a section of the natural map of functors $Div^r_C \to P^r_C$ where the first functor is the functor of relative effective divisors on $C\times T/T$ of degree $r$, and is represented by the $r$-fold symmetric product of $C$, and the second functor is the functor of families of degree $r$ invertible sheaves on $C$ parametrised by $T$, modulo trivial families:
$$
P^r_C(T) = \{ \mathcal{L} \in Pic(C \times T) \mid deg\ \mathcal{L}_t=r \textrm{ for all }\ t \in T\} / q^* Pic(T)
$$
(the natural projections are denoted $p: C \times T \to C$, $q: C\times T \to T$.)
Proposition 4.2 deals with subfunctors of $Div^r_C$ and $P^r_C$. We pick an effective degree $(r-g)$ divisor $D_\gamma$ and define
$$
C^\gamma(T) = \{ D \in Div^r_C(T) \mid h^0(D_t-D_\gamma)=1\ \textrm{ for all }\ t \in T\}
$$
$$
P^\gamma(T) = \{ \mathcal{L} \in Pic^r_C(T) \mid h^0(\mathcal{L}_t \otimes \mathcal{L} _\gamma^{-1})=1\ \textrm{ for all }\ t \in T\}
$$
part b) constructs a section $P^\gamma \to C^\gamma$. Take $\mathcal{L} \in P^\gamma(T)$. By definition of $P^\gamma$ and by Riemann-Roch, $h^1 (\mathcal{L}_t \otimes \mathcal{L}^{-1}_\gamma)=0$. This allows us to apply a base change theorem and coclude that $q_*(\mathcal{L} \otimes p^* \mathcal{L}^{-1} _\gamma)$ is locally free and thus an invertible sheaf on $T$ (call it $\mathcal{M}$). The proof then proceeds to construct a section of $\mathcal{L} \otimes (q^* q_*(\mathcal{L} \otimes p^* \mathcal{L} _\gamma^{-1}))^{-1}$.
In particular, as there is a natural map $q^* q_*(\mathcal{L} \otimes p^* \mathcal{L} _\gamma^{-1}) \to \mathcal{L} \otimes p^* \mathcal{L} _\gamma^{-1}$, one has a canonical global section of $\mathcal{L} \otimes p^* \mathcal{L} _\gamma^{-1} \otimes (q^*\mathcal{M})^{-1}$, and by composing it with the natural map $p^* \mathcal{L} _\gamma^{-1} \to \mathcal{O}_{C\times T}$ one gets the desired.
We did obtain a section $s_\gamma$ of a sheaf equivalent to $\mathcal{L}$ (in the sense of the definition of $Pic^r_C$)
My question is: why does this section give rise to a relative effective divisor, that is, why would $\mathcal{L} \otimes (q^* \mathcal{M})^{-1}/s_\gamma \mathcal{O}_{C\times T}$ be flat over $T$?
Best Answer
Let $N = L \otimes (q^\ast q_\ast (L\otimes L_\gamma^{-1}))^{-1}$. It suffices to show that the zero locus $D \subset C \times T$ of $s \in \Gamma(N)$ is flat over $T$. If $T$ is nice (Noetherian, blah, blah), it then suffices to show that the fiberwise degree of $D$ is constant. Note that the restriction of $N$ to $C \times \{ t \}$ is isomorphic to the restriction of $L$ to $C \times \{ t \}$. Since $L$ is fiberwise degree $r$ by assumption, it follows that $D$ is fiberwise degree $r$.
Oh and you need to check that the section $s$ is fiberwise nonzero.
Well, here's how you do that --- first look at the map $\phi : q^\ast q_\ast (L \otimes p^\ast L_\gamma^{-1}) \to L \otimes p^\ast L_\gamma^{-1}$. What does this map look like on fibers? Well, note that $H^0 (C \times \{ t \} , (L \otimes p^\ast L_\gamma^{-1})|\_{C \times \{ t \}})$ is by assumption 1 dimensional. Hence the restriction of $q^\ast q_\ast (L \otimes p^\ast L_\gamma^{-1})$ to $C \times \{ t \}$ is a trivial line bundle. So, the map $\phi$ on the fiber $C \times \{ t \}$ looks like the map $\mathcal{O}\_{C \times \{ t \}} \to (L \otimes p^\ast L_\gamma^{-1})|\_{C \times \{ t \}}$ corresponding to the one nonzero global section in $H^0 (C \times \{ t \} , (L \otimes p^\ast L_\gamma^{-1})|\_{C \times \{ t \}})$. Clearly this map is not zero.
I'll let you do the rest...