$\DeclareMathOperator{\Pic}{Pic}$ Hello everybody, in a course on the Jacobians of Curves, the lecturer gave the following Motivation for the Construction of the Jacobian:

Let $D$ be a divisor on a proper connected regular curve of genus $g=\dim_k(C,\Omega_C)$ over an algebraically closed field $k$. By the Riemann-Roch Theorem, we have $\ell(D)-\ell(K-D)=\deg(D)+1-g$ where $K$ is a canonical divisor
and $\ell(D)=\dim_k H^0(C,\mathcal{O}_C(D))$.
$\bullet$ If $\deg(D)=g$, then $\ell(D)\geq 1$ and hence $\mathcal{O}_C(D)$ has non-vanishing global sections.
$\bullet$ If furthermore $\ell(K-D)=0$, then $\mathcal{O}_C(D)$ is a one dimensional $k$-vector space.
$\bullet$ If $g\leq 1$, then $\ell(K-D)=0$ is automatic, but for general $g$ it may depend on $D$.
$\bullet$ It is possible to construct an effective Divisor $D=\sum_{i=1}^g p_i$ where $p_i$ are some closed (non-generic) points of $C$, such that $\deg(D)=g$ and $\ell(K-D)=0$.

I understand all of the above assertions, but not the following one:
By the upper semi-continuity theorem - this is the fact that $y\mapsto \dim_k H^p(X_y,\mathscr{F}\mid_{X_y})$ is upper semi-continuous - the above behavior is generic, in the sense that if we just choose the $p_i$ randomly, this will almost surely give us a divisor $D$ with $\ell(K-D)=0$.

This is the assertion that I don't understand. To be more specific, I don't see what is the sheaf that I need to take for $\mathscr{F}$? I mean, we could consider $X=C$, and $y=p$ and for $\mathscr{F}$ we could take $\mathcal{O}_C(-p)$ to obtain an assertion about $-\ell(p)$ but then $\mathscr{F}$ is dependent of $p$!? I guess the idea is to change $p_i$ randomly and show that it does almost never affect the value of $\ell(\sum_{i=1}^g p_i)$, right?

For the sake of completeness, Id like to point out, that the above is a motivation for identifying an open subscheme of $\Pic_{C/k}^g$ with an open subscheme of the "scheme of effective divisors of degree $g$" which is constructed using quotiengs by the action of the symmetric group that is acting on th $g$-fold product $C\times\cdots\times C$ by permuting the factors.

Thank You for Your efforts, appreciate,
greetings SDIGR

  • $\begingroup$ You should take the sheaf $\mathscr F$ to be a flat family of line bundles that corresponds to a varying family of effective divisors, typically on the space $C\times C \to C$. Consider a fixed divisor $D$ such that for generic $p \in C$ you have $h^0(C,\mathcal O_C(K-(D+p))) = 0$. Then as $p \in C$ varies, $\mathcal O_{C\times C}(D+p)$ is flat over $C$, and its global sections are uppersemicontinuous (for finitely many $p$, there will be an extra section). Actually, the vanishing is unnecessary, but it makes the situation a bit cleaner. $\endgroup$ – Tabes Bridges Aug 17 at 20:52

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