# Motivation for Construction of Jacobian

$$\DeclareMathOperator{\Pic}{Pic}$$ 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

• 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. – Tabes Bridges Aug 17 at 20:52