= Solution
Put $n=3g-3$. To specify the <Abel map of an algebraic curve> into $X=\operatorname{Pic}^{g-1}(C)$, use
$$
\alpha(z_1,\ldots,z_n)=\left[\sum_{j=1}^n z_j-K_C\right].
$$
The canonical-class twist is necessary: an untwisted sum has degree $3g-3$, rather than $g-1$. Let $L=\mathcal O_C(2K_C)$. The <Riemann-Roch theorem> and <Serre duality> give $h^0(L)=n$ and $h^1(L)=0$.
On $C\times C^n$, write $\Gamma_i$ for the graph of the $i$th coordinate, $\mathcal D=\sum_i\Gamma_i$, and $\pi$ for projection to $C^n$. The locally free sheaf $E_L=\pi_*(L|_{\mathcal D})$ has rank $n$; at repeated points it records values and jets. Evaluation gives a morphism of rank-$n$ <vector bundles>
$$
H^0(C,L)\otimes\mathcal O_{C^n}\longrightarrow E_L.
$$
Its kernel at a tuple is $H^0(C,2K_C-\sum z_j)$. By the <Riemann-Roch theorem>, this space has the same dimension as $H^0(C,\sum z_j-K_C)$, since both <line bundles> have degree $g-1$. Thus its <determinant> vanishes exactly along $\alpha^*\Theta$. This is also an equality of <Cartier divisors> with multiplicities: the evaluation complex computes the <sheaf cohomology> of $L(-\mathcal D)$; <Serre duality> identifies its determinant equation, up to an invertible factor, with the defining theta evaluation equation for $K_C\otimes L^{-1}(\mathcal D)$.
To calculate the determinant line, set $\mathcal D_i=\Gamma_1+\cdots+\Gamma_i$. Restriction to the extra graph gives
$$
0\longrightarrow L|_{\Gamma_i}\otimes\mathcal O_{\Gamma_i}(-\mathcal D_{i-1})\longrightarrow L|_{\mathcal D_i}\longrightarrow L|_{\mathcal D_{i-1}}\longrightarrow0.
$$
Projection is exact on these finite relative divisors. On $\Gamma_i\cong C^n$, the first term is $\operatorname{pr}_i^*L\otimes\mathcal O(-\sum_{j<i}\Delta_{ij})$, where $\Delta_{ij}=\{z_i=z_j\}$. Taking <determinants> inductively yields the <evaluation determinant on a product of curves>:
$$
\det E_L\cong\bigotimes_i\operatorname{pr}_i^*L\otimes\mathcal O\left(-\sum_{i<j}\Delta_{ij}\right).
$$
The determinant of the fixed space $H^0(C,L)$ contributes only a constant line. Therefore
$$
\boxed{\alpha^*\Theta\sim2\sum_{i=1}^{3g-3}\operatorname{pr}_i^*K_C-\sum_{i<j}\Delta_{ij}.}
$$
The diagonal subtraction is essential: ordinary evaluation <determinants> vanish when columns coincide, whereas restriction to a repeated-point <Cartier divisor> uses jets. The restriction exact sequences account for those factors in every characteristic.
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