= Solution
Identify $\operatorname{Pic}^{g-1}(X)$ with the <Jacobian variety> using the base point and the translation of Solution 4. A <line bundle> $L$ of degree $g-1$ corresponds to the point $z=u(D)-\kappa$ for any <divisor> representation $L\simeq\mathcal O_X(D)$. The <Riemann singularity theorem> is
$$
\boxed{\operatorname{mult}_{z}\Theta=h^0(X,L).}
$$
In particular, the <theta divisor> is singular exactly at the degree-$g-1$ <line bundles> with at least two independent sections. We prove the equality, including the upper bound on multiplicity.
Choose a fixed effective <divisor> $A$ of degree $N\ge g$, and a local <holomorphic> family $L_\eta$ representing nearby points of $\operatorname{Pic}^{g-1}(X)$. Such a family can be constructed locally from the <holomorphic exponential sequence>, by varying line-bundle transition functions through exponentials of <Čech cohomology> representatives. By <Serre duality>, $H^1(L_\eta(A))$ is dual to $H^0(K_X\otimes L_\eta^{-1}(-A))$, whose degree is $g-1-N<0$, so it vanishes. The <Riemann-Roch theorem> gives $h^0(L_\eta(A))=N$. By <cohomology and base change for line bundles on a curve>, these spaces form a <holomorphic vector bundle> $E$ of <rank> $N$. The spaces $H^0(L_\eta(A)|_A)$ form another <holomorphic vector bundle> $F$ of <rank> $N$. Evaluation gives a <holomorphic> square matrix $T(\eta)$ locally, with exact sequence
$$
0\longrightarrow H^0(L_\eta)\longrightarrow E_\eta
\xrightarrow{T(\eta)}F_\eta\longrightarrow H^1(L_\eta)\longrightarrow0.
$$
Consequently $\det T(\eta)=0$ exactly when $L_\eta$ has a nonzero section, namely on the support of the <theta divisor>.
At the point $L$, put $r=h^0(L)=h^1(L)$, the equality being <Riemann-Roch theorem>. An invertible minor of size $N-r$ persists near this point. <Holomorphic> row and column operations, or the <Schur complement>, give
$$
\det T(\eta)=a(\eta)\det M(\eta),\qquad a(0)\ne0,
\qquad M(0)=0,
$$
where $M$ is an $r\times r$ <holomorphic> matrix. This is the <Schur complement presentation of a theta singularity>. Its linear term in a tangent direction $\xi\in H^1(\mathcal O_X)$ is the <cup product> map
$$
dM_0(\xi):H^0(L)\longrightarrow H^1(L),\qquad s\longmapsto\xi\smile s.
$$
To see this concretely, vary transition functions $g_{ij}$ of $L$ to $g_{ij}(1+\epsilon\xi_{ij})$. A section $s_i$ lifts to $s_i+\epsilon s'_i$ precisely when its first-order compatibility equation can be solved; the obstruction is the cocycle $\xi_{ij}s_j$. This is $\xi\smile s$. Passing to the <kernel> and <cokernel> of $T(0)$ gives exactly the derivative of the reduced matrix $M$, proving the assertion without assuming a multiplicity formula.
Before extracting the order of $\det M$, we verify that $\det T$ is a reduced equation of the <theta divisor>. Generic points of $W_{g-1}$ have $h^0(L)=1$ by Solution 3. At such a point, both $H^0(L)$ and $H^0(K_X\otimes L^{-1})$ are one-dimensional. Their nonzero sections $s,t$ have nonzero product $st\in H^0(K_X)$. Choose $\xi\in H^1(\mathcal O_X)$ with $\langle\xi,st\rangle\ne0$ under <Serre duality>. The scalar $dM_0(\xi)$ is nonzero, so $\det T$ has a simple zero generically on the <irreducible> <theta divisor>. Since its support is exactly that <irreducible> <hypersurface>, it defines the reduced <divisor>. Solution 4 independently proved the same reduced structure for the <Riemann theta function>. On the smooth ambient <Jacobian variety>, local equations of the same reduced <divisor> differ by a nonvanishing <holomorphic> unit. Hence the orders of the local theta equation and $\det M$ agree everywhere.
Every entry of $M$ vanishes at the point, so each term in its <determinant> has order at least $r$. This proves $\operatorname{mult}_z\Theta\ge r$. To obtain equality, we exhibit an <invertible cup-product direction for a line bundle on a curve>. Set $V=H^0(L)$ and $W=H^0(K_X\otimes L^{-1})$, both of dimension $r$. There are $r$ distinct points $p_1,\ldots,p_r$ at which evaluation is an <isomorphism> for $V$: choose them inductively, since a surviving nonzero section has only finitely many zeros. The same holds for $W$. The nonvanishing of each evaluation <determinant> defines a nonempty open subset of the <irreducible> variety $X^r$. Their intersection, also avoiding the diagonals, is therefore nonempty. Choose points in it and <local coordinates> and compatible trivializations of $L$ and $K_X\otimes L^{-1}$ at each point.
Let $\xi$ be the <connecting homomorphism> class of the <meromorphic> <principal parts> $\lambda_i/t_i$ at these points, with all $\lambda_i\ne0$. The <residue> form of <Serre duality> gives
$$
\langle\xi\smile s,t\rangle=\langle\xi,st\rangle
=\sum_{i=1}^r\lambda_i s(p_i)t(p_i).
$$
If $S$ and $U$ are the two invertible evaluation matrices, the matrix of this pairing is $S^t\operatorname{diag}(\lambda_1,\ldots,\lambda_r)U$, and is invertible. Thus the <cup product> map $\xi\smile-:H^0(L)\to H^1(L)$ is an <isomorphism>. Along the corresponding analytic parameter line,
$$
M(\epsilon\xi)=\epsilon\,dM_0(\xi)+O(\epsilon^2),\qquad
\det M(\epsilon\xi)=\epsilon^r\det(dM_0(\xi))+O(\epsilon^{r+1}),
$$
with nonzero leading coefficient. The order is at most $r$, completing the proof of the <Riemann singularity theorem>.
This argument also identifies the <tangent cone to a theta divisor>: its degree-$r$ equation is $\det(\xi\smile-)$, which is not the zero polynomial. For $r=1$, the tangent hyperplane is $\langle\xi,st\rangle=0$, in agreement with the annihilator description of the <derivative of the Abelian sum map>. For $r\ge2$ all first derivatives vanish, proving singularity. In <genus> $1$ the <theta divisor> is a single smooth point of multiplicity $1$. In <genus> $0$ the effective degree-$-1$ locus is empty, so there is no singularity statement at a point of it.
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