 Solution

Let $X$ be a smooth projective curve over $\mathbb{C}$, let $D={\sum }_p{m}_{p}p$ be an effective divisor on an algebraic curve of degree $d$, and identify a translate of the Jacobian variety with ${\mathrm{Pic}}^d(X)$. The precise derivative theorem iswhere ${\delta }_D$ is the connecting homomorphism ofThe target is $T_{u_d(D)}\mathrm{Jac}(X)=H^1(X,{\mathcal{O}}_X)=H^0(X,K_X{)}^{\ast }$ under Serre duality. Its dual is restriction of holomorphic differential forms to the length-$d$ closed subscheme $D$:In particular, this includes derivatives of the local coefficients at repeated points, rather than simply evaluating once for each point of the support.

Here is a local proof of all these identifications. Near a point of multiplicity $m$, take a coordinate $z$ vanishing at that point. Nearby effective divisors are represented by monic polynomials of degree $m$. A first-order deformation has equationThese coefficients are the elementary symmetric coordinates on the symmetric product of a curve; they give $m$ independent tangent directions even at a repeated point. Dividing the variation of the equation by the original equation identifies the tangent direction with the principal part $v(z)/z^m$. Doing this at each point gives $H^0({\mathcal{O}}_D(D))$, a space of dimension $d$.

To see the derivative of the line bundle directly, use the local frame $1/(z^m-\epsilon v)$ for ${\mathcal{O}}_X(D_{\epsilon })$ and the frame $1$ outside the small disks. Relative to the frame $1/z^m$ at $\epsilon =0$, its transition multiplier changes byFirst-order changes of line bundle transition functions have the form $1+\epsilon c_{ij}$; changing frames adds a Čech coboundary. Thus the tangent space to the Picard group is $H^1({\mathcal{O}}_X)$, and the derivative is exactly the Čech cocycle of these principal parts, namely ${\delta }_D$.

This agrees with differentiating the Abel integrals, including the sign. If $\omega =f(z)dz$ and $F^{\prime }=f$, the sum of $F$ at the roots of $z^m-\epsilon v$ has derivativeFor example, expand $F={\sum }_{k\ge 1}{f}_{k-1}{z}^k/k$. The first-order Newton identities give ${\partial }_{\epsilon }\sum z_i^k=k{v}_{m-k}$ for $1\le k\le m$, and zero for $k>m$, yielding exactly the residue above. Equivalently the same formula follows by differentiating the contour integral for the sum over roots. Serre duality pairs the principal part class with $\omega$ by the sum of these residues. Locally this pairing is perfect between the principal parts $z^{-1},\dots ,z^{-m}$ and the jets $1,z,\dots ,z^{m-1}$ times $dz$. This proves the asserted dual restriction map.

The long exact sequence in sheaf cohomology now gives the kernel and rank without any reduced-support assumption:By duality,This is the geometric form of the derivative of the Abelian sum map. For distinct points, the image is spanned by the evaluation vectors of the canonical map; at repeated points, the corresponding osculating directions must also be included. The Riemann-Roch theorem confirms the rank as $g-h^0(K_X(-D))$. When $h^0(D)=1$, the Abelian sum map is an immersion at $D$. For $d=g-1$ and $h^0(D)=1$, its image is a hyperplane annihilated by the unique nonzero holomorphic differential form vanishing along $D$, which will be the tangent hyperplane to the theta divisor.