= Derivative of the Abelian sum map
{title2=$d u_D:H^0(\mathcal O_D(D))\to H^1(\mathcal O_X)$}
For an effective <divisor on an algebraic curve> $D$ on a <smooth projective curve>, the <Zariski tangent space> to its <symmetric product of a curve> is $H^0(\mathcal O_D(D))$. The derivative of the <Abelian sum map> is the <connecting homomorphism> of $0\to\mathcal O_X\to\mathcal O_X(D)\to\mathcal O_D(D)\to0$. Its dual under <Serre duality> is restriction of <holomorphic differential forms> to $D$, including all jets at repeated points. Hence its rank is $\deg D+1-h^0(\mathcal O_X(D))$, and its image annihilates $H^0(K_X(-D))$.
Back to article page
