Symmetric product of a curve
= Symmetric product of a curve
{title2=$C^{(d)}$}
The <scheme> $C^{(d)}$ parametrizes effective degree-$d$ <Cartier divisors> on a smooth <algebraic curve>. Near a <Cartier divisor> $\sum m_iP_i$, its completed local ring is a power-series ring in the elementary symmetric coordinates of $m_i$ local parameters at each $P_i$. It is therefore smooth of dimension $d$, even when the <characteristic of a field> divides $d!$. For projective $C$ it is projective.