# effective divisors on a curve and upper semi-continuity

Let consider a smooth projective curve $X$ over $\mathbb{C}$. We consider the scheme that classifies effective divisors of degree $d$, which is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the symmetric group.

We consider the map which associates to an effective divisor D of degree $d$ the sum of its multiplicities greater than two, $\sum\limits_{x,m_{x}(D)\geq 2}m_{x}(D)$.

Is this function is upper-semicontinuous?

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Yes, of course it is. – Dan Petersen Aug 9 '13 at 21:50

The space $X^d/S_d$ has a stratification with strata $X_\lambda$ indexed by partitions $\lambda$ of $d$. The stratum $X_\lambda$ is in the closure of the stratum $X_\mu$ if and only if $\lambda$ is a subpartition of $\mu$ (i.e. one can split the summands of $\lambda$ into groups such that each summand of $\mu$ is a sum of summands of $\lambda$ in one of the groups). Clearly, the number of summands equal to $1$ can only decrease when one goes to a stratum in the closure. Since the function in question can be rewritten as $d$ minus the number of summands equalt to $1$, the semicontinuity follows.