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?



effective divisors on a curve and upper semi-continuity
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.


