Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be moduli space of Higgs bundle of rank $r\geq2$ and degree $d$ on $X$, $W=\bigoplus_{i=2}^{r}H^{0}(X,K_{X}^{\otimes i})$, and $H\colon \mathcal{M}\rightarrow W$ be the Hitchin map.
$X_{s}$ denotes a spectral curve corresponding to $s\in W$.
$\mathcal{D}$ and $\mathcal{D}_{0} \subseteq W$ is defined by
\begin{equation}
\mathcal{D}=\{s\in W\mid X_{s} \,\,\mbox{is a singular curve.}\}
\end{equation}
\begin{equation}
\mathcal{D}_{0}=\{s\in \mathcal{D}\mid X_{s} \,\,\mbox{is irreducible and has a unique simple node.}\}
\end{equation}
I want to show,
$\mathcal{D}_{0}$ is a non empty open set in $\mathcal{D}$.
Choose $s_{r}\in H^{0}(X,K_{X}^{\otimes r})$ s.t. has a uniquie double zero, then $X_{s}$ defined by $y^{r}+s_{r}$ has a unique simple node. This is possible by Riemann-Roch. So, non emptyness is clear.
This paper says that, by upper semi-continuous theorem $\mathcal{D}_{0}\subseteq \mathcal{D}$ is open. But I don't know what kind of upper semi-continuous is applied. I think $\operatorname{length}(\Omega_{X_{s}/X})=1$ means $X_{s}$ has a unique node. But, I don't know how to show this is an open condition in $\mathcal{D}$ and I'm stuck here.
Any help will do. Thanks in advence.
