Let $X$ be an elliptic surface over $\PP^1$ with $\kappa(X)=1$, and
$M$ be the moduli scheme of rank-two stable sheaves on $X$ with
$c_1=0$.
We look into defining equations of $M$ at its singularity $E$.
When the restriction of $E_{\eta}$
to the generic fiber of $X$ has no rank-one subsheaf,
\textcolor{red}{$E$ is a canonical singularity of $M$ (that is a "good" singularity),}
if the number of multiple fibers of $X$ is a few.
Consequently we calculate the Kodaira dimension of $M$
when $X$ has just two multiple fibers, \textcolor{red}{and}
one of its multiplicities equals $2$ and $\chi(\sO_X)=1$.
On the other hand, when $E_{\eta}$ has a rank-one subsheaf,
it may be insufficient to look at
only the degree-two part of defining equations to judge whether $E$ is good singularity.
Research papers (academic journals)
