The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces

Abstract

Let $X$ be an elliptic surface over ${\bf P}^1$ with $\kappa(X)=1$, and $M=M(c_2)$ be the moduli scheme of rank-two stable sheaves $E$ on $X$ with $(c_1(E),c_2(E))=(0,c_2)$ in $\operatorname{Pic}(X)\times\mathbb{Z}$. We look into defining equations of $M$ at its singularity $E$, partly because if $M$ admits only canonical singularities, then the Kodaira dimension $\kappa(M)$ can be calculated. We show the following. (A) $E$ is at worst canonical singularity of $M$ if the restriction of $E_{\eta}$ to the generic fiber of $X$ has no rank-one subsheaf, and if the number of multiple fibers of $X$ is a few. (B) We obtain that $\kappa(M)=\{1+\dim(M)\}/2$ and the Iitaka program of $M$ can be described in purely moduli-theoretic way for $c_2\gg 0$, when $\chi({\mathcal O}_X)=1$, $X$ has just two multiple fibers, and one of its multiplicities equals $2$. (C) 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 at worst canonical singularity or not.