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

## TL;DR

This paper investigates the singularities and Kodaira dimension of moduli spaces of stable sheaves on elliptic surfaces with specific properties, providing conditions for canonical singularities and explicit calculations of the Kodaira dimension.

## Contribution

It establishes criteria for when the moduli space has canonical singularities and computes its Kodaira dimension in certain cases, linking geometric properties to moduli-theoretic data.

## Key findings

- E is at worst canonical if the restriction of E to the generic fiber has no rank-one subsheaf
- The Kodaira dimension of M is (1 + dim(M))/2 for large c_2 under specific conditions
- Presence of a rank-one subsheaf in E_{ ext{eta}} complicates the analysis of singularities

## 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.