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

Let [Formula: see text] be an elliptic surface over [Formula: see text] with [Formula: see text], and [Formula: see text] be the moduli scheme of rank-two stable sheaves [Formula: see text] on [Formula: see text] with [Formula: see text] in [Formula: see text]. We look into defining equations of [Formula: see text] at its singularity [Formula: see text], partly because if [Formula: see text] admits only canonical singularities, then the Kodaira dimension [Formula: see text] can be calculated. We show the following: (A) [Formula: see text] is at worst canonical singularity of [Formula: see text] if the restriction of [Formula: see text] to the generic fiber of [Formula: see text] has no rank-one subsheaf, and if the number of multiple fibers of [Formula: see text] is a few. (B) We obtain that [Formula: see text] and the Iitaka program of [Formula: see text] can be described in purely moduli-theoretic way for [Formula: see text], when [Formula: see text], [Formula: see text] has just two multiple fibers, and one of its multiplicities equals [Formula: see text]. (C) On the other hand, when [Formula: see text] has a rank-one subsheaf, it may be insufficient to look at only the degree-two part of defining equations to judge whether [Formula: see text] is at worst canonical singularity or not.

On the -purity of isolated log canonical singularities

A singularity in characteristic zero is said to be of dense $F$-pure type if its modulo $p$ reduction is locally Frobenius split for infinitely many $p$. We prove that if $x\in X$ is an isolated log canonical singularity with $\mu (x\in X)\leq 2$ (where the invariant $\mu $ is as defined in Definition 1.4), then it is of dense $F$-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense $F$-pure type in the case of three-dimensional isolated $ \mathbb{Q} $-Gorenstein normal singularities.

ON THE DIVISOR CLASS GROUP OF 3-FOLD SINGULARITIES

In this note we calculate the divisor class number of an isolated canonical singularity [Formula: see text], which is assumed to be nondegenerate with respect to its Newton polyhedron, in terms of a suitable set of monomials whose residue classes form a basis for the Milnor algebra of f.