For elliptic surfaces with two multiple fibers of multiplicities 2 and m at least 3 and with chi equal to 1, every singular point of the moduli space M(c2) is canonical, and for compact M(c2) the Kodaira dimension is (dim M(c2)+1)/2.
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The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces
For elliptic surfaces with two multiple fibers of multiplicities 2 and m at least 3 and with chi equal to 1, every singular point of the moduli space M(c2) is canonical, and for compact M(c2) the Kodaira dimension is (dim M(c2)+1)/2.