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.

中文翻译：

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一些椭圆面上稳定滑轮的小平维数和模奇异性

令 [Formula: see text] 是 [Formula: see text] 上带有 [Formula: see text] 的椭圆曲面，并且 [Formula: see text] 是二阶稳定滑轮 [Formula: see text] 上的模数格式[公式：见正文]与[公式：见正文]中的[公式：见正文]。我们研究在其奇点 [Formula: see text] 处定义方程 [Formula: see text]，部分原因是如果 [Formula: see text] 只承认规范奇点，则可以计算 Kodaira 维数 [Formula: see text]。我们展示以下内容：（一种）[Formula: see text] 如果 [Formula: see text] 对 [Formula: see text] 的一般纤维的限制没有一级子层，并且如果数[公式：见正文]的多纤维是少数。(乙)我们得到[公式：见文]和[公式：见文]的饭鹰程序可以用纯模理论的方式描述[公式：见文]，当[公式：见文]，[公式：见文]只有两条多重纤维，其多重性之一等于[公式：见正文]。（C）另一方面，当 [Formula: see text] 有一个 rank-one 子层时，仅看定义方程的二阶部分可能不足以判断 [Formula: see text] 是否是最坏的规范奇点或不是。