Abstract Some coherent sheaves on projective varieties have a non-reduced versal deformation space; for example, this is the case for most unstable rank 2 vector bundles on ℙ2, see [18]. In particular, some moduli spaces of stable sheaves are non-reduced. We consider some sheaves on ribbons (double structures on smooth projective curves): let E be a quasi locally free sheaf of rigid type and let 𝓔 be a flat family of sheaves containing E. We find that 𝓔 is a reduced deformation of E when some canonical family associated to 𝓔 is also flat. We consider also a deformation of the ribbon to reduced projective curves with two components, and find that E can be deformed in two distinct ways to sheaves on the reduced curves. In particular some components M of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and M appears as the “limit” of varieties with two components, whence the non-reduced structure of M.

中文翻译：

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多条曲线上滑轮的非约化模空间

摘要 射影簇上的一些相干子具有非约化的通用变形空间；例如，这是 ℙ2 上最不稳定的 2 阶向量丛的情况，参见 [18]。特别是，稳定滑轮的一些模空间是非约化的。我们考虑一些带上的滑轮（光滑射影曲线上的双重结构）：让 E 是刚性类型的准局部自由滑轮，让 𝓔 是包含 E 的滑轮的平面族。我们发现𝓔 是 E 的减少变形，当一些与 𝓔 相关的规范家族也是扁平的。我们还考虑了带变形为具有两个分量的简化投影曲线，并发现 E 可以以两种不同的方式变形为简化曲线上的滑轮。