algebraic geometry中的墨菲定律：不良变形空间
Murphy's Law in algebraic geometry: Badly-behaved deformation spaces
摘要：
我们考虑这样一个问题：“一个物体的变形空间能有多坏？”答案似乎是：“除非有某种先验原因，否则变形空间可能尽可能地糟糕。我们证明了这对一些重要的模空间。更准确地说，Z上的有限型奇点（直到光滑参数）出现在：射影空间中曲线的希尔伯特方案;光滑射影一般型曲面（或高维曲面）的模空间，有节点和尖点的平面曲线，稳定层，孤立的三重奇点，等等。对象本身不是病态的，事实上是尽可能好的：曲线是光滑的，表面有非常充足的典范丛，稳定层是秩为1的无挠，奇点是正常的和科恩-麦考利等，这证明了芒福德的哲学，即使是模空间的良好行为的对象应该是任意坏的，除非有一个先验的原因。因此，人们可以在射影空间中构造一条平滑曲线，该曲线的变形空间具有任意给定数量的分量，每个分量具有任意给定的奇异类型，沿着各种相关的子方案具有任意给定的非约化行为。类似地，我们可以给出F_p上的一个曲面，该曲面提升到Z/p^7而不是Z/p^8。(Of当然，结果在全纯范畴中也成立。）直接从障碍理论计算变形空间通常是困难的。我们通过光滑态射将它们与更易处理的变形空间联系起来，从而规避了这一点。基本出发点是姆涅夫的普遍性定理。
英文摘要：
We consider the question: ``How bad can the deformation space of an object be?'' The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.'' We show this for a number of important moduli spaces. More precisely, every singularity of finite type over Z (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior along various associated subschemes. Similarly one can give a surface over F_p that lifts to Z/p^7 but not Z/p^8. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mnev's Universality Theorem.
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