Murphy’s law in algebraic geometry: Badly-behaved deformation spaces
Murphy’s law in algebraic geometry: Badly-behaved deformation spaces
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代数几何中的墨菲定律：表现不佳的变形空间
DOI：
10.1007/s00222-005-0481-9
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发表时间：
2004
影响因子：
3.1
通讯作者：
R. Vakil
中科院分区：
文献类型：
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作者：
R. Vakil
我们考虑的问题是：“一个物体的变形空间能有多糟糕？”答案似乎是：“除非有某种先验原因，否则变形空间可能会尽可能地糟糕。”更确切地说，ℤ上的每个有限类型奇点(直到光滑参数)都出现在：射影空间中曲线的希尔伯特格式；光滑射影一般类型曲面(或高维变种)的模空间，带节点和尖点的平面曲线，稳定层，孤立的三重奇点，等等。对象本身并不是病态的，实际上是尽可能好的：曲线是光滑的，曲面是自同构的，并且有非常充足的标准丛，稳定的层是无挠的，奇点是正规的和Cohen-Macaulay的，等等。这证明了Mumford的哲学，即使是行为良好的对象的模空间也应该是任意差的，除非有相反的先验原因。因此，人们可以在射影空间中构造一条光滑的曲线，其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点类型，具有任何给定的非约化行为。类似地，可以给出$\mathbb{F}_{p}$上的曲面提升到ℤ/p7而不是ℤ/p8。(当然，这些结果也适用于全纯范畴。)直接从障碍理论计算形变空间通常是困难的。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，从而绕过了这一点。其基本出发点是曼涅夫的普适性定理。
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 are automorphism-free and 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. Similarly one can give a surface over $\mathbb{F}_{p}$ that lifts to ℤ/p7 but not ℤ/p8. (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 Mnëv’s universality theorem.