Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

By Ravi Vakil. 2004

251 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/699ae867c5ebab3273513196a270876d7c7c3065

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Citations in the corpus

Toric vector bundles, branched covers of fans, and the resolution property (2006, background): “t total Chern class c(∆,Ψ), and it is also smooth. The main result of this section, roughly stated, is that the moduli of rank three toric vector bundles have arbitrarily bad singularities. Following [Vak04], we deﬁne a singularity type to be an equivalence class of pointed schemes, where the equiva”
Fixed point loci of moduli spaces of sheaves on toric varieties (2008, unknown): “Vakil has shown that the moduli space of Gieseker stable sheaves satisfies Murphy’s Law, meaning every singularity type of finite type over Z appears on the moduli space [34].”
A Superficial Working Guide to Deformations and Moduli (2011, background): “Using an interesting result of M’nev about line configurations, Vakil ([Va06]) was able to show that ‘any type of singularity’ can occur for the Gieseker moduli space (his examples are such that S = X and S has no automorphisms, hence they produce the desired singularities also for the local moduli ”
Matroid Theory for Algebraic Geometers (2014, background): “Murphy’s Law, due to Vakil [98], is a theorem that states that certain classes of moduli spaces contain every singularity type over Z up to a particular natural equivalence.”
Moduli of toric vector bundles (2007, methodology): “ of M fr Ψ by a natural GL action (Section 3.3). In the ﬁnal section, we apply this construction to show that the moduli of rank three toric vector bundles satisﬁes Murphy’s Law in the sense of Vakil [Vak06]; in other words, these moduli spaces contain every singularity type of ﬁnite type over SpecZ”
A desingularization of the main component of the moduli space of genus-one stable maps into ℙn (2006, background): “P; d/ behaves quite badly [19].”
Brill-Noether theory for curves of a fixed gonality (2017, background): “The general tropical realization problem for tropical stable maps also satisfies a version of Mnëv universality, due to Vakil’s Law, see [51] and [45, Theorem D & Remark 3.”
Communication lower bounds and optimal algorithms for programs that reference arrays - Part 1 (2013, methodology): “With the next lemma, we use a standard trick of replacing composite terms with single applications of the basic operations to put a general Diophantine set in a standard form (see, e.g., [24]).”