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VersalDeformations :: VersalDeformations

VersalDeformations -- calculating versal deformations and local Hilbert schemes

Description

This package provides tools for calculating tangent and obstruction spaces as well as power series solutions for deformation problems involving isolated singularities and projective schemes.

A basic description of the package's approach to deformation problems can be found at the documentation node for versalDeformation. For details and mathematical background see

  • Jan Stevens, Computing Versal Deformations, Experimental Mathematics Vol. 4 No. 2, 1994.

The numerous examples presented in the documentation nodes for versalDeformation and localHilbertScheme are classical deformation problems, considered in the following articles:

  • Klaus Altmann, The versal deformation of an isolated Gorenstein singularity, Inventiones Mathematicae Vol. 128 No. 3, 1997.
  • Dustin Cartwright and Bernd Sturmfels, The Hilbert scheme of the diagonal in a product of projective spaces, International Mathematics Research Notices Vol. 2010 No. 9.
  • Ragni Piene and Michael Schlessinger, On the Hilbert scheme compactification of the space of twisted cubic curves, American Journal of Mathematics, Vol. 107 No. 4, 1985.
  • Henry Pinkham, Deformations of variety with G_m action, Asterisque 20, 1974.

The author thanks Jan Christophersen for helpful hints, especially regarding the computation of T2.

Author

Version

This documentation describes version 0.7 of VersalDeformations.

Source code

The source code from which this documentation is derived is in the file VersalDeformations.m2.

Exports

  • Functions and commands
  • Symbols
    • CorrectionMatrix -- determines the first order deformations used in correcting liftings
    • HighestOrder -- sets the order to which we compute
    • PolynomialCheck -- checks if power series solution terminates
    • SanityCheck -- checks if lifting solves deformation equation
    • SmartLift -- chooses lifting to avoid obstructions at next order
    • TimeLimit -- sets time limit for lifting calculation
  • Other things
    • CT -- cotangent cohomology
    • t -- deformation parameter