Constructing_Surfaces

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title: Constructing Surfaces permalink: wiki/Constructing_Surfaces/ layout: wiki

Things to do in SurfacesInP4.m2

• Write a function to compute the hilbert polynomial of the ideal of the surface from its hilbert polynomial (hilbIdealFromHilbStructureSheaf ).
• Write a function to compute the hilbert polynomial using the invariants of the surface (hilbertPolynomialFromInvariants).
• Write a function to find a plausible cohomology table in a given range from the hilbert polynomial of the surface (guessCohomologyTable).
• Write a function to find windows of the minimal cohomology table (findMonadWindows).
• Write a function to construct a monad of the ideal sheaf of the surface (guessDifferentials).
• Write a function to construct a surface in projective fourspace with prescribed invariants (constructSurface).
• Modify constructSurfac to write a command to construct a space curve with prescribed invariants (constructSpaceCurve).
• Write a function to compute the image of the surface in projective fourspace under the first adjunction map (firstAdjoint ).
• Write a function to compute the image of the next adjunction map.
• Write a quick test for smoothness of the surface.
• Make the list of existing surface in projective fourspace.
  • Most of the known surfaces were reconstructed (see surfaces_examples.m2).
• BUG: The field is assumed to be K (fixed)