Algebraic Surfaces

New Features:

• A basic surface type Srfc has been added for two-dimensional geometrically integral schemes over a field.
• Invariants, covariants and contravariants of cubic surfaces are accessible via
  ClebschSalmonInvariants, LinearCovariantsOfCubicSurface,
  ClassicalCovariantsOfCubicSurface, ContravariantsOfCubicSurface. Further, the degree 100 skew-invariant of a cubic surface can be computed via SkewInvariant100.
• New functionality has been added for general surfaces but applying mainly to ordinary projective models with very restricted singularity. This includes the following.
• IsNormal, a test for normality and HasOnlySimpleSingularities, which tests whether the surface only has simple (A-D-E type) singularities. The latter will also give the user the singularity type (e.g. "A3") for all geometric singular points if required.
• For ordinary projective surfaces which have only simple singularities (or are at least Gorenstein for some of the functions), computation of basic invariants with GeometricGenus, Plurigenus, Irregularity, ChernNumber (for c₁² and c2 ) and HodgeNumber (for the individual hⁱ, j ).
• KodairaEnriquesType to compute the Kodaira dimension as well as the more specific type of an ordinary projective surface in the Kodaira-Enriques classification (rational, ruled, K3 etc.). The surface must again must have at worst simple singularities.
• Computation of minimal models of ordinary projective surfaces, relying heavily on adjunction and pluricanonical maps. The intrinsics differ slightly by Kodaira dimension and type so there is MinimalModelRationalSurface, MinimalModelRuledSurface,
  MinimalModelKodairaDimensionZero etc. The Kodaira dimension one case also returns a fibration map to a smooth projective curve, presenting the minimal model as a globally minimal elliptic fibration. These intrinsics require the original surface to be non-singular except in the general type (Kodaira dimension 2) case, when simple singularities are allowed.
• For ordinary projective general type surfaces with at most simple singularities, computation of the full canonical model (in weighted projective space) or canonical coordinate ring with
  CanonicalWeightedModel and CanonicalCoordinateIdeal.
• Computation of random nonsingular surfaces in $\bf P^{4}_{}$ from a number of the families described in the paper of Decker, Ein and Schreyer. For example, RandomRationalSurface_d10g9 and
  RandomEllipticFibration_d7g6.

Changes:

• The code to work with (singular) hypersurfaces in $\bf P^{3}_{}$ has been modified so that many of the arguments of intrinsics are now Srfc types rather than polynomials.
• The old Plurigenus intrinsic has been renamed PlurigenusOfDesingularization.
• The old ClassifyProjectiveSurface has been renamed ClassifyRationalSurface and returns more descriptive data.