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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
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
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.
Next: Toric Varieties
Up: Algebraic Geometry
Previous: Algebraic Curves