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Sheaves
New Features:
- A new package is included providing functionality for working
with coherent sheaves on ordinary projective schemes. These
are naturally represented by graded modules over the polynomial
ring, that is, the coordinate ring of the ambient of the base scheme.
- The first major task that the package deals with is the computation
of the maximal (separated) graded module attached to the sheaf starting
from the defining module. The aim was to do this efficiently in reasonable
generality. The maximal module is the direct sum of global sections
of all Serre twists of the sheaf and is needed for several applications.
- The basic assumption is that the exact support of the sheaf S - a
subscheme of the base scheme - has all irreducible components of the
same dimension > 0 and that S has no non-generic associated points
on this support. The implementation computes the maximal module via a
double dual calculation treating the defining module as a module over
the polynomial ring giving a linear Noether normalisation of the
coordinate ring of the exact support of S.
- There are a number of basic constructors of sheaves, including one
for the canonical (dualising) sheaf of an equidimensional, locally
Cohen-Macaulay scheme. Further construction operations include tensors,
direct sums, tensor powers, Homs and duals.
- The initial focus, in terms of functionality, as well as the
computation of important cohomological invariants of varieties, has been
on invertible sheaves (or divisors) and the explicit computation of their
associated rational maps into projective space. There is an intrinsic
DivisorMap for this, which also returns the image of the map. The
map is computed from the ``global section submodule" of S, which in
turn comes from the maximal module. It is naturally computed and returned
as a MapSchGrph, the new type of scheme graph map. This gives a
method of computing important maps like canonical, anticanonical or
adjunction maps on general varieties.
- For a base scheme X and an effective (Cartier) divisor D on X defined
as a closed subscheme by an ideal I, DivisorToSheaf computes an
invertible sheaf corresponding to the class of D. Here, computing the
explicit divisor map is essentially the same as computing the Riemann-Roch
space if X is a variety. In fact, the Riemann-Roch space can be recovered
during the computation of the associated sheaf in a usually more compact
form than from later computation with the divisor map of the sheaf. Thus
we provide a RiemannRochBasis intrinsic that returns a basis in
explicit form (as a sequence of polynomials on the ambient and a
denominator) as well as the associated sheaf. The computation relies on
the fact that, for appropriate r > 0, the
invertible sheaf of D is isomorphic to the rth Serre twist of
the ideal sheaf defining a complementary divisor to D in rH where
H is a hyperplane divisor.
- There is some basic functionality for homomorphisms between sheaves on
the same base scheme, kernels, images etc.
- There are tests intrinsics IsLocallyFree, which tests for local
freeness and also returns the degree, IsArithmeticallyCohenMacaulay,
which tests whether the maximal module of the sheaf S on X is
a Cohen-Macaulay module as a graded module over the coordinate ring of X
(if S is the structure sheaf of X, this just tests whether X is
arithmetically Cohen-Macaulay in its current projective embedding), and
IsIsomorphic for whether two sheaves on the same X are isomorphic.
- The type for a coherent sheaf is ShfCoh and for a sheaf homomorphism
ShfHom.
Next: Algebraic Surfaces
Up: Algebraic Geometry
Previous: Schemes
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