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This chapter describes the Magma functionality for working with coherent
sheaves on ordinary projective schemes. The emphasis in this initial version
is on invertible sheaves and on computing associated cohomological invariants
and explicit divisor maps. Important examples include canonical and anticanonical
maps and adjunction maps on varieties of arbitrary dimension. The tools
provided in Magma enable the user to compute these in a general and
reasonably efficient way. There is also functionality for computing an
invertible sheaf corresponding to the class of an effective Cartier divisor
given as a closed subscheme as well as a basis for the Riemann-Roch space of
that divisor as ambient rational functions.
The correspondence between divisors (or their classes) and invertible sheaves
will be expanded in later releases.
A standard reference for the definition and basic properties of coherent
sheaves on Noetherian schemes is Section 5, Chapter II of [Har77].
The package is based on Magma's functionality for graded modules over
polynomial rings and relies heavily on Gr{öbner basis computations. A
coherent sheaf is represented by a graded module over the coordinate ring of
the ambient projective space. The key difference between the category of sheaves
and the category of modules is that a sheaf is not represented uniquely.
However, there is a unique maximal graded module representing it, which
is finitely generated (with certain provisos). For certain algorithms -- computing
cohomology, for example -- any module representing the sheaf may be used. However
for other calculations, such as explicit Riemann-Roch spaces or divisor maps,
the full maximal module, containing the full space of global sections of the
sheaf and its small Serre twists, is often required.
One of the basic operations, therefore, is the computation of the maximal
module of a sheaf from its initial defining module. We have tried to do this
efficiently in reasonable generality. The basic condition is that the support
of the sheaf has irreducible components all of the same non-zero dimension. This
will be described in more detail in the function descriptions that follow. The
user does not have to explicitly make a call to perform the computation, but it
may be carried out in the background and the result stored by several other
functions.
A coherent sheaf Para is defined by a graded module M over the polynomial ring
R = k[x0, ..., xn] and a subscheme X of Prjn = Proj(R) on which
M is supported. That is, the defining ideal I ⊆R of X annihilates
M. In some contexts, X is unimportant and it doesn't matter whether Para is
thought of as a sheaf on X or on Prjn. In other cases, X plays a role:
we can test whether Para is locally free as a sheaf on X or take its dual.
The sheaf Para is just the coherent sheaf tilde(M) on X as described in
Prop. 5.11, Section 5, Chapter II of [Har77], with M considered
as a graded module over the homogeneous coordinate ring of X.
Sheaves are of type ShfCoh.
There is also a type ShfHom for homomorphisms between sheaves supported
on the same scheme X.
The algorithms used in the package are based on a number of computational
commutative algebra tricks well-known to the experts.
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