The general creation function for sheaves takes a graded module representing
the sheaf and a scheme X on which it is supported. Special constructors are
provided in the cases of the structure sheaf of X and the canonical sheaf of
X, when X is locally Cohen-Macaulay and equidimensional. The user may also
ask for Serre twists of a given sheaf. Other constructions deriving new sheaves
from existing sheaves will be described in later sections.
Given an ordinary projective scheme X, this function returns the canonical
sheaf KX for X. The scheme X should be an ordinary projective scheme
which is equidimensional and locally Cohen-Macaulay. That is, all of the primary
components of X should have the same dimension and its local rings should
all be Cohen-Macaulay rings. These conditions aren't checked by Magma as
the necessary computations can be very expensive in general. A non-singular
variety always satisfies these conditions, and many singular normal varieties
do also. For example, any curve or normal surface will be equidimensional and
locally Cohen-Macaulay. The stronger condition of being arithmetically
Cohen-Macaulay, can be checked by invoking the intrinsic IsArithmeticallyCohenMacaulay
with the structure sheaf of X as argument.
Under these conditions, X has a canonical sheaf KX, defined up to isomorphism,
which acts as a dualising sheaf. See Section 7, Chapter III of [Har77]
and Chapter 21 of [Eis95] for the module-theoretic background.
For non-singular varieties, the canonical sheaf is the usual one: the highest
alternating power of the sheaf of Kahler differentials. The function returns
the canonical sheaf of X. It is computed from the dual complex to the minimal
free resolution of the coordinate ring of X.
If the intrinsic is invoked with an additional integer argument n, it returns
the nth Serre twist (see below) of the canonical sheaf KX(n). For a
non-singular variety of dimension d, the map into projective space corresponding
to KX(d - 1) is the important adjunction map.
We construct some sheaves associated with the smooth cubic surface
defined by x
3 + y
3 + z
3 + t
3 in P
3.
> P<x,y,z,t> := ProjectiveSpace(Rationals(),3);
> R := CoordinateRing(P);
> X := Scheme(P,x^3+y^3+z^3+t^3);
> OX := StructureSheaf(X);
We first examine the underlying graded module of the structure sheaf.
> Module(OX);
Reduced Module R^1/<relations>
Relations:
[x^3 + y^3 + z^3 + t^3]
Observe that the canonical sheaf KX of X is isomorphic to the
twist OX( - 1) of the structure sheaf.
> KX := CanonicalSheaf(X);
> Module(KX);
Reduced Module R^1/<relations> with grading [1]
Relations:
[x^3 + y^3 + z^3 + t^3]
> Module(StructureSheaf(X,-1));
Reduced Module R^1/<relations> with grading [1]
Relations:
[x^3 + y^3 + z^3 + t^3]
Note that the module column weights are the negations of the Serre twist indices!
> Module(Twist(OX,-1));
Reduced Module R^1/<relations> with grading [1]
Relations (Groebner basis):
[x^3 + y^3 + z^3 + t^3]
The equations x=z, y=t define an (exceptional) line in X. We can get its
structure sheaf as a sheaf on X using the basic
Sheaf constructor.
The associated invertible sheaf (L)(Y) of Y as a divisor on X
can be obtained from the
DivisorToSheaf intrinsic described later in
the chapter.
> IY := ideal<R|[x+z,y+t]>; // ideal of line
> OY := Sheaf(QuotientModule(IY),X);
> Module(OY);
Graded Module R^1/<relations>
Relations:
[x + z],
[y + t]
> Scheme(OY);
Scheme over Rational Field defined by
x^3 + y^3 + z^3 + t^3
Maximize: BoolElt Default: false
Given an ordinary projective scheme X, this function returns the sheaf of
1-differentials on X, Ω1X/k. The function computes the natural
representing module for the sheaf coming from the embedding of X in
projective space (see Section 8, Chapter II of [Har77]). If the
parameter Maximize is true, then the maximal module representing
this sheaf is computed and used to define it (see next section).
Maximize: BoolElt Default: false
For an ordinary projective scheme X, this function returns the sheaf of tangent
vectors for X. The function computes the natural representing module for these
sheaves coming from the embedding of X in projective space (see Section 8, Chapter
II of [Har77]). If the parameter Maximize is true, then the maximal
module representing this sheaf is computed and used to define it (see next section).
Combining either of the above intrinsics with the IsLocallyFree intrinsic,
this gives an alternative method for checking non-singularity on varieties that
are known to be (locally) Cohen-Macaulay. It is best to use the sheaf of
differentials since that is generally easier to compute. This approach can be
much faster for varieties having high codimension than the usual Jacobian method.
The projective space P should be ordinary projective 4-space Prj4 over a field.
The function returns the locally free rank 2 sheaf on P which represents the
Horrocks-Mumford bundle (see [HM73]). The scheme of vanishing of a general
global section of this sheaf is a two dimensional Abelian variety in P.
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