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Now we describe computation of adjoint spaces. Let S ⊂PE3 be a surface defined by a homogeneous irreducible polynomial F ∈E[x0, x1, x2, x3] of degree d and ΩE(S) | E the vector space of rational differential forms of the function field (over the ground field E of characteristic zero). We can consider ΩE(S) | E a constant sheaf of OOS-modules. Let Ui ⊂S be the affine open subsets of the standard covering w.r.t. this choice of variables.
Let ωS0 ⊂ΩE(S) | E^ 2 be the subsheaf which is locally generated on Ui by
(frac(∂F/ ∂xj)(xid - 1)) - 1 bigwedge_(k ∈{0, ..., 3} - {i, j}) d frac(xk)(xi)
(for an arbitrary choice of j != i). By sending this generator to xid - 4 one finds that ωS0 isomorphic to OOS(d - 4). Further let FFS, m ⊂(ΩE(S) | E^ 2) tensor m be the subsheaf of those forms whose pullbacks are regular on some desingularization of S. It is called the sheaf of m-adjoints. It is in fact well-defined, i.e., doesn't depend on any specific desingularization, and one can show FFS, m ⊆(ωS0) tensor m. For more details we refer to [BS08].
Now since FFS, m is a coherent sheaf on the projective scheme S ⊂PE3 it can be defined by its associated graded module MS, m and by the above discussion FFS, m is isomorphic to a subsheaf of OOS(m(d - 4)). MS, m is thus naturally a graded submodule of (E[x0, x1, x2, x3]/< F >)(m(d - 4)). The n-th graded piece of MS, m, a linear subsytem of the standard
linear system of degree n + m(d - 4) homogeneous polynomials on S, corresponds to
global sections of the Serre twist FFS, m(n). This, under pullback, corresponds
to the space of global sections of the twisted m-adjoint sheaf
(ωX) tensor m(n) for any desingularization X of S, where
(n) now signifies twisting by the n-th tensor power of LL, the
invertible sheaf on X which gives the map into projective space projecting X
down onto S.
These adjoint linear systems immediately give the plurigenera of any
desingularization X as well as an explicit representation of the important
twisted m-adjoint maps into projective space as rational maps from S
(defined by the sequence of homogeneous polynomials forming a basis of the
adjoint system). These maps are used to take any rational hypersurface to a standard
model, as described in the next section.
All functions in this section (and several in the following sections) allow the
user to enter precomputed formal desingularization data. It is a good idea to
do this if performing several operations on the same hypersurface to avoid
repeated computation of this desingularization.
HomAdjoints(m,n,p) : RngIntElt, RngIntElt, RngMPolElt -> SeqEnum
FormalDesing: SeqEnum Default: 0
SetVerbose("Classify", n): Maximum: 1
Arguments are an irreducible homogeneous polynomial p ∈E[x0, ..., x3] of degree d
(where E is a number field) and natural numbers m and n s.t. n + m(d - 4) ≥0. Let S be the surface in P3 defined by p.
Returns a basis for the vector space of the degree-n graded summand of the graded ring associated to FFS, m (i.e., Γ(S, OOS(n) tensor FFS, m)) as a subspace of the homogeneous forms in E[x0, x1, x2, x3] of degree
n + m(d - 4) (see above).
A precomputed formal desingularization (as returned by a call to hfil
ResolveProjectiveSurface) may be passed as parameter FormalDesing. The
desingularization passed in can be computed with the AdjComp
parameter set to true. The default value for FormalDesing is the
integer 0, in which case a formal desingularization needs to be computed
during function execution.
The function computes the adjoint space as a linear subspace of homogeneous
polynomials of the appropriate degree by using the formal divisor morphisms
of the formal desingularization to give additional linear conditions at the
singular places of S. This is explained fully in [BS08].
FormalDesing: SeqEnum Default: 0
For hypersurface S in P3, returns the geometric genus of (any)
desingularization of S. The function just computes the dimension of
the (1, 0) adjoint space.
As for HomAdjoints, a precomputed desingularization (of the defining
polynomial of S) can be passed in via the FormalDesing parameter.
FormalDesing: SeqEnum Default: 0
For hypersurface S in P3, returns the m-th plurigenus of (any)
desingularization, X, of S. This is the dimension of the global
sections of the sheaf (ωX) tensor m and is just computed
as the dimension of the (m, 0) adjoint space.
As for HomAdjoints, a precomputed desingularization (of the defining
polynomial of S) can be passed in via the FormalDesing parameter.
FormalDesing: SeqEnum Default: 0
For hypersurface S in P3, returns the arithmetic genus of (any)
desingularization of S. This is computed from a simple formula
involving the geometric genus and second plurigenus, coming from
the Riemann-Roch theorem.
As for HomAdjoints, a precomputed desingularization (of the defining
polynomial of S) can be passed in via the FormalDesing parameter.
We compute several adjoint spaces a surface. We precompute a formal
desingularization and use it for the calls to HomAdjoints.
> Q := Rationals(); P<x,y,z,w> := PolynomialRing(Q, 4);
> F := w^3*y^2*z+(x*z+w^2)^3;
> desing := ResolveProjectiveSurface(F : AdjComp := true);
> HomAdjoints(1, 0, F : FormalDesing := desing);
[]
> HomAdjoints(1, 1, F : FormalDesing := desing);
[
x*z*w + w^3
]
> HomAdjoints(1, 2, F : FormalDesing := desing);
[
x^2*z^2 - w^4, x^2*z*w + x*w^3, x*y*z*w, x*z^2*w + z*w^3,
x*z*w^2 + w^4, y*z*w^2, y*w^3
]
> HomAdjoints(1, 3, F : FormalDesing := desing);
[
x^3*z^2 - x*w^4, x^2*y*z^2, x^2*z^3 - z*w^4,
x^3*z*w + x^2*w^3, x^2*y*z*w, x*y^2*z*w, x^2*z^2*w - w^5,
x*y*z^2*w, x*z^3*w + z^2*w^3, x^2*z*w^2 + x*w^4, x*y*z*w^2,
y^2*z*w^2, x*z^2*w^2 + z*w^4, y*z^2*w^2, x*y*w^3, y^2*w^3,
x*z*w^3 + w^5, y*z*w^3, y*w^4
]
>
> HomAdjoints(2, 0, F : FormalDesing := desing);
[]
> HomAdjoints(2, 1, F : FormalDesing := desing);
[]
> HomAdjoints(2, 2, F : FormalDesing := desing);
[
x^2*z^2*w^2 + 2*x*z*w^4 + w^6
]
> HomAdjoints(2, 3, F : FormalDesing := desing);
[
x^3*z^2*w^2 + 2*x^2*z*w^4 + x*w^6, x^2*y*z^2*w^2 - y*w^6,
x^2*z^3*w^2 + 2*x*z^2*w^4 + z*w^6,
x^2*z^2*w^3 + 2*x*z*w^5 + w^7, x*y*z^2*w^3 + y*z*w^5,
x*y*z*w^4 + y*w^6, y^2*z*w^4
]
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