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Adjoint Systems and Birational Invariants

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].

GeometricGenusOfDesingularization(S) : Sch -> RngIntElt
    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.

Plurigenus(S,m) : Sch, RngIntElt -> RngIntElt
    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.

ArithmeticGenusOfDesingularization(S) : Sch -> RngIntElt
    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.


Example AlgSrf_adj_ex (H108E5)

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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