|
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
Del Pezzo surfaces are a special type of non-singular projective surface.
A good reference for their general properties is [Man86].
A Del Pezzo surface X has a degree 1 ≤d ≤9. For d ≥3, the
standard representation is as a degree d surface in Pd for which a
hyperplane section is an anti-canonical divisor. When we talk about Del
Pezzos in this section, we mean a surface in that anti-canonical form.
Del Pezzo surfaces are birationally equivalent to the projective plane
P2 over an algebraically-closed field. That is, there exists an
invertible scheme map from P2 to X : a parametrization.
Most of the functions in this section deal with determining the existence
of and explicit computation of parametrizations for X over a number
field.
The significance of Del Pezzo surfaces comes from adjunction theory
(see [SvdV87]). The adjunction map for surfaces is a
general construct that contracts certain lines, known as exceptional lines,
to points. Repeated application of the adjunction map to a rational surface
results in a reduction to a surface in one of a small number of families,
including the Del Pezzos.
Hence the parametrization problem for general
rational surfaces reduces via adjunction (a purely algebraic construction)
to parametrization of surfaces in the specific families, which is an
arithmetic problem (ie dependent on the ground field). This is the surface
analog of the simpler situation for curves. Any rational curve can be
algebraically reduced to the projective line or a plane conic and the
parametrization of plane conics is also an arithmetic problem.
Thus, the parametrization of Del Pezzo surfaces is an important component
in that of general rational surfaces. General parametrization code for
rational hypersurfaces in P3 is described in the chapter on
Algebraic Surfaces and it makes use of the routines here.
If X is parametrizable with d ≥3, then we can blow down (contract)
exceptional lines on it to arrive at a surface with d=6,8 or 9
(except, possibly, when d=5). The functions described here deal with
parametrization in those cases using computational methods based around
the Lie algebra of the automorphism group of X. For the theory behind
these algorithms, see [dG06], [dGP] and [HS06].
In one of the examples, we parametrize a cubic hypersurface (degree 3
Del Pezzo) by blowing down to a degree 6 surface.
Although reduction to degree 9 is always possible, for d=7 it is more
efficient to work directly using the Lie algebra method. There is now
also an intrinsic for d=7 provided by Josef Schicho. Schicho has also
written code for the d=5 case, minimal or not, that uses a more
geometrical method and is included as an intrinsic.
Furthermore, the degree 5 - 8 intrinsics also now cover degenerate cases of
singular Del Pezzos in their anticanonical projective embeddings. The
additional code for the singular cases comes from Josef Schicho again.
A general intrinsic ParametrizeDelPezzo for parametrizing any Del Pezzo surface
of degree d ≥1
in its anticanonical weighted embedding (for d = 1, 2 the anticanonical divisor
is no longer very ample, but gives an ample embedding into weighted projective
space) is described in the Algebraic Surfaces chapter. This blows down
exceptional lines to reach degree d ≥5 and then calls one of the intrinsics
here. It is more efficient to call the appropriate intrinsic directly if you
are already starting in the d ≥5 case.
Set the verbose printing level for the Del Pezzo parametrizing functions.
Currently the legal values for v are true, false, 0, 1 and
2 (false is the same as 0, and true is the same as 1).
Let X be a degree 9 Del Pezzo surface anticanonically embedded in
9-dimensional projective space. For this function the base field
should be Q. X is defined by 27 degree 2 polynomials. The
function performs only basic checks that the input X is valid.
If X is parametrizable over Q then there is a parametrization
φ: P2 -> X which is everywhere-defined and given
by cubic polynomials in the variables of P2. The function
returns whether such a φexists and, if so, φalso.
Let X be a degree 8 Del Pezzo surface anticanonically embedded in
8-dimensional projective space over a number field.
X is defined by 20 degree 2 polynomials. The
function performs only basic checks that the input X is valid.
For degree 8, there are two types of non-singular Del Pezzo surface, the
second type splitting into subfamilies:
1) X isomorphic to P2 with a single rational
point blown up.
2i) X isomorphic to T1 x T2 with Ti Galois
twists of P1.
2ii) X isomorphic to a Galois twist of P1 x P1
where Galois acts transitively on the two P1 factors.
In case 1), X is always parametrizable.
In case 2i), X is parametrizable <=> both Ti are
trivial twists of P1 <=> X is isomorphic to
P1 x P1.
In case 2ii), there is an infinite family of parametrizable Xa
classified by a ∈Q^ * /Q * 2. Xa is isomorphic (properly,
not just birationally) to the surface in P3 given by the
equation x02 - ax12=x2x3.
The main function determines whether X is parametrizable over Q.
If so, there is a parametrization φ: P2 -> X (given
by cubic polynomials in case 1 and by degree 4 polynomials in case
2) and this is also returned.
The intrinsic now also handles the degenerate case of a singular
degree 8 Del Pezzo. This case is recognised directly from the
Lie algebra computation which is part of the main routine and an
appropriate adaptation of the general method is used.
In this example, we parametrize an anticanonical sphere X2, in the
above notation. This is obviously an artificial illustration, as we
start with X2 in the form F=x02 - 2x12 - x2x3=0, which is
trivial to parametrize directly! The surface in this form is embedded
anticanonically in P8 by any 9-dimensional vector space complement
of <F> in the 10-dimensional linear system of all degree 2 polynomials
in x0 ... x3. The parametrizing map is undefined at precisely 2
points of the plane. Geometrically, the map consists of a blowup of
these 2 points followed by a blowdown of the line joining them.
> P3<x0,x1,x2,x3> := ProjectiveSpace(Rationals(),3);
> X2 := Scheme(P3,x0^2-2*x1^2-x2*x3);
> L := LinearSystem(P3,2);
> L := LinearSystemTrace(L,X2);
> P8<x1,x2,x3,x4,x5,x6,x7,x8,x9> := ProjectiveSpace(Rationals(),8);
> X := map<X2->P8|Sections(L)>(X2); X;
Scheme over Rational Field defined by
x1^2 - 2*x4^2 - x4*x8,
x1*x2 - 2*x4*x5 - x5*x8,
x2^2 - 2*x4*x7 - x7*x8,
x1*x3 - 2*x4*x6 - x5*x9,
x2*x3 - 2*x4*x8 - x7*x9,
x3^2 - 2*x4*x9 - x8*x9,
-x1*x5 + x2*x4,
-x1*x6 + x3*x4,
-x1*x7 + x2*x5,
-x1*x8 + x3*x5,
-x4*x7 + x5^2,
-x1*x8 + x2*x6,
-x1*x9 + x3*x6,
-x4*x8 + x5*x6,
-x4*x9 + x6^2,
-x2*x8 + x3*x7,
-x5*x8 + x6*x7,
-x2*x9 + x3*x8,
-x5*x9 + x6*x8,
-x7*x9 + x8^2
> boo,prm := ParametrizeDegree8DelPezzo(X);
> boo;
true
> prm;
Mapping from: Prj: P2 to Sch: X
with equations : -1/4*U*V*W^2
-1/16*V*W^3
2*U^2*V*W - 4*V^3*W
-1/8*U^2*W^2
-1/32*U*W^3
U^3*W - 2*U*V^2*W
-1/128*W^4
1/4*U^2*W^2 - 1/2*V^2*W^2
-8*U^4 + 32*U^2*V^2 - 32*V^4
> bs := ReducedSubscheme(BaseScheme(prm)); bs;
Scheme over Rational Field defined by
U^2 - 2*V^2,
W
Let X be a degree 7 Del Pezzo surface anticanonically embedded in
7-dimensional projective space over a number field. X can be a
degenerate (singular) Del Pezzo here. X is always parametrizable
over the base field and this intrinsic returns such a parametrisation
without reduction to degree 8 or 9 but directly from the Lie Algebra
method.
ExistenceOnly: BoolElt Default: false
Let X be a degree 6 Del Pezzo surface anticanonically embedded in
6-dimensional projective space. For this function the base field K
may be Q or a number field. X is defined by 9 degree 2 polynomials.
The function performs only basic checks that the input X is valid.
NB: This intrinsic only handles the non-singular case. For a
singular (degenerate) degree 6 Del Pezzo, use the intrinsic that
follows.
The connected component of the automorphism group of X is a
2-dimensional torus over K. For any of the possible tori, there
is a family of degree 6 Del Pezzos which correspond to principal
homogeneous spaces of the torus up to isomorphism. X is
parametrizable <=> it corresponds to the trivial
homogeneous space of its torus.
The function determines whether a parametrization
φ: P2 -> X exists over K and returns one when this
is the case. The degree of the polynomials defining a "minimal" φ(one which is undefined at the smallest number of points) is 3,4 or 6
depending on the torus type. The parametrization returned is always of
this minimal degree.
X is parametrizable if and only if it contains a point over K.
Furthermore, it satisfies the local-global principle: it has a point over
K <=> it has a point over each p-adic completion of
K. (These statements are also true for degree 8 and 9 Del Pezzos)
The ExistenceOnly option allows the function to just perform
this local solubility check, deciding upon the existence of a parametrization
without explicitly constructing one. Depending on the torus type,
simultaneous norm equations over a degree 6 field extension of K or
a single norm equation over a degree 3 extension of K may have to be
solved to construct a parametrization. This is a hard computation, especially
if K is not Q, whereas the pure existence check is quite fast.
Degree6DelPezzoType2_2(K,pt) : FldNum, Pt -> Sch
Degree6DelPezzoType2_3(K,pt) : FldNum, Pt -> Sch
Degree6DelPezzoType3(K,pt) : FldNum, Pt -> Sch
Degree6DelPezzoType4(K,K1,pt) : FldNum, Fld, Pt -> Sch
Degree6DelPezzoType6(K,pt) : FldNum, Pt -> Sch
These functions generate the parametrizable degree 6 Del Pezzo
surface X whose (connected) automorphism group is the torus T,
which comes from field data K, and which contains point pt.
The point pt must be in 6-dimensional projective space over the base field
k of number field K. It's first projective coordinate may not
be 0 and, depending on the torus type, certain of its other
coordinates must also be non-zero.
The torus types and corresponding fields K for the various
functions are as follows (pt = [a0, ..., a6]):
Type2_1. K/k should be a quadratic extension. T(k)=K^ *
and T acts on P6 to give an X with degree 3 minimal parametrization.
pt satisfies not( a1=a2=0 or a3=a4=0 or a5=0 or a6=0 ).
Type2_2. K/k should be a quadratic extension. T(k)=K^ *
and T acts on P6 to give an X with degree 4 minimal parametrization.
pt satisfies not( a1=a2=0 or a3=a4=0 or a5=a6=0 ).
Type2_3. K/k should be a quadratic extension.
T(k)=K * NK/k=1 x K * NK/k=1.
pt satisfies not( a1=a2=0 or a3=a4=0 or a5=a6=0 ).
Type3. K/k should be a cubic extension.
T(k)=K * NK/k=1.
pt satisfies not( a1=a2=a3=0 or a4=a5=a6=0 ).
Type4. K/k and K1/k should be distinct quadratic
extensions. T(k)=L * NL/K=1 where L is K.K1.
pt satisfies not( a1=a2=a3=a4=0 or a5=a6=0 ).
Type6. K/k should be a degree 6 extension which contains
cubic and quadratic subextensions K3 and K2. For simplicity, the
precise condition is that the generator y=K.1 must have minimal polynomial
of the form x6 + 2ax4 + a2x2 - d and then K3=k(y2) and K2=k(y3 + ay).
T(k)=K * NK/K3=NK/K2=1.
pt satisfies not( a1=a2=a3=a4=a5=a6=0 ).
This variant for parametrizing a degree 6 Del Pezzo also handles the
degenerate (singular) case. Note however, that it doesn't recognise
singularity from the Lie algebra computation as occurs for degrees 7
and 8. It tests for singularity at the start using the generic
non-singularity computation that can be very slow. Therefore for
known non-degenerate Del Pezzos of degree 6, it is always better to
use the above ParametrizeDegree6DelPezzo directly.
That is also used here, if X turns out to be non-singular. Otherwise,
projection from a singular point to P5 reduces the problem to that
of parametrizing a rational scroll.
In the next example, we start with a degree 3 Del Pezzo surface - a
non-singular hypersurface in P3 - which contains the 3 disjoint lines
x=y=0, z=t=0 and x=z, y=t. These are blown down to give a degree 6
Del Pezzo, the parametrisation of which gives a parametrisation of the
original surface. As well as demonstrating the degree 6 code, this
gives a nice example of blowing down exceptional lines on surfaces,
something for which more general code will be added at a future date.
> R3<x,y,z,t> := PolynomialRing(Rationals(),4,"grevlex");
> P3 := Proj(R3);
> //equation of the degree 3 surface:
> F := -x^2*z + x*z^2 - y*z^2 + x^2*t - y^2*t - y*z*t + x*t^2 + y*t^2;
> X3 := Scheme(P3,F);
> // get the ideal defing the union of the 3 lines:
> I1 := ideal<R3|[x,y]>;
> I2 := ideal<R3|[z,t]>;
> I3 := ideal<R3|[x-z,y-t]>;
> I := I1*I2*I3;
> I := Saturation(I);
General surface theory tells us that if H is the hyperplane divisor
on X3, then the blowing down is given by the projective map associated
to the divisor H + L1 + L2 + L3, where the Li are our 3 lines. We need
the global sections of the sheaf of this: if L1 + L2 + L3 ~2H - D
(linear equivalence of divisors) for an effective divisor D, then
the space of global sections"is" the degree 3 graded part of the
ideal of D (mod the equation of X3). The ideal ID of a suitable
D is computed by requiring that
ID ∩I = (F, F2) with F2 a degree 2 polynomial in I.
> F2 := Basis(I)[5]; F2;
y*z - x*t
> ID := ColonIdeal(ideal<R3|[F,F2]>,I);
> ideal<R3|[F,F2]> eq (ID meet I);
true
> // get basis of degree 3 graded part of ID
> ID3 := ID meet ideal<R3|Setseq(MonomialsOfDegree(R3,3))>;
> B3 := MinimalBasis(ID3);
> B3;
[
y*z*t - x*t^2,
z^3 - z^2*t + t^3,
y*z^2 - x*z*t,
x*z^2 - x*z*t + y*t^2,
y^2*z - x*y*t,
x*y*z - x^2*t,
x^2*z - x^2*t + y^2*t,
x^3 - x^2*y + y^3
]
> // and a complementary subspace of F
> F in ideal<R3|Remove(B3,7)>;
false
> B3 := Remove(B3,7);
> // now map to the degree 6 Del Pezzo
> P6<a,b,c,d,e,f,g> := ProjectiveSpace(Rationals(),6);
> blow_down := map<X3->P6|B3>;
> X6 := blow_down(X3);
> Dimension(X6); Degree(X6);
2
6
We also need the inverse of blow_down. The general IsInvertible
function could be used here but again the general theory tells us that
the inverse is given by linear equations and it is faster to find
them directly by a Grobner basis + linear algebra computation. We
omit this for brevity and just assume the result.
> X3toX6 := iso<X3->X6|B3,[f,e,c,a]>;
> // now parametrise X6
> boo,prm := ParametrizeDegree6DelPezzo(X6);
> boo;
true
> p2toX3 := Expand(prm*Inverse(X3toX6));
> p2toX3;
Mapping from: Projective Space of dimension 2
Variables : $.1, $.2, $.3 to Sch: X3
with equations : -77/9*$.1^3 + 59/6*$.1^2*$.2 + 10/3*$.1^2*$.3 + 8/9*$.1*$.2^2 -
73/18*$.1*$.2*$.3 - 113/18*$.1*$.3^2 - 59/9*$.2^3 + 383/18*$.2^2*$.3 -
259/9*$.2*$.3^2 + 329/18*$.3^3
253/18*$.1^3 - 193/6*$.1^2*$.2 - 17/3*$.1^2*$.3 + 695/18*$.1*$.2^2 -
244/9*$.1*$.2*$.3 + 353/18*$.1*$.3^2 - 151/9*$.2^3 + 185/9*$.2^2*$.3 -
41/9*$.2*$.3^2 - 79/9*$.3^3
-11/6*$.1^3 + 37/6*$.1^2*$.2 + 10/3*$.1^2*$.3 - 28/3*$.1*$.2^2 + 4*$.1*$.2*$.3 -
7*$.1*$.3^2 + 8/3*$.2^3 + 8/3*$.2^2*$.3 - 11/2*$.2*$.3^2 + 9/2*$.3^3
11/18*$.1^3 + 8/3*$.1^2*$.2 - 1/6*$.1^2*$.3 - 28/9*$.1*$.2^2 - 2/9*$.1*$.2*$.3 -
59/18*$.1*$.3^2 - 2/9*$.2^3 + 34/9*$.2^2*$.3 - 53/18*$.2*$.3^2 + 53/18*$.3^3
and inverse
-884/23043*x^3 + 884/23043*x^2*y - 884/23043*y^3 - 4436/23043*x*y*z -
4334/23043*y^2*z + 6902/23043*x*z^2 - 3560/7681*y*z^2 - 4420/23043*z^3 +
4436/23043*x^2*t + 4334/23043*x*y*t + 3778/23043*x*z*t + 4420/23043*y*z*t +
4420/23043*z^2*t - 4420/23043*x*t^2 + 6902/23043*y*t^2 - 4420/23043*t^3
-442/23043*x^3 + 442/23043*x^2*y - 442/23043*y^3 - 3544/23043*x*y*z -
6808/23043*y^2*z + 8800/23043*x*z^2 - 4392/7681*y*z^2 - 6290/23043*z^3 +
3544/23043*x^2*t + 6808/23043*x*y*t + 4376/23043*x*z*t + 8744/23043*y*z*t +
6290/23043*z^2*t - 8744/23043*x*t^2 + 8800/23043*y*t^2 - 6290/23043*t^3
-884/23043*x^3 + 884/23043*x^2*y - 884/23043*y^3 - 458/23043*x*y*z -
5660/23043*y^2*z + 5828/23043*x*z^2 - 2854/7681*y*z^2 - 3910/23043*z^3 +
458/23043*x^2*t + 5660/23043*x*y*t + 2734/23043*x*z*t + 6208/23043*y*z*t +
3910/23043*z^2*t - 6208/23043*x*t^2 + 5828/23043*y*t^2 - 3910/23043*t^3
and alternative inverse equations:
...
Let X be a degree 5 Del Pezzo surface anticanonically embedded in
5-dimensional projective space over a number field. X can be a
degenerate (singular) Del Pezzo here. X is always parametrizable
over the base field and this intrinsic returns such a parametrisation
without reduction to higher degree.
X has a finite automorphism group in this case, so the Lie Algebra
method cannot be applied. However, there is a more geometric method
using projections that works well for degree 5 and that is used here.
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|
