Del Pezzo Surfaces#

Introduction#

This section contains a collection of geometric and arithmetic routines for Del Pezzo surfaces in their anti-canonical embeddings (the full weighted projective anticanonical embedding for degrees 1 and 2).

There are routines for creation, parametrization, minimisation and reduction, construction, computation of invariants for degree 3, and point-counting for degree 3 surfaces over a finite field.

There is a specialised type for Del Pezzos, SrfDelPezzo, which is a subtype of type Srfc. Some intrinsics use this type for arguments while some use the more general Srfc (or even Sch).

Creation of General Del Pezzos#

DelPezzoSurface(P, L): Prj, List -> SrfDelPezzo#
DelPezzoSurface(S): Setq[Pt] -> SrfDelPezzo#
DelPezzoSurface(Z): Clstr -> SrfDelPezzo#: The Del Pezzo surface of degree \(9-d\), embedded by its anticanonical system, which arises by blowing up the projective plane \(P\) in the \(d\) points. The arguments are either plane \(P\) and a list of points \(L\), the set of points \(Q\), or the length \(d\) zero-dimensional scheme \(Z\) defining the points. If the points are not in sufficiently general position (so that the anticanonical image is not smooth) then an error is reported.

DelPezzoSurface(f): RngMPolElt -> SrfDelPezzo#: The argument \(f\) should be a degree three homogeneous polynomial in a 4-variable polynomial ring \(P\) (with grevlex ordering). Creates the degree 3 Del Pezzo surface with defining polynomial \(f\) inside \({\bf P}^3 = \rm{Proj}(P)\). If this surface is not smooth an error is reported.

IsDelPezzo(Y): Sch -> BoolElt, SrfDelPezzo, MapSch#: Returns true if and only if scheme \(Y\) in ordinary projective space is an abstract Del Pezzo surface. If so, it also returns the image \(X\) of the standard (pluri-)anticanonical embedding of \(Y\), and the map \(Y \rightarrow X\). Note that this can be a computationally very heavy function if \(Y\) is in a reasonably high-dimensional ambient.

Some Auxiliary Intrinsics#

CollinearPointsOnPlane(P2, S): Prj, SetIndx[Pt] -> SeqEnum#: Argument \(P2\) should be an ordinary projective plane over a field and \(S\) an indexed set of distinct points on \(P2\). Returns a sequence containing all subsets (as enumerated sequences) of \(3\) or more points of \(S\) that are collinear.

PointsInGeneralPosition(P2, S): Prj, SetIndx[Pt] -> BoolElt, SeqEnum, SeqEnum, SeqEnum#

NonGenPts: BoolElt                    Default: false

Argument \(P2\) should be an ordinary projective plane over a field and \(S\) an indexed set of \(8\) or fewer distinct points on \(P2\). Returns whether the points in \(S\) are in general position. This means that no \(3\) are collinear, no \(6\) lie on a conic and, if there are \(8\) points, they do not lie on a cubic with a double point at one of them.

If the NonGenPts parameter is set to true and the points are not in general position, also returns \(3\) sequences containing respectively the subsets of \(3\) or more collinear points, the subsets of \(6\) or more co-conic points and the full set of \(8\) points if they fail to satisfy the cubic condition (all as enumerated sequences).

EckardtPoints(X): SrfDelPezzo -> Sch#: Argument \(X\) here should be a degree \(3\) Del Pezzo surface. Computes the set of Eckardt points of \(X\), which are points where \(3\) coplanar lines of the surface meet. The result is returned as a zero-dimensional (or empty) scheme consisting of this set of points.

Parametrization of Del Pezzo Surfaces#

Del Pezzo surfaces are a special type of non-singular projective surface. A good reference for their general properties is [Manin, 1986]. A Del Pezzo surface \(X\) has a degree \(1 \le d \le 9\). For \(d \ge 3\), the standard representation is as a degree \(d\) surface in \({\bf P}^d\) 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 \({\bf P}^2\) over an algebraically-closed field. That is, there exists an invertible scheme map from \({\bf P}^2\) to \(X\) : a parametrization. Most of the functions in this section are concerned with the existence of parametrizations of \(X\) over a number field.

The significance of Del Pezzo surfaces comes from adjunction theory (see [Sommese and van der Ven, 1987]). 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. For a non-singular ordinary projective surface \(Y\), the intrinsic MinimalModelRationalSurface is now available to produce the birational map from \(Y\) to one of the special, terminal cases that include 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 \({\bf P}^3\) is described in the previous section and it makes use of the routines described here.

If \(X\) is parametrizable with \(d \ge 3\), then we can blow down (contract) exceptional lines on it to arrive at a surface with \(d=5\), \(6\), \(8\) or \(9\). 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 [de Graaf, M. Harrison, J. Pilnikova and J. Schicho, 2006], [de Graaf and Pilnikova, n.d.], [Harrison and Schicho, 2006] and [Gonzalez-Sanchez et al., 2012]. 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 the more geometrical method described in the above reference and a corresponding intrinsic is provided.

Furthermore, the degree \(5 - 8\) intrinsics now also cover degenerate cases of singular Del Pezzos in their anticanonical projective embeddings. The additional code for the singular cases is also due to Josef Schicho.

A feature (from V2.17) is the provision of special case code for degree 3 and degree 4 singular Del Pezzos. For some of these, it is not possible to blow down any exceptional lines over the base field but the surface is still parametrizable. Additionally, it is usually much more efficient to handle these cases directly without blowing down curves to get to higher degree.

A general intrinsic ParametrizeDelPezzo for parametrizing any Del Pezzo surface of degree \(d \ge 1\) 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 previous section. This blows down exceptional lines to reach degree \(d \ge 5\) and then invokes one of the intrinsics described in this section. It is more efficient to call the appropriate intrinsic directly if the starting point is either the \(d \ge 5\) case or the \(d = 3,\ 4\) singular cases.

SetVerbose("ParamDP", v): MonStgElt, RngIntElt#: 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 \(1\)).

ParametrizeDegree9DelPezzo(X): Sch -> BoolElt, MapIsoSch#

Let \(X\) be a degree 9 Del Pezzo surface anticanonically embedded in 9-dimensional projective space. For this function the base field should be \({\bf Q}\). The surface \(X\) is defined by 27 degree 2 polynomials. The function performs only basic checks that the input \(X\) is valid.

If \(X\) is parametrizable over \({\bf Q}\) then there is a parametrization \(\phi: {\bf P}^2 \rightarrow X\) which is everywhere-defined and given by cubic polynomials in the variables of \({\bf P}^2\). The function returns whether such a \(\phi\) exists and, if so, \(\phi\) also.

In V2.19, LLL reductions have been applied to give simpler bases for the initial Lie algebra and the corresponding associative algebra. Also, the algorithm for splitting the latter has been changed to the one used with degree 3 homogeneous spaces of elliptic curves. This has greatly speeded up the computation.

ParametrizeDegree8DelPezzo(X): Sch -> BoolElt, MapSch#

Let \(X\) be a degree 8 Del Pezzo surface anticanonically embedded in 8-dimensional projective space over a number field. The surface \(X\) 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:

$X$ is isomorphic to ${\bf P}^2$ with a single rational point blown up.

2i) \(X\) is isomorphic to \(T_1 \times T_2\) with \(T_i\) Galois twists of \({\bf P}^1\).

2ii) \(X\) is isomorphic to a Galois twist of \({\bf P}^1 \times {\bf P}^1\) where Galois acts transitively on the two \({\bf P}^1\) factors.

In case 1), \(X\) is always parametrizable.

In case 2i), \(X\) is parametrizable \(\Leftrightarrow\) both \(T_i\) are trivial twists of \({\bf P}^1\) \(\Leftrightarrow\) \(X\) is isomorphic to \({\bf P}^1 \times {\bf P}^1\).

In case 2ii), there is an infinite family of parametrizable \(X_a\) classified by \(a \in {\bf Q}^*/{\bf Q}^{*2}\). The scheme \(X_a\) is isomorphic (properly, not just birationally) to the surface in \({\bf P}^3\) given by the equation \(x_0^2-ax_1^2=x_2x_3\).

The main function determines whether \(X\) is parametrizable over \({\bf Q}\). If so, there is a parametrization \(\phi: {\bf P}^2 \rightarrow X\) (given by cubic polynomials in case 1 and by degree 4 polynomials in case 2) and this is also returned.

The intrinsic also handles the degenerate case of a singular degree 8 Del Pezzo surface. 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.

Example: Del Pezzo Ex 8 (ex-2e5909)#

In this example, we parametrize an anticanonical sphere \(X_2\), in the above notation. This is obviously an artificial illustration, as we start with \(X_2\) in the form \(F=x_0^2-2x_1^2-x_2x_3=0\), which is trivial to parametrize directly! The surface in this form is embedded anticanonically in \({\bf P}^8\) by any 9-dimensional vector space complement of \(<F>\) in the 10-dimensional linear system of all degree 2 polynomials in \(x_0 \ldots x_3\). 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

Del Pezzo Surfaces#

Introduction#

Creation of General Del Pezzos#

Some Auxiliary Intrinsics#

Parametrization of Del Pezzo Surfaces#

Minimization and Reduction of Surfaces#

Cubic Surfaces over Finite Fields#

Construction of Cubic Surfaces#

Invariant Theory of Cubic Surfaces#

Invariants#

Covariants#

Contravariants#

Interaction of Covariants and Contravariants#

The Pentahedron of a Cubic Surface#