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We refer to K3 surfaces that can be represented as the desingularisation of
a double cover of the projective plane P2 ramified over a sextic curve
as degree 2 K3 surfaces. This condition on K3 surfaces is equivalent to the
existence of an irreducible divisor of self-intersection 2. The divisor map
of such a divisor is the required double cover of P2. Conversely, the pullback
of a general line in a double-cover presentation gives an irreducible divisor
of self-intersection 2.
If we take a (normal) double cover of the plane ramified over a reduced sextic (degree 6)
curve, the desingularisation gives a K3 surface if and only if the curve has at worst
simple curve singularities (s-c-s). If the singularities are worse, the resulting double-cover
surface has Kodaira dimension -1. Given the s-c-s condition, the normal double cover
ramified over the sextic has simple singularities precisely over the singularities of
the sextic (so is already non-singular if the sextic is). For more information on
this and more general surface double covers, see Ch. III, Section 7 and Ch. V, Section 22 of
[BHPdV04].
This section describes functionality to work with degree 2 K3 surfaces given by the
(generally singular) normal double-cover model naturally embedded in a weighted
P(1, 1, 1, 3) space (see the basic constructor below). Singularities and the
set of (-2)-curves lying over them are handled by blow-up desingularisation.
The only restriction on the base-field is that the characteristic cannot be
equal to 2.
The current functionality mainly deals with computation of (-2)-curves coming from
lines and conics that split in the double cover (along with those above singular points),
the intersection matrix of a given finite collection of curves and blow-up divisors
on the surface, and elliptic fibrations of the surface related to appropriate (-2)-curve
configurations. We hope to expand this functionality in a number of directions in
future releases.
Currently, there is no special surface subtype for degree 2 K3 surfaces, but we
construct them in a standard form (in weighted P(1, 1, 1, 3) ambients) which the
intrinsics in this section work with. The main constructor below takes a homogeneous
degree 6 polynomial f in 3 variables, checks that it gives a reduced plane curve
with only simple curve singularities, and returns this standard form for the K3 surface
which is the double cover of the plane ramified over f. The singular subscheme of
the surface is also filled in.
Note that later intrinsics assume that singularity properties of the surface argument have
been computed and cached, so this constructor should always be used to create
degree 2 K3 surfaces. The SingularSubscheme intrinsic currently doesn't work
directly for these surfaces because of the weighted ambient.
The argument f should be a homogeneous degree 6 polynomial in a 3-variable homogeneous
polynomial ring over a field of characteristic ≠2. Returns the surface S, which is a
(generally singular) model of the K3 surface that is a double cover of the projective
plane ramified over the f=0 curve Cf.
Cf should be a reduced curve with at worst
simple curve singularities. This is checked, with an error resulting if the conditions
are not satisfied. Simple curve singularities are double and triple points analytically
equivalent to particular forms as listed in Thm. 8.1 of Chapter II, [BHPdV04].
They lie in the usual A-D-E families and a curve singularity of a certain type on Cf
gives a simple surface singularity of the same type on S (e.g. A2 -> A2) lying
over it. Note that Cf does not have to be irreducible. If, for example, it has two
components meeting transversely at a point, that point is a simple A1 singularity of
Cf (node). Nodes are A1 singularities and simple cusps A2 singularities
on a curve.
If f is defined over the base field k then S will be a hypersurface in a weighted projective space
with variables x, y, z, w which have weights 1, 1, 1, 3 respectively. The defining polynomial
will be f(x, y, z) - w2. S is a normal variety giving the normal double cover of the plane
with ramification locus Cf. As mentioned above, this will have simple singularities over
singularities of Cf and the actual K3 surface is the desingularisation of S.
The intrinsic also computes and caches the singular subscheme of S, if it is singular,
or caches that it is non-singular otherwise, before it is returned.
Strictly speaking, the double cover of the plane ramified over Cf is really only defined
as a variety over k up to quadratic twist for a non algebraically-closed base field k.
The particular twist constructed is determined by f in choosing the defining polynomial
f - w2.
We construct the Drell-Yan K3 surface, which is related to certain Feynmann integrals
arising in high-energy physics (see [BFHN]). The ramification locus for this
interesting surface has many singularities which give rise to 14 blow-up (-2)-curves
via desingularisation. We will use this example again later to illustrate the various
intrinsics.
> R<a,b,c> := PolynomialRing(Rationals(),3);
> f := 4*a*b^2*c*(a+c)^2+(a*(b+c)^2+b*(a+c)^2)*(a*(-b+c)^2+b*(c-a)^2);
> S := DegreeTwoK3Surface(f);
> S; Ambient(S);
Surface over Rational Field defined by
x^4*y^2 + 2*x^3*y^3 + x^2*y^4 + 4*x^3*y^2*z + 2*x^3*y*z^2 - 4*x^2*y^2*z^2 +
2*x*y^3*z^2 + 4*x*y^2*z^3 + x^2*z^4 + 2*x*y*z^4 + y^2*z^4 - w^2
Projective Space of dimension 3 over Rational Field
Variables: x, y, z, w
The grading is:
1, 1, 1, 3
> MinimalBasis(SingularSubscheme(S));
[
x*y + x*z,
w,
y^2*z + y*z^2,
x^2*z + y*z^2
]
> HasOnlySimpleSingularities(S : ReturnList := true);
true
[* <(-1 : -1 : 1 : 0), "A", 1>, <(1 : -1 : 1 : 0), "A", 3>,
<(0 : 0 : 1 : 0), "A", 2>, <(0 : 1 : 0 : 0), "A", 4>,
<(1 : 0 : 0 : 0), "A", 4> *]
One of the main aims of the analysis of a K3 surface S is to compute its Neron-Severi group NS(S),
its divisor group up to algebraic equivalence. For a K3, this is a finitely-generated,
torsion-free abelian group of rank ≤20 (for S in characteristic 0) and algebraic
equivalence is the same as rational equivalence or numerical equivalence so
NS(S)=Pic(S)=Num(S).
In many cases, NS(S) is generated by (-2)-curves. Apart from the irreducible blow-up divisors
over singular points, one way to find (-2)-curves on a degree 2 K3 surface is as the
components of the pullbacks of lines and conics in the plane which split in the double covering.
This idea was first exploited by Elsenhans and Jahnel and formulated into an algorithm
by Dino Festi ([Fes]).
We provide an intrinsic to compute the split lines and also some prototype intrinsics to compute
split conics. We hope to add efficient intrinsics to properly compute the split conics for
characteristic 0 degree 2 K3 surfaces in subsequent releases. As it stands, we provide an
intrinsic to compute split conics when S is defined over a (small) finite field. When applied to
reductions mod p for several p, this often provides split conics that can be lifted to Q
(see the example). There is also an intrinsic to check whether a given conic splits or not.
It should be stated that these intrinsics tend to find split curves in the high-rank cases
(rk(NS(S)) is close to 20) but often return nothing for lower-rank cases, where constructing
non-trivial divisor classes is still a hard problem.
These split divisor intrinsics were adapted from code kindly provided by D. Festi.
We also provide an intrinsic that computes the full intersection matrix of the set consisting of
the strict transforms of a given collection of effective divisors on S and the irreducible
blow-up divisors over singularities of the ramification sextic. This, together with the intrinsics
to find split curve divisors, allows the partial, or in some cases complete, computation of NS(S).
TriTangentLines(F) : RngMPolElt -> List
This computes and returns a list of the full set of lines in the plane that split in the degree 2 K3 surface S.
Since this condition is equivalent to the line meeting the ramification sextic with even multiplicity
at each intersection point, we refer to these as tri-tangent lines. Some may be defined over
a finite extension of the base field k of S, in which case only one of the set of Galois-conjugate lines
is returned.
There are two versions, one taking the surface S as argument and the other taking the degree 6 ramification
polynomial F, which should lie in a 3-variable polynomial ring. In the first case, the lines are returned
as linear forms in the first 3 variables of the coordinate ring of the ambient of S, R, and lie in
a base change of R to the field of definition of the line. In the second case, the return values are linear
forms in a base change of the parent polynomial ring of F.
This computes the full set of conics that split in a degree 2 K3 surface S defined over a
finite field k. Since it is largely intended for usage on the mod p reduction of surfaces defined over
Q (or a finite extension of Q), we have included "Modp" in the name and only include a version where
the argument is a degree 6 homogeneous polynomial F in 3 variables defining the ramification locus of S
as a double cover of the plane. As above, the conics that split are precisely those that intersect this sextic
ramification curve with even multiplicity at each intersection point, so we refer to them as six-tangent conics.
As opposed to TriTangentLines, this only computes split conics which are defined over k and not over
finite extensions. It uses an iteration through all possible conics, so overall speed of execution decreases very
quickly with the size of k. Thus, it should only be used over quite small finite fields.
Note that a quadratic factor of F is not counted as a six tangent conic by this intrinsic, even though
it ramifies in the double cover to give a (-2)-curve on S.
The return value is a sequence of homogeneous degree 2 polynomials in the parent polynomial ring of F.
SixTangentConicCheck(F,C) : RngMPolElt, RngMPolElt -> BoolElt
Returns whether a plane conic splits in the degree 2 surface S or not.
There are two versions, one taking the surface S as its first argument
and the second taking the homogeneous degree 6 ramification
polynomial F, which should lie in a 3-variable polynomial ring which MUST
have the grevlex ordering.
In either case the second argument is a homogeneous degree 2 polynomial
C giving the conic equation. In the first case, it should be a polynomial
in the first 3 variables of the coordinate ring of the ambient of S. In the second,
it should lie in the same polynomial ring as F.
IncludeExCurves: BoolElt Default: true
Argument S should be a degree 2 K3 surface in standard form and argument Ds a sequence of effective
divisors given as subschemes of S. No two of Ds should have an irreducible component in common. This
is not checked, nor is whether the D ∈Ds actually form divisors (i.e. have a pure codimension 1
primary decomposition). The expected case is that the divisors are all distinct and irreducible, in which
case there is no problem.
Let seq be the sequence of divisors given by appending any irreducible blow-up curves ("exceptional"
curves) to the sequence of strict transforms on the desingularised surface of the divisors in Ds.
The order in which they are appended is s1 cat s2 cat ... sn, where
si is the sequence of exceptional divisors for the i-th desingularisation data object dsds[i] of
the desingularisation data list dsds returned by ResolveSingularSurface. The order of blow-up
divisors in si is the order in which they are indexed in dsds[i].
The intrinsic returns the #seq-by-#seq intersection matrix for the divisors in seq
on the K3 surface.
To get the intersection matrix of the strict transforms without the appended exceptional curves,
the IncludeExCurves parameter can be set to false (the default is true).
We construct (-2)-curves on the Drell-Yan surface S via split lines and conics and compute the rank
of the subgroup of NS(S) that they generate together with the 14 exceptional curves by using
the intersection matrix intrinsic.
> R<a,b,c> := PolynomialRing(Rationals(),3);
> f := 4*a*b^2*c*(a+c)^2+(a*(b+c)^2+b*(a+c)^2)*(a*(-b+c)^2+b*(c-a)^2);
> S := DegreeTwoK3Surface(f);
> lins := TriTangentLines(S);
> lins;
[*
x - z,
x,
y + z,
y,
y + alpha1*z,
z
*]
Find which lines are defined over Q.
> [IsIdentical(Parent(l),CoordinateRing(Ambient(S))) : l in lins];
[ true, true, true, true, false, true ]
> BaseRing(Parent(lins[5]));
Number Field with defining polynomial u^2 - 3*u + 1 over the Rational Field
The fifth line gives two conjugate split lines defined over a quadratic extension
of Q (Q(Sqrt(5))) and the others are defined over Q. We ignore the
fifth one here, to avoid base extension, and get the (-2)-curves over Q
defined by the others.
> spl := [PrimeComponents(Scheme(S,lins[i])) : i in [1,2,3,4,6]];
> assert &and[#ps eq 2 : ps in spl];
> Ls := [s[1] : s in spl];
> Lsd := [s[2] : s in spl];
> Ls;
[
Scheme over Rational Field defined by
y^2*z + 2*y*z^2 + z^3 + w,
x - z,
Scheme over Rational Field defined by
y*z^2 + w,
x,
Scheme over Rational Field defined by
x^2*z - z^3 + w,
y + z,
Scheme over Rational Field defined by
x*z^2 + w,
y,
Scheme over Rational Field defined by
x^2*y + x*y^2 + w,
z
]
> Lsd;
[
Scheme over Rational Field defined by
y^2*z + 2*y*z^2 + z^3 - w,
x - z,
Scheme over Rational Field defined by
y*z^2 - w,
x,
Scheme over Rational Field defined by
x^2*z - z^3 - w,
y + z,
Scheme over Rational Field defined by
x*z^2 - w,
y,
Scheme over Rational Field defined by
x^2*y + x*y^2 - w,
z
]
Now we see if we can find any split conics by searching mod 5 and mod 7.
> Rp<a,b,c> := PolynomialRing(GF(5),3,"grevlex");
> SixTangentConicsModp(Rp!f);
[
a^2 + a*b + b^2 + a*c + 2*b*c,
a^2 + 2*a*b + 2*a*c + c^2,
4*a^2 + 4*a*b + b^2 + 2*a*c + 3*b*c,
a^2 + b*c,
a^2 + 2*a*b + 2*a*c + b*c
]
> Rp<a,b,c> := PolynomialRing(GF(7),3,"grevlex");
> SixTangentConicsModp(Rp!f);
[
a^2 + a*b + b^2 + a*c + 2*b*c,
a^2 + b*c
]
We suspect that the two mod 7 split conics may be split conics over Q with
the natural lifted equations, since they both occur as mod 5 split conics.
This is indeed true, as we now check.
> x,y,z,w := Explode([Ambient(S).i : i in [1..4]]);
> Cs := [x^2+y*z, x^2+x*y+y^2+x*z+2*y*z];
> [SixTangentConicCheck(S,C) : C in Cs];
[ true, true ]
> splC := [PrimeComponents(Scheme(S,C)) : C in Cs];
> assert &and[#ps eq 2 : ps in splC];
> LsC := [s[1] : s in splC];
> LsCd := [s[2] : s in splC];
> [MinimalBasis(Z) : Z in LsC];
[
[
x^2 + y*z,
x*y^2 - y^2*z - x*z^2 - y*z^2 - w
],
[
x^2 + x*y + y^2 + x*z + 2*y*z,
y^3 + x*y*z + 4*y^2*z + x*z^2 + 3*y*z^2 + w
]
]
> [MinimalBasis(Z) : Z in LsCd];
[
[
x^2 + y*z,
x*y^2 - y^2*z - x*z^2 - y*z^2 + w
],
[
x^2 + x*y + y^2 + x*z + 2*y*z,
y^3 + x*y*z + 4*y^2*z + x*z^2 + 3*y*z^2 - w
]
]
We now get the intersection matrix for the 28 = 10 + 4 + 14 (-2)-curves consisting of
the split line and conic divisors and the 14 blow-up curves, all of which are defined over Q.
> Ls cat:= LsC;
> Lsd cat:= LsCd;
> imat := IntersectionMatrixOnDegree2K3(S, Ls cat Lsd);
> imat;
[ -2 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 ]
[ 0 -2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 ]
[ 0 1 -2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 ]
[ 1 0 0 -2 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 ]
[ 0 0 0 0 -2 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 ]
[ 0 0 0 1 0 -2 0 0 0 0 0 1 2 1 1 0 1 0 1 0 0 1 0 0 0 0 0 0 ]
[ 1 0 0 1 0 0 -2 0 1 0 0 2 1 3 1 1 0 0 0 1 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 0 0 0 -2 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 ]
[ 0 0 0 0 0 0 1 0 -2 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 ]
[ 0 0 0 0 0 0 0 0 1 -2 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 ]
[ 0 0 0 0 0 0 0 1 0 0 -2 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 ]
[ 0 0 0 0 1 1 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 ]
[ 0 0 0 0 1 2 1 0 0 0 1 0 -2 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 ]
[ 0 1 0 0 2 1 3 1 0 0 1 0 0 -2 1 0 1 0 1 0 0 0 0 0 0 0 0 0 ]
[ 0 0 1 0 0 1 1 0 0 1 0 0 1 1 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
[ 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 -2 0 1 0 0 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 -2 1 0 0 0 0 0 0 0 0 0 0 ]
[ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 -2 0 0 0 0 0 0 0 0 0 0 ]
[ 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -2 1 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 -2 0 0 0 0 0 0 0 0 ]
[ 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -2 0 1 0 0 0 0 0 ]
[ 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 -2 0 1 0 0 0 0 ]
[ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -2 1 0 0 0 0 ]
[ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -2 0 0 0 0 ]
[ 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 1 0 ]
[ 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 1 ]
[ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -2 1 ]
[ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -2 ]
> Rank(imat);
18
This shows that the 28 curves generate a subgroup of
NS(S) of rank 18. In fact, [BFHN] shows that
the rank of the subgroup of NS(S) defined over Q is 18,
that its full rank (over bar(Q)) is 19 and that the full
NS(S) is generated by the above (-2)-curves along with the divisors
of the split line defined over Q(Sqrt(5)) and those of a
split conic also defined over Q(Sqrt(5)). The above curves
probably do generate the full subgroup defined over Q but I haven't
checked this.
Elliptic fibrations of K3 surfaces are everywhere defined maps from the
surface to P1 such that the general fibre is an irreducible, non-singular genus
1 curve. Sometimes these are referred to as genus 1 fibrations with the elliptic
fibration terminology reserved for the stronger condition that there also exists
a section of the fibre map <=> the generic genus 1 fibre has a point rational
over its base-field. However, we do not distinguish here. When there is a section,
uniformly providing a O-point on all fibres, the K3 surface with its fibre map actually
gives the minimal (curve) model over P1 of the generic elliptic curve fibre.
Such fibrations are very useful in analysing arithmetic and geometric properties of
a K3 surface S and finding multiple inequivalent fibrations can also be important,
providing different decompositions of finite index sublattices of the Neron-Severi
lattice and leading to automorphisms of S with interesting properties.
We use a method to compute elliptic fibrations based upon having many (-2)-curves
on the surface at our disposal and finding configurations of these that give bad fibres
of an elliptic fibration. This idea has appeared (in a more theoretical setting)
previously in the works of Shioda. It was discovered independently as a computational
tool by Michael Harrison.
We refer to the configurations of (-2)-curves that give elliptic fibrations as
Kodaira configurations. These are sets of (-2)-curves that intersect in
the right way to give a standard non-irreducible elliptic fibration bad fibre,
when given the correct multiplicity. Each Kodaira configuration on a K3 surface
is the bad fibre of a unique (up to automorphisms of the base P1) elliptic
fibration and two Kodaira configurations give the same fibration if and only if
they are disjoint (see Section 4 of [BFHN]).
The elliptic fibration intrinsics in this section provide the following functionality.
Given the intersection matrix of a collection, Σ, of (-2)-curves on S, there is
an intrinsic that computes all of the maximal disjoint sets of Kodaira configurations
supported on Σ. For a particular Kodaira configuration, consisting of (-2)-curves
that may be either the strict transforms of curves on the singular model of S or
irreducible blow-up curves over the simple singularities, there is an intrinsic
to explicitly compute the associated elliptic fibration map to P1. Additionally,
there is an intrinsic to compute the generic fibre, as a curve over a rational function
field k(t) where k is the base field of S, of an elliptic fibration from its map.
This intrinsic also computes the associated k(t)-rational points on the generic fibre
for a given set of sections of the fibration.
SetVerbose("KodFibs", n): Maximum: 1
The argument imat should be the intersection matrix of an ordered set Σ of
irreducible (-2)-curves on the desingularisation of a degree 2 K3 surface S.
Returns a list containing all of the maximal disjoint sets of Kodaira configurations
supported on Σ (see the introductory section). The list is in 1-1 correspondence
with equivalence classes of elliptic fibrations on S that have at least one bad fibre
supported entirely on curves in Σ. The set of Kodaira configurations in the list
entry gives all of the bad fibres of the corresponding fibration that are supported
on curves in Σ. Here, we say that two elliptic fibration maps to P1 are
equivalent if they simply differ by an automorphism of P1 and that two elliptic
fibrations are equivalent if the associated equivalence classes of elliptic fibration
maps are the same. Equivalent elliptic fibrations clearly have the same set of fibres.
The list entry for a maximal disjoint set of configurations is a sequence of the form
[e1, ..., er] where the ei are the configurations given in the form of a pair
< s, l > where s is the a sequence of indices of the elements of Σ
forming the components of the fibre and l is a label string giving the type of
the fibre.
The type labels use the extended Dynkin notation for elliptic bad fibres, except that
In is used rather than tilde(An) for multiplicative reduction types. The
code cannot distinguish between types III and I2 or IV and I3 (this doesn't
matter in practice: the fibres contain each irreducible component to multiplicity
1 either way), and the In label will be used in both cases. The possible type labels
are thus In, n ≥2, Dn, n ≥4, and E6, E7, E8, where the
Di and Ei types are usually written with a tilde and represent a configuration
of (-2)-curves forming an extended Dynkin diagram of the type label in terms
of their dual intersection graph. See Ch. V, Sec. 7 of [BHPdV04] for a description
of these configurations.
To know what multiplicities of the components of the fibre occur, it is necessary
to know which nodes of the corresponding extended Dynkin diagram correspond
to which positions in the sequence s. This depends on the type of the configuration.
For an In configuration, whose diagram is a (regular) n-gon, the order of s
gives the cyclic order to which the components occur. That is, component s[1] is
joined to component s[2], component s[2] to component s[3], ... component
s[n - 1] to component s[n] and component s[n] to component s[1].
For an extended Dn configuration, if s=[a1, .., an + 1] then the central axis of the
extended Dynkin diagram is a3 - a4 - ... - an - 1 and a1, a2
join onto the a3 node, an and an + 1 join onto the an - 1 node
[a3 is the central node when n=4].
For an extended E6 configuration, s=[a1, ..., a7] and in the extended Dynkin diagram,
a1 is the central node and a2 - a3, a4 - a5, a6 - a7 are the arms
with a2, a4 and a6 meeting a1.
For an extended E7 configuration, s=[a1, ..., a8] and in the extended Dynkin diagram,
a1 - a2 - ... - a6 - a7 is the main axis with a8 meeting a4.
For an extended E8 configuration, s=[a1, ..., a9] and in the extended Dynkin diagram,
a1 - a2 - ... - a7 - a8 is the main axis with a9 meeting a3.
We continue with the Drell-Yan surface, using the 28 (-2)-curves from the last example as
our Σ and the intersection matrix computed there.
We find a lot of inequivalent elliptic fibrations.
> SetVerbose("KodFibs",1);
> time confs := KodairaConfigurations(imat);
Getting simple paths...
Time: 2.590
Getting In loops...
Time: 0.610
Getting extended Dn configs...
Time: 1.540
Getting extended E6 configs...
Time: 0.070
Getting extended E7 configs...
Time: 0.070
Getting extended E8 configs...
Time: 0.190
Getting disjoint sets of configurations...
Dealing with 16716 configurations.
Done 1000 sets of configurations
Done 2000 sets of configurations
Done 3000 sets of configurations
Done 4000 sets of configurations
Done 5000 sets of configurations
Done 6000 sets of configurations
Done 7000 sets of configurations
Done 8000 sets of configurations
Done 9000 sets of configurations
Done 10000 sets of configurations
Done 11000 sets of configurations
Done 12000 sets of configurations
Done 13001 sets of configurations
Done 14001 sets of configurations
Done 15001 sets of configurations
Done 16001 sets of configurations
Time: 11.460
> #confs;
16402
We have found 16402 inequivalent elliptic fibrations of S!
We examine the configuration data for some of them.
> Max([#e : e in confs]);
3 2022
> // 3 is the maximum number of disjoint Kodaira fibrations in
> // one of the configuration sets.
> // Get the indices in confs of the size 2 and 3 configuration sets
> s2 := [i : i in [1..#confs] | #(confs[i]) eq 2];
> #s2;
282
> s3 := [i : i in [1..#confs] | #(confs[i]) eq 3];
> #s3;
16
> confs[s3[3]];
[
<[ 16, 18, 1, 4, 27, 28, 26, 3 ], "I8">,
<[ 10, 20, 9, 24, 22, 23 ], "D5">,
<[ 5, 14 ], "I2">
]
The elliptic fibration corresponding to the above configuration
has 3 bad fibres which are supported solely on curves in Σ.
They are multiplicative reduction fibres of type I 8 and I 2 and an
additive reduction fibre of extended Dynkin type tilde(D 5) (I * 1
in the alternative bad fibre notation). Using the order of indexing
of our (-2)-curves and the notation E i, j for the jth blow-up
divisor of the ith desingularisation data object dsds[i] of S, we see
that the additive bad fibre as a divisor on the desingularisation
of S is L 3d + E 3, 2 + E 4, 2 + E 4, 3 + 2 * L 2d + 2 * E 4, 4 where
L id is the strict transform of the curve Lsd[i].
The 1968th elliptic fibration will be constructed in the next example.
It's configuration data contains two bad fibres supported on Σ:
an I10 and an I2 fibre.
> confs[1968];
[
<[ 25, 10, 9, 24, 23, 2, 3, 26, 28, 27 ], "I10">,
<[ 6, 13 ], "I2">
]
The first argument S should be a degree 2 K3 surface in standard form. Given a Kodaira fibre F
on the desingularisation X of S, the intrinsic returns the 2-dimensional Riemann--Roch space of
F, H0(X, L(F)), as a sequence B and denominator d. Here B=[f1, f2] where f1, f2, d are all
weighted homogeneous polynomials of the same degree in the coordinate ring of the ambient of S, and a basis
for the R-R space is given by the two rational functions on S, f1/d and f2/d.
The return is in this form for convenience because the divisor map from S -> P1 of |F|, which is
the elliptic fibration map with F as a bad fibre, is then just defined by B.
The components of F may be strict transforms of irreducible curves on S or irreducible blow-up divisors
over the singular points of S. F is specified by the second and third argument as follows. Assume that,
as a divisor on X, it is given by the sum m1 * D1 + ... + mr * Dr + n1 * E1 + ... + ns * Es, where the Di are the irreducible
components of F that are strict transforms of curves on S, the Ei are irreducible blow-up divisors
and the mi, nj are positive multiplicities.
The second argument divlst specifies the (always non-empty) Di part of the divisor sum.
It should be a sequence of (Sch,RngIntElt) 2-element lists where the ith 2-element list is
[ * Zi, mi * ] where Zi is the subscheme of S whose strict transform is Di.
The third argument exdivlst specifies the Ei part of the sum. It should be a sequence of
(Sch, RngIntElt, RngIntElt) 3-element lists. For the ni * Ei term of the sum, if
Ei is the jth irreducible divisor over the singularity represented by desingularisation data
object dsd, the corresponding 3-element list would be [ * dsd, j, ni * ]. If there are no Ei,
this sequence should just be the empty sequence.
A concrete example of using this intrinsic is given below.
The computation uses a simplified version of the standard Magma Riemann--Roch functionality applied
to an ordinary projective embedding of S into ordinary P10, with simplifications coming from
the fact that this is a projectively-normal embedding. The blow-up divisor part is handled
by the linear system divisor restriction intrinsic.
varname: MonStgElt Default: "t"
As usual, the first argument S should be a degree 2 K3 surface in standard form. The second argument
B should be a sequence of two weighted-homogeneous polynomials of the same degree in the coordinate
ring of the ambient of S that define an elliptic fibration map on S. E.g., as returned by the
previous intrinsic. The third argument secs should be a (possibly empty) sequence of irreducible
subschemes of S whose strict transforms to the desingularisation of S give sections of this fibration.
If F is any fibre of the elliptic fibration (e.g. a Kodaira fibre defining it), then the condition that
the strict transform tilde(D) of irreducible D ⊂S is a section is precisely that the intersection number
tilde(D).F = 1. This is easily checked with the intersection matrix intrinsic.
If the base field of S is k, the intrinsic returns a (generally singular) curve C over rational function field
k(t) that is a birational model of the smooth genus 1 generic fibre of the fibration map.
Technically speaking, this is the scheme-theoretic
pullback of the fibration S -> P1 under the generic point inclusion map Spec(k(t)) -> P1.
t corresponds to the rational function B[1]/B[2] on S. C will usually be returned as a plane curve.
A second return value gives the sequence of k(t)-rational points on C corresponding to the sections in secs.
The third return value is a sequence of 4 rational functions on the ambient of C that give the substitutions
for x, y, z, w (the coordinate variables on the ambient of S) that transform S generically into C.
By default, the variable generator t of the rational function field k(t) over which C is defined, will be labelled "t".
To change this, the varname parameter can be used to specify the string used for
the variable labelling.
We construct one of the elliptic fibrations of the Drell-Yan surface coming from the configurations found in the
last example, compute its generic fibre, and use a point on this fibre coming from a section to transform to
a Weierstrass elliptic curve model over Q(t).
The fibration we will construct is the one corresponding to the 1968th configuration found, confs[1968].
It has two bad fibres supported on our set of 28 curves, an I2 and an I10 fibre. Although it is
simpler to construct the fibration from the I2 fibre, we will use the I10 fibre, which contains
blow-up divisors, to better illustrate the Kodaira fibre divisor specification.
In the notation of the last example, the I10 Kodaira fibre is
L2 + L2d + L3 + L3d + E4, 3 + E4, 4 + E5, 1 + E5, 2 + E5, 3 + E5, 4
We also note that L4, L4d, L5, L5d each give sections of the fibration as is
easily deduced from the intersection matrix imat.
> dsds := ResolveSingularSurface(S); // get the desing data objects
> B,F := EllipticFibrationRRSpaceDeg2K3(S,[ [*Ls[2],1*],[*Lsd[2],1*],
> [*Ls[3],1*], [*Lsd[3],1*] ], [ [*dsds[4],j,1*] : j in [3,4] ] cat
> [ [*dsds[5],j,1*] : j in [1..4] ]);
> B; F;
[
x^2 + y*z,
x*y + x*z
]
x*y + x*z
Now we get the generic fibre of the fibration and the points on it corresponding
to the sections listed above.
> C,pts,subst := EllipticGeneralFibreDeg2K3(S,B,[Ls[4],Lsd[4],
> Ls[5],Lsd[5]]);
> C; Ambient(C);
Curve over Univariate rational function field over Rational Field defined by
(t^2 + 2*t + 1)/t^4*a^4 + (-2*t^3 - 2*t^2 - 6*t - 2)/t^4*a^3*c + (t^4 + 8*t^2 +
4*t + 3)/t^4*a^2*c^2 + (-2*t^3 - 2*t^2 - 6*t - 2)/t^4*a*c^3 + (t^2 + 2*t +
1)/t^4*c^4 - b^2
Projective Space of dimension 2 over Univariate rational function field over
Rational Field
Variables: t
Variables: a, b, c
The grading is:
1, 2, 1
> pts;
[ (t : (-t^2 + 1)/t^2 : 1), (t : (t^2 - 1)/t^2 : 1), (1 : (-t - 1)/t^2 : 0),
(1 : (t + 1)/t^2 : 0) ]
> subst;
[
$.1,
(1/t*$.1^2 - $.1*$.3)/($.1 - 1/t*$.3),
$.3,
($.1^3*$.2 - $.1*$.2*$.3^2)/($.1^2 - 2/t*$.1*$.3 + 1/t^2*$.3^2)
]
C is given as a y 2= quartic curve in P(1, 2, 1) over Q(t).
We transform to a Weierstrass form using pts[4] as a base point
> E,Emp := EllipticCurve(C,pts[4]);
> E;
Elliptic Curve defined by y^2 + (-2*t^7 - 2*t^6 - 6*t^5 - 2*t^4)/(t^4 + 6*t^3 +
4*t^2 + 2*t + 1)*x*y + (-4*t^19 - 20*t^18 - 52*t^17 - 92*t^16 - 108*t^15 -
76*t^14 - 28*t^13 - 4*t^12)/(t^12 + 18*t^11 + 120*t^10 + 366*t^9 + 555*t^8 +
588*t^7 + 496*t^6 + 324*t^5 + 171*t^4 + 74*t^3 + 24*t^2 + 6*t + 1)*y = x^3 +
2*t^8/(t^4 + 6*t^3 + 4*t^2 + 2*t + 1)*x^2 + (-4*t^24 - 32*t^23 - 112*t^22 -
224*t^21 - 280*t^20 - 224*t^19 - 112*t^18 - 32*t^17 - 4*t^16)/(t^16 +
24*t^15 + 232*t^14 + 1160*t^13 + 3268*t^12 + 5640*t^11 + 7096*t^10 +
7128*t^9 + 5830*t^8 + 3976*t^7 + 2296*t^6 + 1112*t^5 + 452*t^4 + 152*t^3 +
40*t^2 + 8*t + 1)*x + (-8*t^32 - 64*t^31 - 224*t^30 - 448*t^29 - 560*t^28 -
448*t^27 - 224*t^26 - 64*t^25 - 8*t^24)/(t^20 + 30*t^19 + 380*t^18 +
2650*t^17 + 11205*t^16 + 30376*t^15 + 56560*t^14 + 79960*t^13 + 91530*t^12 +
87300*t^11 + 70824*t^10 + 49580*t^9 + 30090*t^8 + 15880*t^7 + 7280*t^6 +
2872*t^5 + 965*t^4 + 270*t^3 + 60*t^2 + 10*t + 1) over Univariate rational
function field over Rational Field
We can now find minimising transformations to greatly simplify this Weierstrass equation
but we will finish here.
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