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Subsections
K3Database(n) : RngIntElt -> SeqEnum
K3Database(t) : MonStgElt -> SeqEnum
A sequence of all K3 surfaces in the K3 database, or only those of codimension
at most n.
Magma Version 2.8 contains K3 surfaces of codimensions 1,2,3 and 4.
If a string is included as the argument, it will be used as the
print value for the variable in the Hilbert numerator of K3 surfaces in
the database. This can be set in any case by assigning a variable
to the univariate polynomial ring over the rationals.
Attach the numbering systems of Alt{i}nok, Fletcher and Reid's
printed versions of the database
in [IF00] and [Alt98] to the elements of DB.
Note that, for historical reasons, Alt{i}nok's numbering in codimension 4
is as follows: there are 142 K3 surfaces numbered from -24 to 121
omitting the numbers 0, 76, 81, 107.
K3SurfaceFromWeights(DB,W) : SeqEnum,SeqEnum -> VSrfK3
Return a sequence of all K3s in the sequence DB having weights sequence W.
The singular version of the intrinsic returns the unique such K3 if it exists.
K3SurfaceFromBasket(DB,B) : SeqEnum,SeqEnum -> SeqEnum
Return a sequence of all K3s in the sequence DB having basket B.
The singular version of the intrinsic returns the unique such K3 if it exists.
Return the K3 surface in the sequence DB of codimension c with number
n in the Alt{i}nok--Fletcher--Reid versions of these lists.
(Alt{i}nok's list of exceptional K3 surfaces --- those without a Type 1
projection --- have been given negative numbers nr: Alt{i}nok's exceptional
number 5 is our AFR number -5.)
Return the K3 surface number i in the sequence DB.
Return true if and only if the baskets and weights of the K3
surfaces X and Y are
identical.
Identifier(X) : VSrfK3 -> RngIntElt
The arbitrary integer assigned to the K3 surface X for the duration of the
current session.
This number is used only for easy identification and can change from session
to session.
An ordered sequence of positive integer weights of the projective space in
which the K3 surface X lies.
There is no guarantee that there really is a model of this X
in the proposed weighted projective space, although it is true for
all K3s in the database which have a positive AFR number.
The basket of singularities of the K3 surface X. Each singularity of the form
oneover(r)(a, r - a) is listed in the basket as [r, a] with r, a coprime
and r >= 2a.
The indices of the singularities of the basket of the K3 surface X.
The codimension of the K3 surface X if embedded in projective space with
weights equal to the weights of X.
An integer, at least -1,
that is one less that the number of 1s in the weights of the K3 surface X.
In the classical theory, this is the genus of a polarising curve on a K3
of the family.
The Hilbert series of the K3 surface X with the weights of X
"multiplied up".
That is, if X is embedded in weighted projective space with weights W
and if P(t) is the Hilbert series of X together with its polarising
divisor then the Hilbert numerator is P(t)⋅Pi(1 - t^w), where
the product is taken over all w in W.
See [Rei00] Section 3.2 for a tutorial in interpreting Hilbert
numerators.
The Hilbert series of the K3 surface X with the weights of W
"multiplied up".
That is, P(t)⋅Pi(1 - t^w) where the product is taken over w in W
and P(t) is the Hilbert series of X.
This is like the intrinsic HilbertNumerator, although in this
case the weights are supplied by the user as the argument W.
The degree of the K3 surface X computed from its genus and basket.
The Hilbert series of the K3 surface X computed by the Riemann--Roch formula using
the genus and basket of X.
A sequence containing the first n coefficients of the Hilbert series
of the K3 surface X when expressed as a power series.
ReidNumber(X) : VSrfK3 -> RngIntElt
FletcherNumber(X) : VSrfK3 -> RngIntElt
AltinokNumber(X) : VSrfK3 -> RngIntElt
The number of the K3 surface X in the printed tables of Alt{i}nok, Fletcher
and Reid. See [IF00] and [Alt98].
Those K3s in Alt{i}nok's exceptional list have been given negative AFR numbers.
EnriquesForm(X) : VSrfK3 -> SeqEnum
NoetherForm(X) : VSrfK3 -> SeqEnum
A sequence containing the degrees of coordinates having particular properties
for the K3 surface X, if they have been computed.
The Nöther form refers to Nöther coordinates, the Enriques form
to the coordinates of a generic projection to codimension 1.
The Hilbert form refers to the final choice of degrees determined by the
database generating routines before local polarisation degrees were
taken into account.
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