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We work implicitly on the resolution of a particular, usually singular,
model of a curve C that has been referred to as tilde(C).
To handle prime divisors at the singularities
properly we use the notion of places of the curve and devote
the first section below to their construction. Places include
the case of prime divisors at points of degree greater than 1, that is,
points whose coordinates do not lie in the base ring. Following this are
sections on constructions of divisors and their basic arithmetic.
It may seem a little strange at first to distinguish places from other
divisors, but in practice when doing arithmetic the difference is not
noticeable.
The most important invariant associated with a divisor D is its
Riemann--Roch space, often denoted by L(D) or H0(D).
This is a vector subspace of the function field of a curve.
Its computation has an enormous number of applications.
One we give as an illustration is the computation of the
canonical embedding of C (in the case that C is non-hyperelliptic).
This section, together with Section Linear Equivalence of Divisors, is devoted to Magma's
facility to work with divisors on curves.
A divisor on a nonsingular curve
C is a formal sum of points
Σ nipi, where each pi∈C and each ni is an integer.
It is clear that divisors form a group under componentwise addition.
This group, and various variants of it, is an important invariant of
the curve C.
For a singular (but still irreducible) curve, one can make a similar
definition in which points are replaced by places, a notion
that makes precise the vague idea that singularities arise in different
ways, both by "gluing" nonsingular points together and by "pinching"
tangent vectors at a particular point.
The sections of this chapter contain very brief discussions
which will help to orient the user who already knows something about
divisors. For a more complete account one should consult an advanced textbook
such as [Har77] Chapter II 6, or [Ful69] Chapter 8.
We start with a description of prime divisors or places.
Then we show how to create divisor groups and divisors in them and go on
to explain the basic arithmetic of divisors.
All of the functions here are based on equivalent functions which apply
to algebraic function fields. All calculations are done in that context
using the functionality of Chapter ALGEBRAIC FUNCTION FIELDS.
A translation is made in the
background for all functions described here. For information about
function fields, their representations and translations see
Section Function Fields. The translations should not be explicitly needed
in the ordinary course of working with curves and their divisors.
Note, however, that divisors of function fields have 4 possible representations,
each having their own advantages and disadvantages. These representations of
the FunctionFieldDivisor of a curve divisor are stated in Section Divisors.
The main concern when working with divisors is with questions of linear
equivalence. In this respect, one should at least have in mind
that the most substantial calculations really make sense on projective
curves. However, the functions below allow constructions using an affine curve
and its points which are reinterpreted in terms of the projective closure.
Section Linear Equivalence of Divisors describes those functions which are related to
linear equivalence of divisors.
A divisor is called principal if it is linearly equivalent to
the zero divisor.
In the case of a curve defined over a finite field, we can compute the
class group, that is, the group of divisors modulo principal divisors.
In fact, we compute a finitely presented abelian group isomorphic to the
class group and a map which transforms elements between the divisor group
and the class group.
For any divisor on any curve, there are functions to compute Riemann-Roch
spaces and a host of related entities.
A place is a point of the resolution of a curve or, equivalently, a
valuation of the function field of a curve.
It is characterised by a point on the curve and, if that point is
singular, some data that distinguishes a single resolved point above it.
The set of places of the curve C.
The curve on which the places in the set of places P lie.
This will be a projective curve.
P ne Q : PlcCrv, PlcCrv -> BoolElt
Returns true if and only if the sets of places P and Q are (not)
equal.
There are
some explicit methods for constructing places.
As we show later, places arise implicitly as components of divisors
and this is a common way of getting hold of them.
p1 + p2 : PlcCrvElt, PlcCrvElt -> DivCrvElt
- p1 : PlcCrvElt -> DivCrvElt
p1 - p2 : PlcCrvElt, PlcCrvElt -> DivCrvElt
k * p : RngIntElt, PlcCrvElt -> DivCrvElt
p div k : PlcCrvElt, RngIntElt -> DivCrvElt
p mod k : PlcCrvElt, RngIntElt -> DivCrvElt
Quotrem(p1, k) : PlcCrvElt, RngIntElt -> DivCrvElt, DivCrvElt
P in S : Any, PlcCrv -> BoolElt
P notin S : Any, PlcCrv -> BoolElt
A sequence of all places of degree m on C, a curve defined over
a finite field, where m is a positive integer.
RandomPlace(C, m) : Crv[FldFin], RngIntElt -> BoolElt,PlcCrvElt
If a place of degree m exists on the curve C over a finite field,
then this returns true
and a single such place, where m is a positive integer.
Otherwise it returns false.
The place corresponding to the nonsingular point p of some curve.
A sequence containing the places corresponding to the point p of some curve.
Return the place of the curve C defined by the ideal I of the coordinate
ring of C.
The Weierstrass places of the curve C, which are the Weierstrass places of
the zero divisor of C.
The place of a curve C determined by the sequence of elements of
the sequence Q, which should all be contained in the function
field of C.
Given a place P of a curve C return
the prime ideal of the coordinate ring of the ambient of C of coordinate
functions which vanish at the place P.
Return two elements of the function field of P which determine the place P.
The sequence containing these two elements can be used as input to
Place to create a place equal to P.
In this example we show how to use rational functions to create
a place on a curve. This is not directly a very geometric operation.
However, it is very useful since a pair of rational functions which
determine a place form a very concise description of that place.
Thus one often uses this method to recreate a given place on a curve
in one step.
We illustrate that by first finding a place and later recreating it
from rational functions.
> P2<x,y,z> := ProjectivePlane(FiniteField(17));
> C := Curve(P2,x^5 + x^2*y^3 - z^5);
> p := C ! [1,0,1];
> Places(p);
[
Place at (1 : 0 : 1)
]
> P := $1[1];
> P:Minimal;
Place at (1 : 0 : 1)
> TwoGenerators(P);
$.1 + 16
$.1^2*$.2
So now we have a place and some rational functions.
As usual, these functions are elements of the function field of the
curve C, so to be able to read them conveniently we assign names
to that function field.
> FC<a,b> := FunctionField(C);
> TwoGenerators(P);
a + 16
a^2*b
We can use this sequence to recreate the place P.
The real convenience of the first line of code below is that it could be in a
different Magma session in which only the curve C has been defined
together with the names a, b of its function field.
(You can confirm this by running the four relevant lines in a separate
Magma session.)
The final line is simply to confirm that we really have created the
same place P as we started with.
> Place([a+16,a^2*b]);
Place at (1 : 0 : 1)
> Place([a+16,a^2*b]) eq P;
true
> Place([a+16,a*b,a^2*b^2]) eq P;
true
Notice that in the final line we create exactly the same place using
more than the two elements that TwoGenerators
returned.
Poles(f) : FldFunFracSchElt[Crv] -> SeqEnum[PlcCrvElt]
A sequence of places of the curve C containing the zeros or poles of
f where f is an element of the function field of C.
Poles(C, f) : Crv, RngElt -> [PlcCrvElt]
A sequence of places of the curve C containing the zeros or poles of
f where f is some function on C, i.e. f is coercible into the
function field of C.
CommonZeros(C, L) : Crv, [FldFunFracSchElt] -> [PlcCrvElt]
Given a sequence L of elements of the function field of some curve C
or a curve C and a sequence L of functions on C,
return the zeros which are common to all elements of L as places of C.
The second argument to the intrinsic Poles(C,f) can be either an
element of the function field of the curve C or an element of the
function field of its ambient space.
> A<x,y> := AffineSpace(GF(2),2);
> C := Curve(A,x^8*y^3 + x^3*y^2 + y + 1);
> FA<X,Y> := FieldOfFractions(CoordinateRing(A));
> FC<a,b> := FunctionField(C);
> Poles(C,X/Y);
[
Place at (1 : 0 : 0)
]
> Poles(C,a/b);
[
Place at (1 : 0 : 0)
]
> $1 eq $2;
true
In particular, we did not use the ambient coordinates x, y in
the arguments.
The projective curve on which the place P lies.
A representative point of the projective model of the curve
underlying the place.
P ne Q : PlcCrvElt, PlcCrvElt -> BoolElt
Returns true if and only if the two places are (not) the same.
The order of vanishing of the function f on the curve C
at the place P of C.
A negative value indicates a pole at P.
The valuation of the function field centred at the place P.
This is a map from the function field to the integers.
The valuation of the differential a at the place P.
The residue of the differential a at the degree 1 place P.
A function on the curve of the place P
having valuation 1 at P.
IsWeierstrassPlace(D, P) : DivCrvElt, PlcCrvElt -> BoolElt
Returns true if and only if the place P (which must have degree 1)
is a Weierstrass place (of divisor D if given).
The residue class field of the place P.
Evaluate the element a in the function field of the curve of the place P,
returning an element of the residue class field of P.
Lift(i, P) : Infty, PlcCrvElt -> FldFunFracSchElt
Lift the element a of the residue class field of the place P
(including infinity) to a function on the curve of P.
The degree of the place P of a curve over the base ring of the curve of P.
GapNumbers(P) : PlcCrvElt -> [RngIntElt]
The gap numbers of the curve C at the degree 1 place P.
Parametrization(C, p) : Crv, PlcCrvElt -> MapSch
Parametrization(C, p, P) : Crv, PlcCrvElt, Crv -> MapSch
Returns a map parametrizing the rational curve C at the rational point p
or place p of degree 1. If p is a singular point on C, then it must
have a unique place above it of degree 1.
If P is also given it must be the projective line
of dimension 1 as a curve (ie of type Crv) and then the domain of the
map will be P.
A curve has an associated
group of divisors which is simply the formal abelian group
generated by the places of the curve C.
Divisors are elements of this group.
In other words, divisors are expressions of the form Σ ni pi
where ni are integers, pi are places of the curve which one
usually assumes to be distinct.
Each term n p is called a summand of d or the component
of d corresponding to p.
The integer n will be called the coefficient or multiplicity of
the summand.
Divisors are created by specifying the curve for which they will a divisor
(if that is not clear) and then giving sufficient data to identify precisely
the divisor in question. This data could be a list of points or places together
with integers, but there are many other creation methods.
Divisors are printed as a linear combination of the places
which support them, if such a combination is known.
However, giving this information can be extremely expensive so often
printing simply refers to the curve.
The group of divisors of the curve C.
This curve may be either an affine or projective curve.
The curve that was used to create the divisor group Div, or its projective
model.
Div1 ne Div2 : DivCrv, DivCrv -> BoolElt
Returns true if and only the divisor groups Div1 and Div2 are (not)
equal.
The divisor group in which the divisor D lies.
The (projective) curve on which the divisor D lies.
Id(D) : DivCrv -> DivCrvElt
D ! 0 : DivCrv,RngIntElt -> DivCrvElt
The zero divisor of the divisor group D of a curve.
Div ! p : DivCrv, Pt -> DivCrvElt
Divisor(p) : PlcCrvElt -> DivCrvElt
Divisor(p) : Pt -> DivCrvElt
The prime divisor in the divisor group Div of the curve C
corresponding to the place or nonsingular point p of some curve C.
Divisor(C, S) : Crv, SeqEnum -> DivCrvElt
Divisor(S) : [<PlcCrvElt, RngIntElt>] -> DivCrvElt
The divisor of the curve
C or the curve of the divisor group D
described by the factorization sequence S.
The sequence should contain tuples of the form <place,integer>.
One can use the intrinsic Divisor(S) to reconstruct a divisor
from concise data related to it.
The intrinsics Support and CanonicalDivisor are
defined below.
> P<x,y,z> := ProjectivePlane(FiniteField(17));
> C := Curve(P,x^5 + x^2*y^3 - z^5);
> F<a,b> := FunctionField(C);
> DivisorGroup(C);
Group of divisors of Curve over GF(17) defined by
x^5 + x^2*y^3 + 16*z^5
> K := CanonicalDivisor(C);
> supp, exps := Support(K);
> Q := [ < RationalFunctions(supp[i]),exps[i] > : i in [1..#supp] ];
> Q;
[
<[ a, 2*a^2*b^2 + 4*a*b ], 2>,
<[ a + 16, a^2*b ], 2>,
<[ a^4 + a^3 + a^2 + a + 1, a^2*b ], 2>,
<[ 1/a, (a + b)/a ], -2>,
<[ 1/a, (a^2 + 16*a*b + b^2)/a^2 ], -2>
]
> K;
Divisor 2*Place at (0 : 1 : 0) + 2*Place at (1 : 0 : 1) + 2*Place at
($.1 : 0 : 1) - 2*Place at (16 : 1 : 0) - 2*Place at (16*$.1 + 1 : 1 : 0)
Now we can reconstruct the divisor K using this sequence.
> Divisor([<Place(f[1]), f[2]> : f in Q]);
Divisor on Curve over GF(17) defined by
x^5 + x^2*y^3 + 16*z^5
> K eq $1;
true
PrincipalDivisor(D, f) : DivCrv, RngElt -> DivCrvElt
PrincipalDivisor(f) : FldFunFracSchElt[Crv] -> DivCrvElt
Divisor(C, f) : Crv, RngElt -> DivCrvElt
Divisor(D, f) : DivCrv, RngElt -> DivCrvElt
Divisor(f) : FldFunFracSchElt[Crv] -> DivCrvElt
The principal divisor corresponding to f, that is, the divisor of the curve
C of zeros and poles of the function field element f, where C is the
curve of the divisor group D if given.
The divisor of the curve C corresponding to the differential a of C.
Divisor(D, X) : DivCrv, Sch -> DivCrvElt
The divisor described by intersection of the curve C with the scheme X,
(where C is the curve of the divisor group D if the group is given instead
of the curve).
Divisor(D, p, q) : DivCrv, Pt, Pt -> DivCrvElt
The principal divisor corresponding to the line through points
p and q (the tangent
line to the curve C there if they coincide) where C is the curve of the
divisor group D if given.
Divisor(D, I) : DivCrv, RngMPol -> DivCrvElt
The divisor of the curve C defined by the ideal I of the ambient
coordinate ring where C is the curve of the divisor group D if given.
The decomposition sequence of D as a sequence of tuples
of the form <place, multiplicity> characterizing the
divisor D.
The sequence of places in the support of D, followed by
their sequence of multiplicities in D.
A curve, its divisor group and some divisors are created.
> P<x,y,z> := ProjectiveSpace(GF(7), 2);
> C := Curve(P,y^2*z - x^3 - x*z^2 - z^3);
> F := FunctionField(C);
> D := Divisor(C,F!(x/z));
> D;
Divisor of Curve over GF(7) defined by
6*x^3 + 6*x*z^2 + y^2*z + 6*z^3
> Decomposition(D);
[
<Place at (0 : 1 : 0), -2>,
<Place at (0 : 6 : 1), 1>,
<Place at (0 : 1 : 1), 1>
]
> D;
Divisor -2*Place at (0 : 1 : 0) + 1*Place at (0 : 6 : 1) + 1*Place at (0 : 1 : 1)
The support of a
divisor is written in the style of the factorization of other objects
in Magma; compare with the factorization of the integer 84 below.
This expression is called the factorization of a divisor and provides
a method of accessing the individual components.
> Factorization(84);
[ <2, 2>, <3, 1>, <7, 1> ]
> Support(D)[2];
Place at (0 : 6 : 1)
One can access the point underlying a given place.
> p := Support(D)[1];
> p;
Place at (0 : 1 : 0)
> RepresentativePoint(p);
(0 : 1 : 0)
A divisor in the canonical divisor class of the curve C.
The ramification divisor of the curve C.
D + E : DivCrvElt,DivCrvElt -> DivCrvElt
D + E : DivCrvElt, PlcCrvElt -> DivCrvElt
D + E : PlcCrvElt, DivCrvElt -> DivCrvElt
D + E : PlcCrvElt, PlcCrvElt -> DivCrvElt
- D : DivCrvElt -> DivCrvElt
- D : PlcCrvElt -> DivCrvElt
D - E : DivCrvElt,DivCrvElt -> DivCrvElt
D - E : DivCrvElt,PlcCrvElt -> DivCrvElt
D - E : PlcCrvElt,DivCrvElt -> DivCrvElt
D - E : PlcCrvElt,PlcCrvElt -> DivCrvElt
n * D : RngIntElt,DivCrvElt -> DivCrvElt
n * D : RngIntElt,PlcCrvElt -> DivCrvElt
D div n : DivCrvElt,RngIntElt -> DivCrvElt
D div n : PlcCrvElt,RngIntElt -> DivCrvElt
D mod n : DivCrvElt,RngIntElt -> DivCrvElt
D mod n : PlcCrvElt,RngIntElt -> DivCrvElt
Basic formal arithmetic of divisors; D and E are divisors (or places)
and n is an integer.
d in D : Any, DivCrv -> BoolElt
d notin D : Any, DivCrv -> BoolElt
D lt E : DivCrvElt, DivCrvElt -> BoolElt
D le E : DivCrvElt, DivCrvElt -> BoolElt
D gt E : DivCrvElt, DivCrvElt -> BoolElt
D ge E : DivCrvElt, DivCrvElt -> BoolElt
The quotient and remainder on dividing the divisor D by the integer n.
The sum of coefficients of the divisor D multiplied by the degrees of the
places of the corresponding components.
IsPositive(D) : DivCrvElt -> BoolElt
Returns true if and only if all coefficients of the divisor D are nonnegative.
Denominator(D) : DivCrvElt -> DivCrvElt
The numerator, respectively denominator, of the divisor D of a curve.
The numerator and denominator are both positive divisors such that
D is the difference between numerator and denominator.
The minimal effective divisors A and B such that the equality
of divisors D = A - B holds.
The sign decomposition of the previous example is calculated.
> P<x,y,z> := ProjectiveSpace(GF(7),2);
> C := Curve(P,y^2*z - x^3 - x*z^2 - z^3);
> F := FunctionField(C);
> D := Divisor(C,F!(x/z));
> Decomposition(D);
[
<Place at (0 : 1 : 0), -2>,
<Place at (0 : 6 : 1), 1>,
<Place at (0 : 1 : 1), 1>
]
> Decomposition(D div 2);
[
<Place at (0 : 1 : 0), -1>
]
> A, B := SignDecomposition(D);
> IsEffective(A);
true
> IsEffective(B);
true
> A - B eq D;
true
D ne E : DivCrvElt, DivCrvElt -> BoolElt
Returns true if and only the divisors D and E are (not) equal.
Note that this means equality in the group of divisors and is not the
same as being linearly equivalent.
Returns true if and only the divisors D and E are linearly equivalent.
Returns true if and only if all coefficients of the divisor D are zero.
Returns true if and only if the divisor D is the divisor of a differential, in which
case also return a differential realising this.
Gcd(D1, D2) : DivCrvElt, DivCrvElt -> DivCrvElt
GreatestCommonDivisor(D1, D2) : DivCrvElt, DivCrvElt -> DivCrvElt
The greatest common divisor of the divisors D1 and D2. This is the divisor
supported on the places common to the support of both divisors with
coefficients the minimum of those occurring in D1 and D2.
Lcm(D1, D2) : DivCrvElt, DivCrvElt -> DivCrvElt
LeastCommonMultiple(D1, D2) : DivCrvElt, DivCrvElt -> DivCrvElt
The least common multiple of the divisors D1 and D2. This is the divisor
supported on all the places in the supports of D1 and D2 with coefficients
the maximum of those occurring in the input divisors.
We find that a given divisor is actually a canonical divisor.
> P<x,y,z> := ProjectiveSpace(GF(7),2);
> C := Curve(P,y^2*z - x^3 - x*z^2 - z^3);
> F := FunctionField(C);
> D := Divisor(C,F!(x/z));
> IsCanonical(D);
true (($.1) ^ 1 * ($.1^3 + $.1 + 1) ^ -1 * ($.1) ^ 1) d($.1)
The printing of the differential in the last line above is not very
clear since names have not been assigned, but nonetheless, it can
be used as an argument to intrinsics.
> _, dd := IsCanonical(D);
> Valuation(dd,Support(D)[1]);
-2
Given a divisor D of a curve C, returns the ideal of the coordinate
ring of the ambient of C of coordinate functions
which cuts out D. The ideal D must be effective.
Given a divisor D of a curve C, returns the cluster (zero-dimensional
scheme) which gives D as a subscheme of C. This is the scheme
defined by Ideal(D). The ideal D must be effective.
Valuation(D,P) : DivCrvElt, PlcCrvElt -> DivCrvElt
The coefficient of the divisor summand of the divisor D corresponding to the
point p or place P.
ComplementaryDivisor(D,P) : DivCrvElt,PlcCrvElt -> DivCrvElt
The divisor after removing from the divisor
D the component corresponding to the point p or place P.
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