Codes, Horn's problem and Gromov-Witten invariants

0.4 MB·English

by  Alberto Besana

CODES, HORN’S PROBLEM AND GROMOV-WITTEN
INVARIANTS
ALBERTO BESANA AND CRISTINA MART´INEZ
Abstract. We study the Horn problem in the context of algebraic
3 codes on a smooth projective curve defined over a finite field, reducing
1 the problem to the representation theory of the special linear group
0 SL(2,F ). WecharacterizethecoefficientsthatappearintheKronecker
q
2 product of symmetric functions in terms of Gromov-Witten invariants
n of the Hilbert scheme of points in the plane. In addition we classify all
a the algebraic codes defined over the rational normal curve.
J

8
1. Introduction
]
O
Finite fields have a remarkable property that finite dimensional vector
C
spaces over them are naturally endowed with a canonical and compatible
.
h field structure. This leads to some interesting notions where the field struc-
t ture and the linear structure are intertwined. Let denote by F the Galois
a p
m field of p elements. Any other field F of characteristic p contains a copy of
F . Any V = F field extension of F is a F vector space of dimension n
[ p pn p p

andan(n−1)−dimensionalprojectivespacePG(n−1,p). Onecanconsider
1
field extensions F of F as q varies through powers of the prime p.
v q p
2 Let us now consider the finite field F with q elements. When the finite
q
5 field F is generated as a vector space over F by a unique element α ∈ F ,
qm q q
6
then the set {1,α,α2,...αm−1} forms a basis of F . In particular F =
1 qm qmn
1. Fq(α). If W is generated by v1,...,vm as an Fq vector space then αW is
generated by mn elements: αv ,...,αv , ...,αn−1v ,...,αn−1v . There
0 1 m 1 m
3 is a Fq−basis B of Fqmn such that each element of B generates Fqmn over Fq.
1 Let V be an n+1 dimensional vector space over the field F , we denote
q
:
v by PG(n,q) or P(V) the n−dimensional projective space over it. The set of
Xi all subspaces of dimension r is called the Grassmannian and it is denoted
r by GFq(r,n) or by PGr(n,q). The dual of an r−space in PG(n,q) is an
a (n−r−1)−space.
Consider the Fq-rational points of GFq(r,n) as a projective system, we
obtain a q−ary linear code, called the Grassmann code, which we denote
C(r,n). The lenght l and the dimension k of C(r,n) are given by the q
(cid:20) (cid:21)
binomial coefficient l = n = (qn+1−1)(qn+1−q)...(qn+1−qr), and k = (cid:0)n(cid:1),
r (qr+1−1)(qr+1−q)...(qr+1−qr) r
q
respectively.
There is a right action of the general linear group GL(n,Fq) on GFq(k,n):
2000 Mathematics Subject Classification. 05E10 (primary) ; 05A15 (secondary) .
Key words and phrases. Algebraic code, symmetric group, partitions.
1
2 ALBERTOBESANAANDCRISTINAMART´INEZ
(1) GFq(k,n)×GL(n,Fq) → GFq(k,n)
(U,A) → UA.
Observe that the action is defined independent of the choice of the repre-
sentation matrix U ∈ Fk×n.
q
Definition 1.1. Let U ∈ GFq(k,n) and G < GL(n,Fq) a subgroup, then
C = {UA| A ∈ G} is an orbit in GF (k,n) of the induced action.
q
In order to classify all the orbits we need to classify all the conjugacy
classes of subgroups of GL(n,F ). In [BM2] we studied cyclic coverings of
q
the projective line that correspond to orbits defined by a cyclic subgroup,
that is a subgroup in GL(n,F ) containing a cyclic subgroup Z for some
q p
prime number. In particular, we showed that any irreducible cyclic cover
can be given by a prime ideal
(ym−(x−a )d1...(x−a )dn) ⊂ F [x,y].
1 n q
This ideal defines an affine curve in A2(F ) which has singularities, if there
q
are some d > 1 for some 1 ≤ k ≤ n. But there exists an unique smooth
k
projective curve birationally equivalent to this affine curve obtained by ho-
mogenizationofthepolinomial. Herewestudytheconnectionbetweenideal
sheaves on F [x,y] and its numerical invariants together with the combina-
q
torics of partitions of n and the representation theory of the special linear
group SL(F ,n).
q
Any cyclic cover of P1 which is simply ramified corresponds to an un-
ordered tuple of n points on P1. We will consider more generally, configura-
tions of n points in d−dimensional projective space PdF ) which generically
q
lie on a rational normal curve and we study the algebraic codes defined on
it.
From now F will be a field with q = pn elements and C a non-singular,
q
projective, irreducible curve defined over F with q elements.
q
Conventions. For d a positive integer, α = (α ,...,α ) is a partition of
1 m
d into m parts if the α are positive and non-decreasing. We set l(α) = m
i
for the length of α, that is the number of cycles in α, and l for the length
i
of α . The notation (a ,...,a ) stands for a permutation in S that sends
i 1 k d
a to a . A curve is an integral scheme of dimension 1, proper over k.
i i+1
A homogeneous symmetric function of degree n over a commutative ring R
(with identity) is a formal power series f(x) = (cid:80) c xα, where α ranges
α α
over all weak compositions of α = (α ,...,α ) of n, c ∈ R and xα stands
1 n α
for the monomial xα1 ·xα2···xαn. We write PGL(2,k) = GL(2,k)/k∗, and
elements of PGL(2,k) will be represented by equivalence clases of matrices
(cid:18) (cid:19)
a b
, with ad−bc (cid:54)= 0.
c d
2. Algebraic codes over finite fields
Let X be a smooth projective curve defined over a finite field F with
q
q elements. The classical algebraic-geometric (AG) code due to Goppa is
defined by evaluating rational functions associated to a divisor D at a finite
CODES, HORN’S PROBLEM AND GROMOV-WITTEN INVARIANTS 3
set of F −rational points. From another point of view, we are considering
q
the evaluation of sections of the corresponding line bundle O (D) on X.
X
Namely, let {P ,...,P } be a configuration of distinct F −rational points
1 n q
of X, the usual algebraic-geometric code is defined to be the image of the
evaluation map:
(2) ϕ : L(D) → Fn
D q
f (cid:55)→ (f(P ),...,f(P )).
1 n
Usingthisdefinition,thenotionofAGcodesiseasilygeneralizedforvarieties
of higher dimension.
Let E be a vector bundle of rank r on X defined over F . One can define
q
the code C(X,P,E) to be the image of the evaluation map:
n
(3) ϕ : H0(X,E) → (cid:77)E ∼= Fn
E Pi Q
i=1
s (cid:55)→ (s(P ),...,s(P )).
1 n
ObservethatC(X,P,E)isanF −linearsubspaceofFn andthusapoint
q qr
of the Grassmannian G (F ).
r,n q
The representation theory of the special linear group SL(n,F ) can be
q
viewed as a form of Gale duality first proven by Goppa in the context of
algebraic coding theory.
One can study linear systems defined over a finite field. A convolutional
code is essentially a linear system defined over a finite field. Convolutional
codes have been studied by graph theoretic methods. In doing so, convo-
lutional codes can be viewed as submodules of Rn where R := F[z] is a
polynomial ring (see [MTR]). The set of convolutional codes of a fixed de-
greeisparametrizedbytheGrothendieckQuotscheme. Ifthedegreeiszero,
these schemes describe a Grassmann variety.
3. Convolutional codes
LetO bethestructuresheafofthecurveX definedoverafieldk andlet
X
K be its field of rational functions, considered as a constant O −module.
X
Following[BGL],wedefineadivisorofrankranddegreedor(r,d)divisoras
a coherent sub O -module of Kr = K⊕r, having rank r and degree d. This
X
set can be identified with the set of rational points of an algebraic variety
Divr,d whichmaybedescribedasfollows. Foranyeffectiveordinarydivisor
X/k
D, set:
Divr,d (D) = {E ∈ Divr,d ;E ⊂ O (D)r},
X/k X,k X
where O (D) is considered as a submodule of Kr.
X
The space of all matrix divisors of rank r and degree d can be identi-
fied with the set of rational points of Quotm parametrizing torsion
OX(D)r/X/k
quotientsofO (D)r andhavingdegreem = r·degD−d. Itisasmoothpro-
X
jective irreducible variety. Tensoring by O (−D) defines an isomorphism
X
between Q (D) = Quotm and Quotm .
r,d OX(D)r/X/k OXr /X/k
4 ALBERTOBESANAANDCRISTINAMART´INEZ
Since the whole construction is algebraic, it can be performed over any
complete valued field, for example, a p−adic field or the ring R = C{x}
of convergent power series. The matrix code A can be diagonalized by ele-
mentary row and column operations with diagonal entries xα1,xα2,...,xαn,
for unique non-negative integers α ≥ ... ≥ α , where x is a uniformizing
1 n
parameter in R.
These matrices are in correspondence with endomorphisms of Rn, with
cokernels being torsion modules with at most n generators. Such a module
is isomorphic to a direct sum
R/xα1R⊕R/xα2R⊕...⊕R/xαnR, α ≥ ... ≥ α .
1 n
The set (α ,...,α ) of invariant factors of A defines a partition α of size
1 n
d = |λ|. Reciprocally, when R = C{x} is the ring of convergent power
series, any partition λ defines a rank one torsion-free sheaf on C by setting
I = (xλ1,xλ2,xλ3,...,xλn). In particular, the ideal sheaf corresponding to
λ
n times
the identity partition (1)n, defines a maximal ideal I = (x, (cid:122).(cid:125).(cid:124).(cid:123) ,x) in
(1)n
C[x].
Question. Which partitions α,β,γ can be the invariant factors of ma-
trices A,B, and C if C = A·B?
In the case of convergent power series, this problem was proposed by
I. Gohberg and M. A. Kaashoek. Denoting the cokernels of A,B and C by
A,B and C respectively, one has a short exact sequence:
0 → A → B → C → 0,
∼
i.e. B is a submodule of C with C/B = A, then such an exact sequence
corresponds to matrices A,B and C with A·B = C.
If we specialize C to be the identity matrix I, by the correspondence be-
tweenpartitionsandidealsheavesabove,theinvariantfactorsoftheidentity
matrix are defined by the partition (1)n, then the question becomes: Which
partitions α,β can be the invariant factors of matrices A, B if A·B = I?
An example with algebraic codes
Definition 3.1. Let D be an effective divisor with disjoint support defined
over a smooth projective curve X and C and C be the corresponding codes
1 2
obtained evaluating non-constant rational functions f(x) and g(x) with non
common roots on X over the support of the divisor D.
Then we define the quotient code of C and C to be the code associated
1 2
to the quotient rational function ϕ = f/g.
Since f and g take the value ∞, they are defined by non constant poly-
nomials f(x) and g(x) in F [x]. The degree of ϕ is defined to be deg(ϕ) =
q
max{deg(f),deg(g)}.
As ϕ is a finite morphism, one may associate to each rational point x ∈
X(F ) a local degree or multiplicty m (x) defined as:
q ϕ
m (x) = ord ψ(z),
ϕ z=0
whereψ = σ ◦ϕ◦σ ,y = ϕ(x),andσ ,σ ∈ PGL(2,F )suchthatσ (0) = x
2 1 1 2 q 1
and σ (y) = 0.
2
CODES, HORN’S PROBLEM AND GROMOV-WITTEN INVARIANTS 5
To each non-constant rational function ϕ over X, one can associate a
matrix A with entries in the ring F [x]. Namely, let us call f := f(x)
q 0
and call f the divisor polynomial g(x), and f the remainder polynomial,
1 2
then by repeated use of the Euclid’s algorithm, we construct a sequence
of polynomials f ,f ,...,f , and quotients q ,...q , K ≤ n . Then the
0 1 k 1 k
quotientmatrixAisdefinedtobethediagonalmatrixwithentriesq ,...,q
1 k
correspondingtothecontinuedfractionexpansionoftherationalfunctionϕ.
Here we include a SAGE code [S] which implements the algorithm.
def euclid(f, g):
r = f % g
q = f // g
while r.degree() >= 0:
yield q
f = g
g = r
r = f % g
q = f // g
Let λ be the partition of the integer k, defining the degree multiplicities
i
ofthepolynomialq . ThentheHornproblemappliedtothissituationreads:
i
Which partitions α,β,γ can be the degree multiplicities of polynomials
q ,q and q such that the corresponding diagonal matrices A,B, and C
A B C
satisfy C = A·B?
As in [BM1] and [BM2], where we considered a variant of the Horn prob-
lem in the context of cyclic coverings of the projective line defined over an
arbitrary field k, the problem is reduced to study the representation theory
of the special linear group SL(n,F ).
q
Representation theory of SL(n,F )
q
To a partition α = (α ,...,α ) is associated a Young diagram. The
1 k
diagram of α is an array of boxes, lined up at the left, with α boxes in the
i
ith row, with rows arranged from top to botton. For example,
is the Young diagram of the partition α = (5,3,3,1) with l(α) = 4 and
|α| = 12. Every Young diagram λ defines four objects which give four
different isomorphic theories:
(1) A representation V of SL(n,F ).
λ q
(2) A representation [λ] of the symmetric group.
(3) A symmetric function s (x ,...,x ) which is the Schur function of
λ 1 n
shape λ in the variables (x ,··· ,x ).
1 n
(4) A Schubert cell X in the Grassmannian.
λ
We define the Schur projection c : (cid:78)dV → (cid:78)dV. Let S be the
λ n
symmetric group of permutations over d elements. Any permutation σ ∈ S
n
acts on a given Young diagram by permuting the boxes. Let R ⊆ S be the
λ n
subgroupofpermutationspreservingeachrow. LetC ⊆ S bethesubgroup
λ n
6 ALBERTOBESANAANDCRISTINAMART´INEZ
(cid:80) (cid:80)
of permutations preserving each column, let c = (cid:15)(τ)στ.
λ σ∈R τ∈C
λ λ
The image of c is a irreducible SL(n,F )−module, which is nonzero iff
λ q
the number of rows is ≤ dimV . All irreducible SL(n,F )−modules can be
λ q
obtained in this way. Every SL(n,F )−module is a sum of irreducible ones.
q
In terms of irreducible representations of SL(n,F ), a partition η corre-
q
sponds to a finite irreducible representation that we denote as V(η). Since
SL(n,F ) is reductive, any finite dimensional representation decomposes
q
into a direct sum of irreducible representations, and the structure constant
cη is the number of times that a given irreducible representation V(η) ap-
λ,µ
pears in an irreducible decomposition of V(λ)⊗V(µ). These are known as
Littlewood-Richardson coefficients, since they were the first to give a com-
binatorial formula encoding these numbers (see [Fu]). In terms of the Hopf
algebra Λ of Schur functions, let s be the Schur function indexed by the
λ
partition λ, we have s ·s = (cid:80) cν s for the product and we get the co-
λ µ ν λµ ν
efficients kη as the structure constants of the dual Hopf algebra Λ∗. These
λµ
are known as Kronecker coefficients, (see [Ma] and [SLL]).
One can stack Kronecker coefficients cν in a 3D matrix or 3-dimensional
λµ
matrix. Intuitivelya3Dmatrixisastackingofboxesinthecornerofaroom.
Theelementsoftheprincipaldiagonalarecalledrectangularcoefficientsand
are indexed by triples (λ,µ,ν) = ((in),(in),(in)) of partitions (in) with all
their parts equal to the same integer 1 ≤ i ≤ n.
Proposition 3.2. Let C be the 3D matrix whose entries are the Littlewood-
Richardson coefficients, and K the 3D matrix of Kronecker coefficients.
Then the matrices are inverse one to each other.
Proof. Sincecν andkν correspondtothestructureconstantsoftheHopf
λµ λµ
algebraofSchurfunctionsanditsdualonerespectively,andtheHopfalgebra
of Schur functions is self-dual (see [SLL]), one gets that the product matrix
C · K is the identity 3D matrix I, that is, the matrix whose rectangular
coefficients are identically 1. Thus both matrices are inverse one to each
other, that is, (cν )−1 = kν . (cid:3)
λ,µ λ,µ
CODES, HORN’S PROBLEM AND GROMOV-WITTEN INVARIANTS 7
Remark 3.3. How we define the product of two 3D matrices? For each
indexν fixed, λandµrun over allpartitions P(n)ofn. Thus the coefficients
(cid:16) (cid:17)
cν are encoded in a matrix of order p(n) × p(n), where p(n)
λ,µ
λ,µ∈P(n)
denotes the number of partitions of n. Thus the product matrix Cν ·Kν is
the standard product of matrices in M (R).
p(n)×p(n)
3.1. Effective computation of Littlewood-Richardson coefficients.
The convex hull in R3 of all triples (λ,µ,ν) with cν > 0 is the Newton
λ,µ
polytope of f(x,y,z) = (cid:80) cν xλyµzν ∈ C[x,y,z]. Here xλ denotes the
λ,µ,ν λ,µ
monomial xλ1···xλn of partition degree λ.
Theorem 3.4. The polynomial f(x,y,z) = (cid:80) cν xλyµzν ∈ C[x,y,z],
λ,µ,ν λ,µ
isthegeneratingseriesfortheGromov-WitteninvariantsN (λ,µ,ν), count-
d,g
ing irreducible plane curves of given degree d and genus g passing through
a generic configuration of 3d−1+g points on P2(C) with ramification type
at 0,∞ and 1 described by the partitions λ,µ and ν and simple ramification
over other specified points with |λ|+|µ|+|ν| = d.
Proof. Whenever the coefficient cν > 0 is positive consider the cor-
λ,µ
responding ideal sheaves I , I and I in C associated to the partitions
λ µ ν
λ,µ and ν respectively. Each ideal sheaf determines a curve in C[x,y]
via homogenization of the corresponding monomial ideals.Thus each co-
efficient represents the number of ideal sheaves on C3 of colength n and
degree d equal to the size of the partition, that is the corresponding 3-
point Gromov-Witten invariant (cid:104)λ,µ,ν(cid:105) of the Hilbert scheme Hilb
0,3,d n
of n = 2d−1+|ν|+|µ|+|λ|+g distinct points in the plane, or the rela-
tiveGromov-WitteninvariantN (λ,µ,ν)countingirreducibleplanecurves
d,g
of given degree d and genus g passing through a generic configuration of
3d−1+g points on P2(C) with ramification type at 0,∞ and 1 respectively,
described by the partitions λ,µ and ν of n, (see [BM1]). (cid:3)
Remark 3.5. The Euler characteristic of each ideal sheaf is fixed and co-
incides with the Euler characteristic χ of the polyhedra described in R3 by
the convex hull of all triples (λ,µ,ν) with cν > 0, that is, the Newton
λ,µ
polytope of f(x,y,z) = (cid:80) cν xλyµzν ∈ R[x,y,z]. Thus each coefficient
λ,µ,ν λ,µ
represents the number of ideal sheaves on C3 of fixed Euler characteristic
χ = n and degree d equal to the size of the partition, that is the correspond-
ing Donaldson-Thomas invariant of the blow-up of the plane P1×(C2) with
discrete invariants χ = n and degree d.
Remark3.6. TheHilbertschemeHilb ofnpointsintheplaneC2parametriz-
n
ing ideals J ⊂ C[x,y] of colength n contains an open dense set in the
Zariski topology parametrizing ideals associated to configurations of n dis-
tinctpoints. MoreoverthereisanisomorphismHilb ∼= (C2)n/S . Inpartic-
n n
ular, as we showed in [BM1], any conjugacy class in the symmetric group S
n
determines a divisor class in the T−equivariant cohomology H4n(Hilb ,Q),
T n
for the standard action of the torus T = (C∗)2 on C2. The T−equivariant
cohomology of Hilb has a canonical Nakajima basis indexed by P(n). The
n
map λ → J is a bijection between the set of partitions P(n) and the set of
λ
T −fixed points HilbT ⊂ Hilb .
n n
8 ALBERTOBESANAANDCRISTINAMART´INEZ
Denote the series (cid:104)λ,µ,ν(cid:105)Hilbn of 3-point invariants by a sum over curve
degrees:
(cid:88)
(cid:104)λ,µ,ν(cid:105)Hilbn = qd(cid:104)λ,µ,ν(cid:105)Hilbn.
0,3,d
d≥0
Corollary 3.7. Let H be the divisor class in the Nakajima basis correspond-
ing to the tautological rank n bundle O/J → Hilb with fiber C[x,y]/J over
n
J ∈ Hilb andν thecorrespondingpartition.Thenwecanrecoverinductively
n
in the degree d, all the Littlewood-Richardson coefficients (cν ) .
λ,µ λ,µ∈P(n)
Proof. The non-negative degree of a curve class β ∈ H (Hilb ,Z) is
2 n
defined by d = (cid:82) H. Then via the indentification of cν with the 3-point
β λ,µ
Gromov-Witten invariant (cid:104)λ,H,µ(cid:105)Hilbn where [λ],[µ] are the corresponding
0,3,d
classes in H4n(Hilb ,Q) associated to the partitions λ and µ in P(n), we
T n
proceed by induction on the degree d as in section 3.6 of [OP]. (cid:3)
Remark 3.8. If we choose the partition ν to be the empty partition ∅, we
recover the relative Gromov-Witten invariants N (λ,µ) studied by Fomin
d,g
and Mikhalkin in [FM], and by Caporaso and Harris in [CH].
4. Configurations of points over a rational normal curve
Assume V is a vector space of dimension n+1 over a field k equipped
with a linear action, that is, G acts via a representation G → GL(V). We
denote by SdV the d−th symmetric power of V.
Consider the d−Veronese embedding of Pn
(4) PV∗ → PSdV∗
v (cid:55)→ vd,
mapping the line spanned by v ∈ V∗ to the line spanned by vd ∈ SdV∗. In
coordinates, ifwechoosebases{α,β}forV and{[ n! ]αkβd−k}forSdV∗
k!(n−k)!
and expanding out (xα+yβ)d, we see that in coordinates this map may be
given as
[x,y] → [xd,xd−1y,xd−2y2,...,xyd−1,yd].
Goppa recognized that the Gale transform of a configuration of n distinct
pointssupportedonarationalnormalcurveinPd isaconfigurationofpoints
supported on a rational normal curve on Pn−d−2.
In particular, the homogeneous coordinate ring for the natural projective
embbeding of the GIT quotient (Pd)n//SL is the ring of invariants for
d+1
n ordered points in the projective space up to projectivity. Generators
for this ring are given by tableau functions, which appear in many areas
of mathematics, particularly representation theory and Schubert calculus.
Consider the hypersimplex:
(cid:88)
(cid:52)(d+1,n) = {(c ,...,c ) ∈ Qn|0 ≤ c ≤ 1, c = d+1},
1 n i i
for any 1 ≤ d ≤ n−3 and choose of linearization c ∈ (cid:52)(d+1,n), there is a
morphism
ϕ : M¯ → (Pd)n// SL ,
0,n c d+1
CODES, HORN’S PROBLEM AND GROMOV-WITTEN INVARIANTS 9
sending a configuration of distinct points on P1 to the corresponding config-
uration under the dth Veronese map.
The symmetric power SymnC of the curve C is the quotient of the
d d
configurationspaceCn ofnunorderedtuplesofpointsontherationalnormal
d
curve C by the symmetric group S . Furthermore, we can identify the set
d n
ofeffectivedivisorsofdegreedonC withthesetofk−rationalpointsofthe
d
symmetric power SymnC, that is, SymnC represents the functor of families
of effective divisors of degree n on C.
Why codes on the rational normal curve? By definition, the rational
normal curve C is the image by the d−Veronese embedding of PV∗ = P1
d
where V is a 2-dimensional vector space. The action of PGL(2,k) on Pd
preserves the rational normal curve C . Conversely, any automorphism of
d
Pd fixing C pointwise is the identity. It follows that the group of automor-
d
phisms of Pd that preserves C is precisely PGL(2,k). Thus the problem
d
of classifying codes on the rational normal curve is reduced to study finite
groups of the projective linear group PG(2,k) or the symmetric group S .
n
In more concrete terms, one can consider the action of finite subgroups of
S on configuration of points on the rational normal curve C .
n d
Proposition 4.1. IfweconsiderthesetoforbitsofCn bytheactionoffinite
d
subgroups of the symmetric group S , we get all possible divisor classes in
n
the group Divn(C ) of degree n divisors on C .
d d
Proof. Since the symmetric group S is generated by 3 elements, a re-
n
flection of order 2, a symmetry of order 3 and a rotation of order n, we get
all the divisor classes by quotienting the configuration space Cn of n points
d
on the rational normal curve, by the cyclic group generated by the rotation,
or one of the triangle groups, the dihedral group D , the alternated groups
n
A , A or the symmetric group S . (cid:3)
4 5 4
5. Notion of collinearity on the rational normal curve
Definition 5.1. An incidence structure S on V is a triple (P,B,I), where
P is a set whose elements are smooth, reduced points in V, B is a set whose
elements are subsets of points called blocks (or lines in several specific cases)
endowed with a relation of collinearity, and an incidence relation I ⊂ P×B.
If (P,L) ∈ I, then we say that P is incident with L or L is incident with P,
or P lies in L or L contains P.
When the collinearity relation is a symmetric ternary relation defined on
triples (p,q,r) ∈ P×P×P by the geometric condition (p,q,r) ∈ B if either
p+q+r is the full intersection cycle of C with a k−line l ⊂ Pn(k) with the
d
rightmultiplicities,orelseifthereexistsak−linel ⊂ V suchthat,p,q,r ∈ l,
then the triple (p,q,r) is called a plane section.
(1) For any (p,q) ∈ P2(V∗), there exists an r ∈ P(SdV∗) such that
(p,q,r) ∈ l. The triple (p,q,r) is strictly collinear if r is unique with
this property, and p,q,r are pairwise distinct. The subset of strictly
collinear triples is a symmetric ternary relation. When k is a field
algebraically closed of characteristic 0, then r is unique with this
property, and we recover the euclidean axioms.
10 ALBERTOBESANAANDCRISTINAMART´INEZ
(2) Assume that p (cid:54)= q and that there are two distinct r ,r ∈ P with
1 2
(p,q,r ) ∈ B and (p,q,r ) ∈ B. Denote by l = l(p,q) the set of all
1 2
such r(cid:48)s, then l3 ∈ B, that is any triple (r ,r ,r ) of points in l is
1 2 3
collinear. Such sets l are called lines in B.
If V is a 3-dimensional vector space defined over the finite field F ,
p
then the projective plane P2(F ) on V is defined by the incidence struc-
p
ture PG(2,p) = (P(V),L(V),I).
Definition 5.2. (1) A (k;r)−arc K in PG(2,p) is a set of k−points
such that some r, but not r + 1 of them are collinear. In other
words, some line of the plane meets K in r points and no more
than r−points. A (k;r)−arc is complete if there is no (k+1;r) arc
containing it.
(2) A k−arc is a set of k points, such that, every subset of s points with
s ≤ n points is linearly independent.
LetqdenotesomepoweroftheprimepandPG(n,p)bethen−dimensional
projective space (F )n+1 ∼= F , where n ≥ 2.The normal rational curve C is
p q
defined as:
(cid:110) (cid:91) (cid:111)
Vn := F (1,x,x2,...,xn)| x ∈ F {∞} .
1 q q
If q ≥ n+2, the NRC is an example of a (q+1)−arc. It contains q+1
points, and every set of n + 1 points are linearly independent. For each
a ∈ (F )n+1, the mapping:
p
F (x ,...,x ) → F (a0x ,...,anx ),
p 0 n p 0 n
describes an automorphic collineation of the NRC.
All invariant subspaces form a lattice with the operations of ”join” and
”meet”.
For j ∈ N, let Ω(j) = {m ∈ N|0 ≤ m ≤ n,(cid:0)m(cid:1) (cid:54)= 0modp}. Given
j
(cid:83) (cid:83)
J ⊂ {0,1,...,n}, put Ω(J) = Ω(j), Ψ(J) := {j,n−j}.
j∈J j∈J
Both Ω and Ψ are closure operators on {0,1,...,n}. Likewise the projec-
tive collineation F (x ,x ,...,x ) → F (x ,x ,...,x ) leaves the NRC
p 0 1 n p n n−1 0
invariant whence Λ has to be closed with respect to Ψ.
Proposition 5.3. Each subspace invariant under collineation of the NRC,
is indexed by a partition in P(t). If the ground field k is sufficiently large,
then every subspace which is invariant under all collineations of the NRC,
is spanned by base points kc , where λ ∈ P(t).
λ
Proof. Let
Et := {(e ,e ,...,e ) ∈ Nn+1|e +e +...+e = t},
n 0 1 n 0 1 n
be the set of partitions of t of n parts and let Y be the (cid:0)n(cid:1)−dimensional
t
vector space over F with basis
p
{c ∈ F : (e ,e ,...,e ) ∈ Et}.
e0,e1,...,en q 0 1 n n
Let’s call Vt the Veronese image under the Veronese mapping given by:
n
n
(cid:88) (cid:88)
F ( x b ) → F ( c xe0xe1···cen), x ∈ F .
p i i p e0,...,en 1 n i p
i=0 Et
n