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CODES, HORN’S PROBLEM AND GROMOV-WITTEN INVARIANTS ALBERTO BESANA AND CRISTINA MARTÍNEZ 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 ALBERTOBESANAANDCRISTINAMARTÍNEZ (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 representation 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, configurations 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, convolutional 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 identified 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 ALBERTOBESANAANDCRISTINAMARTÍNEZ 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 matrices 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 polynomials 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 problem 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 ALBERTOBESANAANDCRISTINAMARTÍNEZ (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 coefficient 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 corresponding 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 ALBERTOBESANAANDCRISTINAMARTÍNEZ 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 corresponding 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 configuration 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 ALBERTOBESANAANDCRISTINAMARTÍNEZ (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 projective 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