On projective classification of plane curves

Transcription

1 Global and Stochastic Analysis Vol. 1, No. 2, December 2011 Copyright c Mind Reader Publications On projective classification of plane curves Nadiia Konovenko a and Valentin Lychagin b a Odessa National Academy of Food Technologies ONAFT, 112, Kanatna Str., 65039, Odessa, Ukraine konovenko@ukr.net b University of Tromsø, Tromsø, Norway valentin.lychagin@uit.no Received by the Editorial Board on October 15, 2011 Abstract Algebra of projective differential invariants and description of projective classes of regular smooth plane curves are found. A classification and projective normal forms for special classes of plane curves are given. Key words: plane curve, W-curve, elliptic curve, projective differential invariant, projective curvature, Study differential 2010 Mathematics subject classification: 53A20 53A55 1. Introduction In this paper we investigate projective equivalence of smooth curves on the real projective plane. Some of these results are classical and known. Mainly they go back to Halphen s dissertation [2]. The considered curves are smooth but they have singularities in projective sense. For example, singular, from the projective point of view, are points on a curve, where tangent line has second order contact, i.e. inflection or flex points. Another example gives points on a curve, where osculating quadric has 5-th order contact, i.e. Monge points. 241

2 We describe SL 3 R-orbits of the projective action on jets of smooth plane curves up to 5-th order and classify all possible projective singularities. The level of 5-th jets taken for the only reason: starting from 6-jets regular orbits have trivial stabilizers, and from the level of 7-jets first differential invariants come up. This orbit classification gives projective classification, or projective normal forms for smooth curves, up to 6-th order jets. To find the complete algebra of differential invariants we reproduce the Study derivation in suitable for us form. It allows us to prove that algebra of polynomial differential invariants which separates regular orbits can be obtained from the projective curvature by taking of higher Study derivatives Theorem 4. As a by-product of this theorem we get two results: normal forms curves up to 10-th jets Theorem 6 and projective classification of germs regular smooth curves Theorem 5. The rest of the paper devoted to special classes of plane curves. The popular and trivial classes plane curves such as straight lines and quadrics are singular from the projective point of view. The next class is the class of W -curves, introduced by Klein and Lie, we collected some known properties of these curves see, [3] and [4] for more details. The more important property of these curves, in light of Theorem 5, is that they are not regular. They are curves of constant projective curvature. The first regular curves can be found in cubics and theorem 16 repeats the known result that projective classes of regular cubics can be described by one parameter. We give explicit formula to find this parameter. The second important class of regular curves delivers by extremals of Study functional cf. [7]. The corresponding Euler equation has order 10 and therefore projective classes of such extremals can be described by two parameters. Theorem 16 gives a constructive way to compute them. 2. Jets of curves Let P 2 be the real projective plane and let J k be the manifold of k - jets of non-parametrized curves on the plane P 2. If L P 2 is a plane curve we denote by [L] k a J k the k -jet of the curve at the point a L. We denote by π k,l : J k J l, k > l, the natural projections: π k,l : [L] k a [L] l a. The structure of jet - manifolds can be described as follows: obviously J 0 = P 2, and the fibres π1,0 1 a, a P 2, of the projection π 1,0 are projectivizations of the tangent planes P T a P 2 = P 1, and fibres π 1 k,k 1 [L] k 1 a, when k 2, are affine lines. 242

3 The vector spaces associated with them are S k τ a ν a, where τ a = T a L - cotangent space, and ν a = T a P 2 /T a L - normal space to a curve. Let x, u be an affine chart on the plane and let x, u, u 1,..., u k be the natural coordinates in the space of k -jets. Here u i [L] k a = i h x i b, if L = L h def = {u = h x} is a graph of function h in a neighborhood of point a = b, h b. In these coordinates the affine action is given by tensors and has the form θ = λ k! dxk u S k τ a ν a, x, u, u 1,..., u k 1, u k x, u, u 1,..., u k 1, u k + λ, where u = u modt a L. Any smooth curve L P 2 determines curves L k J k, k-th prolongations of L, formed by points [L] k a where point a runs over curve L. The action of projective group SL 3 R can be prolonged in manifolds J k in the natural way: φ k : [L] k a [φ L] k a where φ is a projective transformation. 3. Model curves The use of model curves is based on the following observation. Assume that we have a class M of plane curves which is projectively invariant and such that for any point x k J k there is a unique curve L = L x k M such that x k = [L] k a, for a = π k x k. Then points x k+1 = [L x k ] k+1 a can be taken as basic points in the affine line π 1 k+1,k x k and the corresponding section m : J k J k+1 we consider as the zero section in the line bundle π k+1,k : J k+1 J k. Let now L P 2 be an arbitrary curve. Then curves L k+1 J k+1 and m L k J k+1 differs on element Θ L S k+1 TL ν L. The last tensor is a projective differential invariant of order k + 1 in the sense that φ Θ φl = ΘL, for arbitrary projective transformation φ. Below we ll realize this scheme for different classes of projective curves. 243

4 3.1 Straight Lines Let M be the class straight lines on the projective plane. Obviously, for any point x 1 J 1 there is a unique line L x 1, such that x 1 = [L x 1 ] 1 a. Therefore, the above construction leads us projective differential invariant of order 2 Θ 2L S 2 T L ν L. If L = L h is the graph of function u = h x in the affine coordinates, then the restriction of tensor Θ 2 on this curve has the form: We write in the jet coordinates and Θ 2L = h x dx2 2! u. dx 2 Θ 2 = u 2 u 2! Θ 2L = Θ 2 L 2 h Denote by Π 2 J 2 the submanifold, where Θ 2 = 0. Then the points Π 2 L = Π 2 L 2 are precisely inflection or flex points on the curve, i.e. points where tangent lines have 2-rd order contact with the curve. 3.2 Quadrics Let M be now the class of quadrics on the projective plane. In affine coordinates each such quadrics is defined by the equation: Q 2 = a 11 u 2 + 2a 12 xu + a 22 x 2 + 2a 1 u + 2a 2 x + a 3 = 0. Taking derivatives of Q 2 up to order 5 and eliminating a s coefficients we arrive at equation 9 u5 u u u 2 u 4 u 3 u2 = 0.. If we really will consider only quadrics u 2 0, then we get the Monge equation: 9 u 5 u u u 2 u 4 u 3 = 0, or In other words, for any point x 4 Q x 4 such that [Q x 4 ] 4 a = x 4. u 5 = 5u 3u 4 40 u 2 9 u 3 3 u 2 2. J 4 \ π 1 4,1 Π 2 there is a unique quadric 244

5 Therefore, as above, for any curve L we have projective differential invariant Θ 5L S 5 T L ν L, where or Θ 5L = Θ 5 = h 5 5 h3 h h 2 9 u 5 5u 3u u 2 9 h 3 3 h 2 2 u 3 3 u 2 2 dx 5 5! dx 5 u 5! u, in jet coordinates in the domain J 5 \ π5,2 1 Π 2. Denote by Π 5 J 5 \ π5,2 1 Π 2 the submanifold, where Θ 5 = 0. Then the points Π 5 L = Π 5 L 5 we call Monge points. They are the points where osculating quadrics have 5-th order contact with the curve. 3.3 Cubics Let M be now the class of cubics on the projective plane. In affine coordinates each such quadrics is defined by the equation: Q 3 = a 111 u 3 + 3a 112 u 2 x + 3a 122 ux 2 + a 222 x 3 +a 11 u 2 + 2a 12 xu + a 22 x 2 + 2a 1 u + 2a 2 x + a 3 = 0. Taking derivatives of Q 3 up to order 9 and eliminating a s coefficients we arrive at equation see, for example, [6]: u 2 P 7 u 9 + P 8 = 0, where P 8 is a polynomial of degree 10 and order 8, P 7 = detm 7 is a a polynomial of degree 8 and order 7, and 120 u 3 30 u 4 6 u 5 u 6 u 7 /7 360 u u 3 30 u 4 6 u 5 u 6 M 7 = u u 3 10 u 3 u 4 2 u 3 u 5 + 5u 2 4 / u u 3 u 2 30 u 4 u u 3 6 u 5 u u 3 u u u 3 u 2 60 u u 4 u 2 245

6 more explicitly P 7 = u 2 u 3 6 u u 2 5 u 3 u 4 u u 2 5 u 3 u 5 u u 2 4 u 3 2 u u 2 4 u 3 u 4 2 u u 2 5 u 4 u u 2 3 u 3 3 u 4 u u 2 6 u u 2 3 u 3 2 u u 2 4 u 3 3 u u 2 4 u u u 2 5 u 4 2 u u 2 2 u 3 5 u u 2 2 u 3 4 u u 2 4 u 3 2 u 4 u u 2 6 u 5 u 7. Therefore, for cubics. u 9 = P 8 u 2 P 7 In other words, for any point x 8 J 8 \ π8,2 1 Π 2 π8,7 1 Π 7, where Π 7 = P7 1 0 J 7, there is a unique cubic Q x 8 such that [Q x x ] 8 a = x 8. Therefore, as above, for any curve L we have projective differential invariant Θ 9L S 9 TL ν L, where Θ 9 = u 9 + P 8 dx 9 u u 2 P 7 9! in jet coordinates in the domain J 9 \ π9,2 1 Π 2 π9,7 1 Π 7. Denote by Π 9 J 9 \ π9,2 1 Π 2 π9,7 1 Π 7 the submanifold, where Θ 9 = 0. Then the points Π 9 L = Π 9 L 9 we call Monge cubic points. They are the points where osculating cubics have 9-th order contact with the curve. 3.4 General Polynomial forms Let M be now the class of curves on the projective plane, having degree n. Each such polynomial is defined by k k + 3 /2 coefficients. Hence, taking derivatives up to order j = k k + 3 /2 and eliminating these coefficients we arrive at equation of the form P j u j + P j 1 4. SL 3 R-orbits in the jet spaces 4.1 J 2 -orbits The action of the projective group on the manifold of 1-jets is obviously transitive. 246

7 The stabilizer of point 0, 0, 0 J 1 is formed by matrices a 11 a a 22 0 a 31 a 32 a 33. Action of such matrices on the fibre of projection π 2,1 : J 2 J 1 has the form: A 2 : 0, 0, 0, u 2 0, 0, 0, a 3 11 u 2. Therefore, there is the only one open regular orbit Π 20 = J 2 \ Π 2 and the singular orbit Π 2 : J 2 = Π 20 Π 2, and points can be taken as representatives. 4.2 J 3 -orbits p 20 = 0, 0, 0, 1 Π 20, p 2 = 0, 0, 0, 0 Π 2 Consider the action of the stabilizer of point 0, 0, 0, 1 from the regular orbit Π 20 on the fibre of projection π 3,2 : J 3 J 2. This stabilizer is formed by matrices a 11 a a 2 11a 1 a 31 a 32 a 33 and their action is the following affine action: where A 3 : 0, 0, 0, 1, u 3 0, 0, 0, 1, α A u 3 + β A, α a 33 a 1 11, β 3 a 11 a 31 a 12 a 33 a Therefore, Π 30 = π3,2 1 Π 20 is the open regular orbit. The stabilizer of the point 0, 0, 0, 0 from the singular orbit Π 2 formed by matrices a 11 a a 22 0 a 31 a 32 a 33 which act in the following way A 3 : 0, 0, 0, 0, u , 0, 0, 0, a 33 u a

8 Therefore, the preimage π3,2 1 Π 2 of the singular orbit is a union of two orbits Π 32 = {x, u, u 1, 0, 0} and Π 31 = π3,2 1 Π 2 \ Π 31 and J 3 has the following decomposition of SL 3 R-action: where Π 30 is the regular open orbit. The following points J 3 = Π 30 Π 31 Π 32, p 30 = 0, 0, 0, 1, 0 Π 30, p 31 = 0, 0, 0, 0, 1 Π 31, p 32 = 0, 0, 0, 0, 0 Π 32 can be taken as representatives of the orbits. 4.3 J 4 -orbits Consider the action of the stabilizer of point 0, 0, 0, 1, 0 from the regular orbit Π 30 on the fibre of projection π 4,3 : J 4 J 3. This stabilizer is formed by matrices a 11 a 11 a 31 a 1 0 a 2 11a 1 a 31 a 32 a 33 with the following affine action: A 4 : 0, 0, 0, 1, 0, u 4 0, 0, 0, 1, 0, a 2 33a 2 11 u 4 + 6a 33 a 32 3a 2 31 a Therefore, Π 40 = π4,3 1 Π 30 is an open regular orbit. The stabilizer of point 0, 0, 0, 0, 1 from the singular orbit Π 31 is formed by matrices a 11 a a 2 11a 1 a 31 a 32 a 33 with the following affine action A 4 : 0, 0, 0, 0, 1, u 4 0, 0, 0, 0, 1, a 33 a 1 11 u 4 + 8a 31 a Therefore, Π 41 = π4,3 1 Π 31 is an orbit. Finally, the stabilizer of point 0, 0, 0, 0, 0 from Π 32 is formed by matrices a 11 a a 22 0 a 31 a 32 a 33 with the following action A 4 : 0, 0, 0, 0, 0, u 4 0, 0, 0, 0, 0, a 3 33a 22 a 4 11 u

9 Therefore, the preimage π4,3 1 Π 32 of the singular orbit Π 32 is a union of two orbits Π 43 = {x, u, u 1, 0, 0, 0} and Π 42 = π4,3 1 Π 31 \ Π 43. Summarizing, we see that there is the only one open and regular orbit Π 40 and three singular orbits Π 41, Π 42 and Π 43 : The following points J 4 = Π 40 Π 41 Π 42 Π 43. p 40 = 0, 0, 0, 1, 0, 0 Π 40, p 41 = 0, 0, 0, 0, 1, 0 Π 41, p 42 = 0, 0, 0, 0, 0, 1 Π 42, p 43 = 0, 0, 0, 0, 0, 0 Π 43 can be taken as representatives of the orbits. 4.4 J 5 -orbits Let s begin with preimage of regular orbit Π 40. The stabilizer of point 0, 0, 0, 1, 0, 0 is formed by matrices a 11 a 11 a 31 a 1 0 a 2 11a 1 a a 2 31 a a 2 33 and has the following action on the fibre: A 5 : 0, 0, 0, 1, 0, 0, u 5 0, 0, 0, 1, 0, 0, a 3 33a 3 11 u 5. Therefore, the preimage π5,4 1 Π 40 of the regular orbit is a union two orbits: the singular one Π 5 and the open regular orbit Π 50 = π5,2 1 Π 20 \ Π 5. The stabilizer of point 0, 0, 0, 0, 1, 0 from the singular orbit Π 41 is formed by matrices a 11 a a 3 11a 2 0 a 32 a 33 and acts in the following way A 5 : 0, 0, 0, 0, 1, 0, u 5 0, 0, 0, 0, 1, 0, a 2 33a 2 11 u 5 10a 12 a 2 33a Therefore, Π 51 = π5,4 1 Π 41 is an orbit. The stabilizer of point 0, 0, 0, 0, 0, 1 from the singular orbit Π 42 is formed by matrices a 11 a a 4 11a 3 a 31 a 32 a

10 and acts in the following way A 5 : 0, 0, 0, 0, 0, 1, u 5 0, 0, 0, 0, 0, 1, a 33 a 1 11 u a 31 a Therefore,Π 52 = π5,4 1 Π 42 is an orbit too. Finally, the stabilizer of point 0, 0, 0, 0, 0, 0 from the singular orbit Π 43 is formed by matrices a 11 a a 22 0 a 31 a 32 a 33 and acts as following A 5 : 0, 0, 0, 0, 0, 0, u 5 0, 0, 0, 0, 0, 1, a 3 33a 6 11 u 5. Therefore, the preimage π 1 5,4 Π 43 is a union of two orbits: and Π 54 = {x, u, u 1, 0, 0, 0, 0} Π 53 = π 1 5,4 Π 43 \ Π 54. Summarizing, we conclude that SL 3 R-action in J 5 has the following orbit decomposition: J 5 = Π 50 Π 5 Π 51 Π 52 Π 53 Π 54, where Π 50 is the only regular open orbit. The following points p 50 = 0, 0, 0, 1, 0, 0, 1 Π 50, p 5 = 0, 0, 0, 1, 0, 0, 0 Π 5, p 51 = 0, 0, 0, 0, 1, 0, 0 Π 51, p 52 = 0, 0, 0, 0, 0, 1, 0 Π 52, p 53 = 0, 0, 0, 0, 0, 0, 1 Π 53, p 54 = 0, 0, 0, 0, 0, 0, 0 Π 54 can be taken as representatives of the orbits. 4.5 J 6 -orbits Let s begin with preimage of regular orbit Π 50. The stabilizer of point p 50 is formed by matrices a 33 a a which act in the following way A 6 : 0, 0, 0, 1, 0, 0, 1, u 6 1 a 31 a 2 31a 1 33 a , 0, 0, 1, 0, 0, 1, u 6 + 3a 31, a 33

11 and therefore Π 60 = π 6,5 Π 50 is an open regular orbit. The stabilizer of the point p 5 is formed by matrices a 11 a 11 a 31 a 1 0 a 2 11a 1 with the following action: 1 a 31 2 a2 31a 1 33 a 33 A 6 : 0, 0, 0, 1, 0, 0, 0, u 6 0, 0, 0, 1, 0, 0, 0, a 4 33a 4 11 u 6. Therefore, preimage π 6,5 Π 5 is a union of three orbits π 6,5 Π 5 = Π + 61 Π 61 Π 62 with the following representatives p + 61 = 0, 0, 0, 1, 0, 0, 0, 1, p 61 = 0, 0, 0, 1, 0, 0, 0, 1, p 62 = 0, 0, 0, 1, 0, 0, 0, 0. The stabilizer of the point p 51 = 0, 0, 0, 0, 1, 0, 0 Π 51 contains matrices a a 3 11a 2 0 a 32 a 33 and acts in the following way A 6 : 0, 0, 0, 0, 1, 0, 0, u 6 0, 0, 0, 1, 0, 0, 0, a 3 33a 3 11 u a 3 11 a 32 a Therefore, Π 63 = π6,5 1 Π 51 is an orbit. The stabilizer of the point p 52 = 0, 0, 0, 0, 0, 1, 0 Π 52 formed by matrices a 11 a a 4 11a 3 0 a 32 a 33 and acts in the following way A 6 : 0, 0, 0, 0, 0, 1, 0, u 6 0, 0, 0, 1, 0, 0, 0, a 2 33a 2 11 u 6. Therefore, π 1 6,5 Π 52 is a union of three orbits π 1 6,5 Π 52 = Π + 64 Π 64 Π

12 with representatives p + 64 = 0, 0, 0, 0, 0, 1, 0, 1, p 64 = 0, 0, 0, 0, 0, 1, 0, 1, p 65 = 0, 0, 0, 0, 0, 1, 0, 0. The stabilizer of the point p 53 = 0, 0, 0, 0, 0, 0, 1 Π 53 formed by matrices a 11 a a 5 11a 4 a 13 a 32 a 33 with the following action A 6 : 0, 0, 0, 0, 0, 0, 1, u 6 0, 0, 0, 0, 0, 0, 1, a 33 a 1 11 u a 31 a Therefore, Π 66 = π 1 6,5 Π 53 is an orbit with representative p 66 = 0, 0, 0, 0, 0, 0, 1, 0. Finally, the stabilizer of the point p 54 = 0, 0, 0, 0, 0, 0, 1 Π 54 is formed by matrices a 11 a a 22 0 a 13 a 32 a 33, which act in the following way: A 6 : 0, 0, 0, 0, 0, 0, 0, u 6 0, 0, 0, 1, 0, 0, 0, a 4 33a 7 11u 6. Therefore, the preimage π 1 6,5 Π 54 is a union of two orbits Π 67 and Π 68 with representatives p 67 = 0, 0, 0, 0, 0, 0, 0, 1 and p 68 = 0, 0, 0, 0, 0, 0, 0, 0 respectively. Summarizing, we get the following result. Theorem 4.1. SL 3 R-action in J 6 splits into the following orbit decomposition: J 6 = Π 60 Π + 61 Π 61 Π 62 Π 63 Π + 64 Π 64 Π 65 Π 66 Π 67 Π 68, with the following representatives p 60 = 0, 0, 0, 1, 0, 0, 1, 0, p + 61 = 0, 0, 0, 1, 0, 0, 0, 1, p 61 = 0, 0, 0, 1, 0, 0, 0, 1, p 62 = 0, 0, 0, 1, 0, 0, 0, 0, p 63 = 0, 0, 0, 0, 1, 0, 0, 0, p + 64 = 0, 0, 0, 0, 0, 1, 0, 1, p 64 = 0, 0, 0, 0, 0, 1, 0, 1, p 65 = 0, 0, 0, 0, 0, 1, 0, 0, p 66 = 0, 0, 0, 0, 0, 0, 1, 0, p 67 = 0, 0, 0, 0, 0, 0, 0, 1, p 68 = 0, 0, 0, 0, 0, 0, 0,

13 As a corollary of this theorem we get the following SL 3 R classification of 6-jets of projective curves. Theorem 4.2. Let L P 2 be a smooth projective curve. Then for any point a L there are projective coordinates x, y such that x a = y a = 0 and the curve can be written in the form y = p x + ε x, where function ε x has seventh order of smallness and polynomial p x has one of the following form: p 60 x = x 2 + x 5, p ± 61 x = x 2 ± x 6, p ± 64 = x 4 ± x 6, p 62 x = x 2, p 63 x = x 3, p 65 = x 4, p 66 = x 5, p 67 = x 6, p 68 = 0, where polynomials p ij correspond to orbits Π ij. 4.6 Stabilizers of regular orbit The open orbit Π 60 = J 6 \ π 6,2 Π 2 \ π 6,5 Π 5 we call regular, and elements of this orbit we also call regular. A point a L on a smooth projective curve we call regular if [L] 6 a Π 60, if not the point is calling singular. It was to note that our definitions differ from the standard ones: both regular and singular points belong to smooth curve, and their singularity has projective nature.remark also that the previous theorem states that the regular orbit is connected even though singular orbits Π 2 and Π 5 have codimension 1. Before to consider differential invariants of projective curves we ll finish this section by description of stabilizers of regular point in J k, when k = 2, 3, 4, 5, 6. Take 2-jet p 20 = 0, 0, 0, 1. Then the stabilizer is a 4-dimensional group and consist of matrices 1 α 0 St 2 = 0 β 1 0 γ δ β, where α, γ, δ R 3, β R\0. For 3-jet p 30 = 0, 0, 0, 1, 0 the stabilizer is a 3-dimensional group and consist of matrices 1 α 0 St 3 = 0 β 1 0 αβ γ β, where α, γ R 2, β R\0. For 4-jet p 40 = 0, 0, 0, 1, 0, 0 the stabilizer is a 2-dimensional group and consist of matrices 1 α 0 St 4 = 0 β 1 0 αβ 1 2 α2 β β, 253

14 where α R, β R\0. For 5-jet p 50 = 0, 0, 0, 1, 0, 0, 1 the stabilizer is a 1-dimensional group and consist of matrices 1 α 0 St 5 = α 1 2 α2 1, and for 6-jet p 60 = 0, 0, 0, 1, 0, 0, 1, 0 the stabilizer is trivial. 5. Projective Invariants 5.1 Relative Invariants As we have seen, functions where P 2 = u 2, P 5 = u 5 5u 3u u 2 9 determine singular orbits Π 2 and Π 5. Therefore, they are relative invariants of the SL 3 R- action. Indeed, it is easy to check that X 2 P 2 = α 2 X P 2, u 3 3 u 2 2 X = 2 a 1,1 x + a 2,2 x + a 1,2 u + a 1,3 a 3,1 x 2 a 3,2 xu x + a 1,1 u + 2 a 2,2 u + a 2,1 x + a 2,3 a 3,1 xu a 3,2 u 2 u is a general element of Lie algebra sl 3 R, and α 2 X = 3 a 1,2 a 3,2 x u 1 3 a 1,1 + 3 a 3,1 x the corresponding 1-cocycle. Also X 5 P 5 = α 5 X P 5, where α 5 X = 6 a 1,2 a 3,2 x u a 3,2 u 9 a 1,1 + 9 a 3,1 x 3 a 2,2. It is also easy to see that the function P 7 is a relative invariant. Indeed, X 7 P 7 = α 7 X P 7, 254

15 where α 7 X = 32 a 1,2 a 3,2 x u 1 40 a 1, a 3,1 x 8 a 2,2 + 8 a 3,2 u. Cocycles α 2, α 5 and α 7 are not independent, and we have 16α 2 + 8α 5 3α 7 = 0. Another relative invariant we can get from the volume form Ω = dx du because X Ω = α 0 X Ω, where and α 0 X = 3 a 1,1 + 3 a 2,2 3 a 3,1 x 3 a 3,2 u, α 0 2α 2 + α 5 = 0. The last relative invariant we get from the contact form ω = du u 1 dx. In this case X 1 ω = α 1 X ω, where 1-cocycle α 1 has the following form and α 1 X = a 1,2 a 3,2 xu 1 + a 1,1 + 2 a 2,2 a 3,1 x 2 a 3,2 u, 2α 0 3α 1 + α 2 = 0. These relations between 1-cocycles allow us to construct the following invariant tensors. Theorem 5.1. The following tensors on jet spaces are SL 3 R-invariants: Function Differential 1-form Differential 2-form Q 7 = P 7 P 8/3 5 P 16/3 2 ω 5 = P 2/3 5 P 5/3 2 C π 1 7,6 Π 60. ω Ω 1 Π 50. Ω 5 = P 5 Ω Ω 2 Π P

16 5.2 Algebra of projective differential invariants Let s denote by τ k and ν k the vector bundles on J k induced by projection π k,1 from the canonical bundles τ 1, ν 1 on J 1, where and τ 1 [L] 1 a = Ta L, ν 1 [L] 1 a = Ta P 2 T a L. As we have seen symmetric differential forms Θ 5 = dx 2 Θ 2 = u 2 u S 2 τ2 ν 2 2! u 5 5u 3u u 2 9 u 3 3 u 2 2 dx 5 u S 5 τ5 ν 5 5! are SL 3 R-invariants. Remark that all bundles τ k and ν k are 1-dimensional. Therefore, there exists a symmetric 3-form σ S 3 τ5 such that Θ 5 = 60 σ Θ 2. We call σ as Study 3-form. This form is obviously SL 3 R-invariant and in affine coordinates can be written as follows σ = P 5 P 2 dx 3. In addition to Study form we introduce a Study derivation as a such total derivation that σ,, = 1. Once more, in affine coordinates this derivation has the form = P 1/3 2 P 1/3 5 d dx. This is SL 3 R-invariant derivation. It is easy to check that invariant Q 7 is an affine function in u 7 having the form Q 7 = P 2/3 2 P 5/3 5 u 7 +. Applying the Study derivation we get an 8-th differential invariant Q 8 = Q 7 = P 2 u P ,

17 and Q 9 = Q 8 = P u P Continue in this way we get differential invariants in each order k 7 : Q k = k 7 Q 7 = P k u P k 2 k These relations show that differential invariants Q 7,..., Q k, separate SL 3 R-orbits in J k if their projection to J 6 coincides with regular orbit Π 60. We call such orbits regular. Let us specify now the notion of differential invariant for this SL 3 R- action. First of all remark that all bundles π k,k 1 : J k J k 1 are affine, when k 2. Therefore, we can talk about functions which are polynomial, or algebraic in derivatives u k, k 2. We say that a function f defined in open and dense domain in manifold k-jets J k is a SL 3 R-differential invariant or simply projective differential invariant of order k if X k f = 0, for any vector field X sl 3 R, and function f is a polynomial with respect to u σ, σ 2, and P ±1/3 2, P ±1/3 5. Theorem Any projective differential invariant of order k is a polynomial of invariants Q 7,..., Q k. 2. The algebra differential invariants separates regular orbits. Proof. We ll use induction in k. Let Q be a differential invariant of order k which is a polynomial of degree n in u k. Then, due to fact that Q k is a polynomial of degree 1 in u k, we can represent Q in the following way Q = a n Q n k + a 1 Q + a 0, where a i are functions on k 1-jets. Applying vector fields X sl 3 R to both sides of this relation, we get X k 1 a n Q n k + + X k 1 a 1 Q k + X k 1 a 0 = 0. Therefore, X k 1 a i = 0 for all i = 0,..., n, X sl 3 R, and we can use induction. 257

18 6. Projective equivalence of plane curves 6.1 SL 3 R- action Let L and L be smooth plane curves, and let L k, L k J k be their prolongations. We say that L and L are projectively equivalent if g L = L, for some element g SL 3 R. Let s Q k L = Q k L k be the value of invariant Q k on the curve L. Function Q 7 L is called projective curvature of the curve. We will consider such curves L that function Q 7 L is a local coordinate on it. It is equivalent that function Q 7 = Q 8 does not vanish on L. To distinguish this situation we say that curve L is a regular at point a L, if its 5-jet belongs to the regular orbit, [L] 5 a Π 50, and Q 8 L does not vanish at point a L. Therefore, in a neighborhood of a regular point function Q 7 L is a local coordinate and Q 8 L = Φ Q 7 L for some smooth function Φ. We call this function defining function of the curve. We say that two plane curves L and L are projectively equivalent at points a L and ã L if there exist a projective transformation φ such that φ a = ã and image φ L of curve L and L coincide in a neighborhood of point ã. Theorem 6.1. Two plane curves L and L are projectively equivalent in neighborhoods of regular points a L and ã L if and only if Q 7 L a = Q 7 L ã and their defining functions coincide in a neighborhood of point Q 7 L a. Proof. The necessity condition is obvious. Let s prove the sufficiency, and let Φ be the defining function. Consider ordinary differential equation Q 8 Φ Q 7 = of the 8-th order. Curves L and L are local solutions of this equation. Relation 5.1 shows that solutions of the above differential equation are uniquely defined by their 8-jets. Values of Q 7 and Q 8 on 8-jets [L] 8 a and [ L] ea 8 equal and therefore these jets belong to the regular orbit. Therefore, φ 8 [L] 8 a = [ L] 8 ea, 258

19 for some projective transformation φ. Moreover, projective transformations are symmetries of differential equation 6.1. Hence, φ L is a solution 6.1 too. But 8-jets of L and φ L at point ã equal. Therefore, due to uniqueness of solutions, L = φ L. Theorem 2 allows us to get normal forms of plane curves in a neighborhood of regular point. Theorem 6.2. Let L be a plane curve and let a L be a regular point. Then there are affine coordinates x, y in a neighborhood a L, such that xa = y a = 0, and curve L is given by equation y = Y x, where Y x = x x k 7x k k 7k k 10 and k i are values of Q i L at point a. 35 k 8 x 8 + x 10 +, k k 9 x 9 + Remark 6.3. If Φ is the defining function of the curve then coefficients k i can be computed as follows: k 8 = Φ k 7, k 9 = Φ k 7 Φ k 7, k 10 = Φ k 7 Φ 2 k 7 + Φ k 7 2 Φ k Cubics As an example of application of the above theorem let s consider cubic curves. As we have seen these curves are solutions of equation u 2 P 7 u 9 + P 8 = 0. The left hand side of the equation is an obviously relative invariant. This invariant can be written in terms of the known invariants as follows: P 2 P Q 7 Q Q Q Q Therefore, if the cubic curve is an irreducible, i.e. is not union of quadric and straight lines, then this curve satisfies the 9-th order differential equation Q 7 Q Q Q Q The leading term of the last equation has the form = Q 7P u P

20 Singularities for this equation in regular orbit belong to hyper surface Q 7 = 0. In other words, on cubic curves we have three types of singularities: general singular points: 5-jet of the curve belongs to a singular orbit, non regular points: invariant Q 8 vanishes, projectively flat points: projective curvature Q 7 vanishes. Theorem 6.4. Two plane connected cubic curves are projectively equivalent if and only if there are regular points on them where projective curvatures and their Study derivatives coincide. Proof. The proof of local projective equivalency is similar to proof of theorem 5. The rest follows from the fact that cubics are algebraic curves. Remark 6.5. This result can be reformulated as follows. Let L be a plane curve. Define a new plane curve I L as a image of the map a L Q 7 L a, Q 8 L a I L, and let I 0 L I L is the image of regular points. Then the above theorem claims, that two connected cubics L and L are projectively invariant if and only if I 0 L = I 0 L. 7. Special curves on projective plane 7.1 W -curves Two classes of projective curve we get from description of singular orbits: straight lines and quadrics. The third class come from the description of singularities for cubics: these are curves of zero projective curvature, or solutions of differential equation of 7-th order: Q 7 = 0. This class is a subclass of so-called W -curves, introduced by Lie and Klein in [4]. Namely, straightforward computations show the following lemma is valid. Lemma 7.1. SL 3 R-orbits are transversal to curve L 8 at points where Q

21 Therefore, curves L for which SL 3 R-orbits of prolongations L 8 have dimension less then 9 should be solutions of the differential equation Q 8 = 0. In other words, such curves have constant projective curvature. They are called W -curves see, [3]. It follows from the above lemma that trajectories of vector fields from our Lie algebra sl 3 are W -curves. Counting dimensions shows that the dimension of the space of solutions passing through a point a P 2 for the differential equation Q 8 = 0 at regular point a P 2 equals 7 that coincide with dimension of nonparametric trajectories of vector fields from sl 3. The 3-rd description of W -curves was proposed by Klein and Lie. Namely, let s take three straight lines l 1, l 2 and l 3 on projective plane which are in general position and a plane curve L. Considering tangent line T a L as a straight line on the plane, we define a number j L a to be equal the value of j-invariant for 4 points [a, T a L l 1, T a L l 2, T a L l 3 ] on T a L. Consider now such curves for which j-number j L is a constant. They are also W -curves. We have the following equivalent description of regular W -curves see also [3]. 1. W -curves are solutions of 8-th order differential equa- Theorem 7.2. tion Q 8 = W -curves are non parametrized trajectories of vector fields from the projective Lie algebra sl W -curves are curves of constant projective curvature. 4. If projective curvature Q 7 L of a curve L is a constant and equal k 0, k then the j-number of this curve equal 3 675, where λ =. k 3 λ Corollary 7.3. If a regular cubic is a W -curve then its projective curvature equals , and j-number equals 7.2 Study extremals Another class of curves we obtain from the Study differential. To this end let s consider the following functional L 3 σ, L 261

22 or in affine coordinates b 3 9 y 5 y y y 2 y 3 y 4 y x dx, a assuming that L is not a straight line, and call it Study functional. Straightforward computations show that the following theorem is valid. Theorem 7.4. Extremals of the Study functional are solutions of the following differential equation of the 10-th order: P 5 P2 2 y 2 Q 10 24Q 7 Q 8 = Corollary 7.5. W -curves are extremals of the Study functional. Corollary 7.6. If a regular cubic L is an extremal of the Study functional then L is the W -curve. 8. Defining functions In this section we consider a behavior of defining functions for cubics and Study extremals. It is worth to note that for straight lines, quadrics and W -curves the defining functions do not exist. 8.1 Cubics Let Φ be the defining function of a cubic L considered in a neighborhood of regular point. Then, applying the Study derivative to the relation Q 8 = Φ Q 7, we get Q 9 = Φ Q 7 Φ Q 7. Relation 6.2 can be rewritten now as a differential equation for defining function Φ τ: τ τφφ Φ Φ = 0. Integrating this equation we get the following relation between invariants Q 7 and Q 8 which depends on arbitrary constant c and has the following form where F = F 3 + cgq 7 9 = 0, Q Q Q Q Q Q Q Q

23 and G = Q Q Q Q Q 7 3 Q Q 7 3 Q Q Q 8 4. In other words, regular cubics are projectively defined by constant c. 8.2 Study extremals Rewriting Euler equation 7.1 in terms of defining function we get the following differential equation Φ 2 Φ + ΦΦ 2 24τΦ = 0. Integrating, we get the following relation between invariants Q 7 and Q 8 : Q 2 8 8Q 3 7 c 1 Q 7 + c 2 = 0, 8.3 which depends on two arbitrary constant c 1 and c 2. Summarizing, we arrive at the following description of non singular cubics and Study extremals. Theorem 8.1. Projective classes of regular cubics are defined by an arbitrary constant c, where F 3 + cgq 7 9 = 0, and expressions for invariants F and G are given in 8.1 and 8.2. Projective classes of regular Study extremals are defined by relation 8.3 depending on two arbitrary constants c 1 and c 2. References [1] J. L. Coolidge, A treatise on algebraic plane curves, Dover Publ., NY, 1959 [2] G. H. Halphen, Sur les invariants différentiels, Paris: Gauthier-Villars, 1878 [3] F. Klein, W. Blaschke, Vorlesungen über höhere Geometrie, Berlin, J. Springer,

24 [4] F. Klein, S. Lie, Uber diejenigen ebenen Curven, weiche aurch ein geschlossenes System von einfach unedlich veilen vertauschbaren linearen Transformationen in sich ubergehen, Math. Annalen, Vol. 4, pp , [5] E.P. Lane, Projective differential geometry of curves and surfaces, The University of Chicago press, Chicago-Illinois, 1932 [6] A. Lascoux, Differential equations of a plane curve, Bull. Sc. Math., Vol. 130, pp , [7] E.J. Wilczynski, Interpretation of the symplest integral invariant of projective geometry, Proc. Nat. Acad. Sci. USA, Vol. 2, No. 4, pp ,