(Σ 1,0,Σ 1,0 ) (Σ 1,0,Σ 1,0,Σ 1,0 ) (Σ 1,1,0, Σ 1,0 ) N 2 F(Σ 1,1,0 ) N 3 F(Σ 1,0 )

Transcription

1 FELIX KLEIN'S PAPER ON REAL FLEXES VINDICATED Felice Ronga Universite de Geneve Abstract. In a paper written in 876 [4], Felix Klein gave a formula relating the number of real exes of a generic real plane projective curve to the number of real bitangents at non real points and the degree, which shows in particular that the number of real exes can't exceed one third of the total number of exes. We show that Klein's arguments can be made rigourous using a little of the theory of singularities of maps, justifying in particular his resort to explicit examples x. Introduction Recently, there has been a renewed interest for enumerative problems over the reals in algebraic geometry [], which were abandoned after a few attempts by the founders of modern algebraic geometry. In general, in algebraic geometry, the number of solutions of an enumerative problem over the reals is bounded by the number of solutions over the complex. So, two natural questions arise : () Is it possible to arrange that all the solutions are real If it is the case, the problem is called fully real in [6]. (2) Which intermediate number of solutions can be obtained Examples. i) According to Bezout's theorem, two plane curves in general position, of degree m and n respectively intersect in m n points. By taking curves consisting of lines, some of them real, some pairwise complex conjugate, one can realise any number of intersection points of the form m n 2k ; with 2k m n ii) There are 27 lines on a smooth, complex cubic surface. Over the reals, there can be 27,5,7 or 3 lines [2]. iii) Using Weierstrass form y 2 x(x )(x ) = for the equation of a smooth cubic plane curve, one sees easily that there are always exactly 3 real exes, whereas there are 9 over the complex. Inspired by a paper of Zeuthen [], F. Klein [4] proves that for a real, smooth, generic curve of degree n, one has : Theorem I. i + 2t = n(n 2) where i denotes the number of real exes and t the number of real bitangents at a pair of complex conjugate points (or bitangents of type t for short). In particular, there are at most n(n 2) real exes, whereas there are 3n(n 2) over the complex. Klein also gives a formula for generic curves with ordinary cusps and double points, that we won't consider here. Klein's method consists in deforming a given curve into a smooth, generic one with n(n 2) exes and no bitangent of type t. During the deformation, non generic curves are encountered, in the neighbourhood of which i and t may change. He proves that they do change only when passing through a curve with a February 26, 997 Typeset by AMS-TEX

2 2 FELICE RONGA degenerate ex, a point at which the tangent has multiplicity 3. When crossing the stratum of such curves, on one side there are curves having a real line bitangent at two close real points, between which there are 2 real exes. Coming close to the stratum, the two tangency points and the two exes come to coincide, and when moving to the other side, the two exes disappear, and the 2 points of tangency become imaginary (an explicit example of such a deformation can be obtained by taking the images by the map (x; y) 7! (x 2 ; y) of circles of diameter centered at the point (t; ), when t varies from =4 to 3=4 { see gure ). During such a deformation, i decreases by 2 and t increases by, and therefore i + 2t remains unchanged. This is in fact the key argument. Figure. Two exes disappear and a bitangent of type t appears The curve with the maximal number of real exes is obtained as a small deformation of a curve consisting of conics if n is even, conics plus a cubic if n is odd. In [8], O. Ya. Viro proves a generalized Klein's formula using integration with respect to Euler characteristic, and a similar formula is proved in [9] by C.T.C. Wall, using in an essential way transcendental topology. Klein's approach is still of interest, because it provides examples of curves where the maximal number of n(n 2) real exes is reached. In addition, his method can be used to show : Theorem II. Curves with any admissible number of real exes exist : i if n is even i = n(n 2) 2k ; i n(n 2) ; i 3 if n is odd Our goal is to interpretate Klein's arguments in terms of assertions on the singularities of the universal map naturally associated to the incidence relation of a curve of degree n and a line (see x 2). In particular, the key argument reduces essentially to observe that on the celebrated swallow-tail singular locus ([7], page 82 and our gure 2), the curve of double points has two parts, one where 2 real sheets of the surface intersect, and another, which is a continuation of the previous, but is usually not drawn, where 2 imaginary sheets of the surface intersect; the 2 parts are separated by the most singular point of the surface, where the curve of double points and the cuspidal edges meet. x. First strata We begin by recalling some general facts about the singularities of maps that we shall use. We refer to [3] for the material needed, but will recall all the basic facts. The singular loci k. We work simultaneously in the real and complex case. Let U C n be an open subset and f : U! C n be a holomorphic map, or U R n and f : U! R n a C map. We set (f) = fx 2 U j dim (Ker(df x )) = g : If x 2 (f), using the implicit function theorem we may assume that after a change of coordinates in the surce and target of f, it can be written : f(x ; : : : ; x n ) = (x ; : : : ; x n ; g(x ; : : : ; x n ))

3 FELIX KLEIN'S PAPER ON REAL FLEXES VINDICATED 3 N 2 (Σ,,Σ, ) N 2 (Σ,,Σ, ) F(Σ,,, ) F(Σ, ) N 3 (Σ,,Σ,,Σ, ) F(Σ,, ) N 2 (Σ,,, Σ, ) F(Σ,, ) Figure 2. A freehand sketch of the swallow tail and the interpretation of the various strata in terms of the bers of the n (x ) =. We shall say that f is -transversal if Note that the equation of (f) near x is 9 i 2 f; : : : ; ng n (x) 2 n (x ) 6= : and therefore -transversal means that the above equation is of maximal rank, and hence it denes a smooth hypersuface near x. We dene now the notion of k-transversal and the ag of subvarieties k (f) k (f) (f) U where the symbol k is an abbreviation for ;:::;. We set k (f) = {z} k x 2 (x) 8 ` = : : : ; k and say that f is k (f)-transversal if the above set of equations is of maximal rank. Note that Ker (df x ) equals the x n -axis, and 2 (x ) = if and only if n dim Ker d f (f) =, and similarly k (f) = n x 2 k (f) dim Ker d f o k (f) = :

4 4 FELICE RONGA Finally, we set k; (f) = k (f) n k+ (f) If f : X n! Y n is now a proper C or holomorphic map between smooth manifolds, the singular loci k(f) X and the k-transversality can be dened with the previous denitions in local coordinates. If f is k-transversal for all k, then we have a ag of smooth subvarieties : n (f) n (f) (f) X with k(f) a smooth submanifold of codimension k of X, k(f) = ; for k > n and : k (f) = x 2 k (f) j dim Ker d f k (f) x = : A more formal and conceptual approach can be found for example in [3], Part II, x 3 (our k; is written S k there). It was proved by B. Morin (see [3], Part II theorem 4.), that if f : X! Y is k-transversal and x 2 k; (f), then there exist coordinates systems on X and Y, sending x and f(x ) to the origin, in which f has the form : f(x ; : : : ; x n ) = x ; : : : ; x n ; x k+ n + x k xn k + x k2 xn k2 + + x x n Self intersections : the singular loci M h f; k ; ; : : : ; k h ;. Let f : X n! Y n be transversal to k(f) for all k. We set M h f; k ; ; : : : ; k h ; = n x 2 k ; (f) 9 x 2 2 k 2 ; (f); : : : ; x h 2 k h ; (f); with x; x 2 : : : ; x h all dierent and f(x) = f(x 2 ) = = f(x h ) and shall say that f is M h k ; ; : : : ; k h ; transversal if the vector spaces Im df x T k ; x ; : : : ; Im df xh T k h ; x h are in general position in T W f (x) ; that is, their intersection has codimension k + + k h. More notation : we set N h f; k ; ; : : : ; k h ; = f M h f; k ; ; : : : ; k h ; and denote by M h M h f; k ; ; : : : ; k h ; the subset of M h f; k ; ; : : : ; k h ; consisting of points that do not belong to higher self-intersections : n f; k ; ; : : : ; k h ; = x 2 k ; (f) 9 x 2 2 k ; 2 (f); : : : ; x h 2 k ; h (f);. Remark. Consider a map of the form : with x; x 2 : : : ; x h all dierent, f(x) = f(x 2 ) = = f(x h ) and f(x ) = f(x); x 6= x; x ; : : : x h =) df x is bijective f : U! R n+k ; f(x; u) = (g(x; u); u) where x 2 R n, u 2 R k, g : U! R n, and set g u (x) = g(x; u). It is immediate that if, for a xed u, g u is h transversal in a neighborhood of x for h k, then f is h-transversal in a neighborhood of (x; u) for h k, and that h (g u ) = h (f) \ (R n fug) o o

5 FELIX KLEIN'S PAPER ON REAL FLEXES VINDICATED 5 in a neighborhood of x, and a similar remark holds for the self-intersections M h k ; ; : : : ; k h ;..2 Example. All the types of singularities and their mutual positions that we shall need are those of codimension at most 3; they can all be contemplated on the following example. Let We have df = ; (F ) x 3 3 x 3 4x 3 + 2x 3 x + x 2 F (x ; x 2 ; x 3 ) = (x ; x 2 ; x x 2 3 x + x 3 x 2 ) : (x ; x 2 ; x 3 ) 2 (x x 2 3 x + x 3 x A ; (F ) = (x ; x 2 ; x 3 ) j x 2 = 4x 3 3 2x 3 x = = (6x 2 3 ; 8x3 3 ; x 3) ; ;;; (F ) = fg : (F ) can be parametrized by : (x ; x 3 ) 7! (x ; 4x 3 3 2x 3x ; x 3 ), and F (F ) can be parametrized by : (x ; x 3 ) 7! (x ; 4x 3 3 2x x 3 ; 3x 4 3 x x 2 3) and from there one can calculate that M 2 (F; ; ; ; ) = ( (x ; ; x 3 ) j x 6= ; x = 2x 2 3 ), and so its closure M 2 (F; ; ; ; ) can be parametrized by x 3 7! (2x 2 3; ; x 3 ). The restriction of F to M 2 (F; ; ; ; ) composed with the latter parametrisation reads : Also, F ; (F ) is parametrized by x 3 7! (2x 2 3; ; x 4 3) x 3 7! (6x 2 3 ; 8x3 3 ; 3x4 3) For a complete picture, we still need to produce the singular strata M 2 ( ;; ; ; ) and M 3 ( ; ; ; ; ; ). This can be done by taking the disjoint union of F and the map G(u ; u 2 ; u 3 ) = (u 2 ; u 2; u 3 ). The only singular locus of G is ; (G) = f(; u 2 ; u 3 )g, whose image G( ; (G)) = f(; u 2 ; u 3 )g meets transversally F ( ;; (F )) and M 2 ( ; (F ); ; (F )) at (; 8(= p 6); = p 6) and (; ; =4) respectively (see gure 2). x 2. The universal incidence relation map for curves of degree n and lines Let C [x ; x ; x 2 ] n and R[x ; x ; x 2 ] n denote the vector spaces of homogeneous polynomials of degree n with complex and real coecients respectively. We denote by : Pn;C = P (C [x ; x ; x 2 ] n ) and Pn;R = P (R[x ; x ; x 2 ] n ) the associated projective spaces. Let P 2 C and P2 R denote respectively the complex and real projective plane. For [f] 2 Pn;C, V (f) = [x] 2 P 2 C j f(x) = denotes the variety dened by f. When we mean either complex or real spaces, we drop the subscripts C and R. Denote by P 2 C, respectively P 2 R the dual projective planes, namely the space of lines of P 2, or equivalently P 2 C = P ;C, P 2 R = P ;R. We dene the Zariski open subsets of Pn;C and Pn;R respectively : P n;c = [f] 2 Pn;C V (f) does not contain a line ; P n;r = P n;c \ Pn;R : For n 3, P n;r is connected, because the subspace of P n;r of curves containing a line has codimension 2 + = n : dim (P n;r ) dim P 2 P (n);r = n(n + 3) 2 (n )(n + 2) 2

6 6 FELICE RONGA We assume henceforth that n 3, without any loss of exes. The universal incidence relation is dened to be the variety : and the universal incidence relation map W = ([x]; []; [f]) 2 P 2 P 2 P n (x) = ; f(x) = : W! P 2 P n is the restriction to W of the natural projection P 2 P 2 P n! P 2 P n. For [x] 2 P 2 xed, the set of equations (x) = ; f(x) = in P 2 P n is of maximal rank, from which it follows that W is a smooth variety. Denote by T : P 2 P n! P n the natural projection and set S = T : W! P n. We have a commutative diagram of maps : W S &! P 2 P n. T P n For [f] 2 P n, set W f = S ([f]) ; f = Wf : W f! P 2 We can view the map as the family of maps f, [f] 2 P n. The ber of f over the line [] is constituted by the intersection V (f) \ V (). Studying the singularities of the map f amounts to study the possible special intersections of a line and the curve V (f) : the theorem below shows that if V (f) is smooth, f ( ( f ) is the dual curve of V (f), N 2 ( f ; ; ) is the set of bitangents, and ; ( f ) identies to the set of exes of V (f). 2. Theorem. the restriction of f to the line ` has a zero ([x]; []; [f]) 2 k; () () of multiplicity exactly k + at x The map is transversal to the k 's and to the M h ( k ; : : : ; k h )'s. Proof. We may alternatively consider as a family of maps ` : W`! P n, where W` = f`g P n parametrized by ` 2 P 2 : W! P 2 P n S &. T P 2 W` is smooth, since the natural map W`! P 2 is a ber bundle. We will show that for each `, ` has the required transversality properties and interpretation of the singular loci, and the theorem will follow from remark.. Note that for xed `, ` is an extension of the universal family of polynomials of degree n on the line `, as shown by its local expression ~, and it is therefore no surprise that it satises the required transversality properties. Anyway, we give a formal proof now. Let z = ([x ]; `; [f ]) 2 W. First we introduce local coordinates on the various projective spaces and on W. We may assume that [x ] = [ : : ], [ ] = [ : : ]. Take ane coordinates (u; v) 7! [u : v : ] near x ; f can be written X f (u; v) = a i;j ui v j : i;jn Since f (x ) =, a ; = ; since f doesn't contain the line f[u : : ]g dened by, we have : f (u; ) = a k; uk+ + X k+<hn a h; uh with a k; 6=, for some k n ;

7 FELIX KLEIN'S PAPER ON REAL FLEXES VINDICATED 7 and so we may assume that a k+; =. The (a i;j) i;jn;(i;j)6=(k+;) are ane coordinates on Pn near [f ], that we will write as follows : [a ; ; ; (a i;j ) (i;j)2i ], where I = f(i; j) j i; j n; (i; j) 6= (k + ; ); (; )g. Dene : X F (u; v; (a i;j ) (i;j)2i ) = u k+ + a i;j u i v j (i;j)2i so that, if for u ; v xed we set a ; = F (u ; v ; a i;j ), the X polynomial f(u; v) = a ; + u k+ + a i;j u i v j (i;j)2i vanishes at (u ; v ). The map u; (ai;j ) (i;j)2i 7! [u : : ]; [F (u; ; ai;j ); ; (a i;j ) (i;j)2i ] denes local coordinates on W` near z, using which ` reads : X ~ ` loc : u; (a i;j ) (i;j)2i 7! u k+ Set Since and g u; (a i;j ) (i;j)2i = u k+ in i6=k+ a i; u i ; (a i;j ) (i;j)2i X in ; i6=k+ Ker (d loc (;;a i;j )) = f(; ; ; )g a i; u g (;;a i;j ) = `! ; ` k k+ k+ (;;a i;j ) = (k + )! we see that [x ]; `; [f ] 2 k; (`) and that ` is k; transversal. It follows also that 8 < X T k (f) = z : f = uk+ + a h; u h + k<hn X in;<jn a i;j u i v j 9 = ; which is the space of polynomials in two variables u; v of degree n, with a k+; =, whose restriction to ` vanishes of order at least k at x, and transversality to M h (f; k ; : : : ; k h ) follows from the fact that given distinct x ; : : : ; x h 2 C and positive integers k ; : : : ; k h such that k + + k h + n, the linear conditions imposed on polynomials of degree n in one variable to have roots of multiplicity at least k i + at x i, i = ; : : : ; h are linearly independent. 2.2 Corollary. There exists a non-empty Zariski open subset n;c P n;c such that for f 2 n;c, f : W f! P 2 presents only the singularities ;, ;;, M 2 ( ; ; ; ), to which it is transversal. Proof. For a smooth subvariety X W, denote by S(X) crit the set of non regular values of the map SjX : X! P n;c. Set K = S ; () crit [ S ;; () crit [ S ;;; () crit [ [ Then K P n;c. For the real case, we set <h<n <k ;:::;k h <n is algebraic of codimension at least, and we may take n;c = P n;c n K n;r = n;c \ P n;r : S M h ; k ; ; : : : ; k h ; crit : The set K in the proof of the corollary above is the "catastropy set" of the family of maps f : W f! P 2 P n, and n is the set of "generic curves". For our purposes, we need to analyse more in detail the set K. We do this in the next paragraph.

8 8 FELICE RONGA x3 Controlling flexes and bitangents Let WR = W \ P 2 R P 2 R P n;r, R = jwr : WR! P 2 R P n;r. Consider the map R : ; (R)! P n;r which is the restriction to ; (R) of the projection of SR : WR! P n;r. The bers R ([f]) identify to the set of real exes on V (f), which might be degenerated. The restriction of R to R ( n;r) is a covering map, but n;r is not connected. We can view as the "control map" for real exes; we must analyse how the bers of change as we move from one connected component of n;r to another. Note that it follows from 2. that ; () is smooth. For the next propositions we work simultaneously in the real and complex case again. Let [x] 2 P 2 be an ordinary double point on the curve dened by f, [`] a line through [x] simply tangent to one of the 2 branches of the curve through x. Then the intersection of ` and V (f) at [x] has multiplicity exactly 3, and so ([x]; `; [f]) 2 ;; (). We denote by the set of such ([x]; `; [f]) and by its closure in W. 3. Proposition. Let z = ([x]; `; [f]) 2 ; () () d z is not bijective () z 2 [ ;; () (2) If z 2, can be written in suitable local coordinates near z as (z ; : : : ; z N ) 7! (z 3 ; z 2 ; : : : ; z N ), and therefore in the real case it is a local homeomorphism at those points. We rst need a lemma, that we work out over the real, leaving to the reader to check the validity of the complex analogue, in which case C must be replaced by holomorphic. Let U R N be open and : U! R N a C map. For x 2 U, the k-th derivative of at x is a homogeneous form of degree k : d k x : R N! R N : Consider the composition : d k e x : Ker(d x ) R N dk x! N R! R N =Im(d x ) = Coker(d x ) : It is readily checked that if d h e x =, < h k, then d k+ e x is intrinsic, which means that it is aected only by the linear part of the local dieomorphisms near x and (x); the conditions d h e x =, < h k also are preserved by coordinate changes. This ensures that the hypothesis in the next lemma are preserved by coordinate changes. In fact, d k e x is a restriction of the k-th intrinsic derivate introduced by Porteous [5]. 3.2 Lemma. Let U R N be open and : U! R N be a C map. Assume that () () = fx 2 U j dim (Ker (d x )) = g is smoooth in a neighborhood of x 2 (). (2) d h e x =, h k, x 2 (), and d k+ e x 6=,. Then there exist coordinate systems near x and (x ) in which reads : (x ; : : : ; x N ) 7! (x k+ ; x 2 ; : : : ; x N ) Proof. We may assume that x = and that (x ; : : : ; x N ) = (g(x ; : : : ; x N ); x 2 ; : : : ; x N ). If x 2 (), Ker(d x ) = f(; ; : : : ; )g, Coker(d x ) ' f(; ; : : : ; )g and d h e x identies h (x). Let u(x) be a local equation of () near. k (x) = for x 2 (), there exists a fonction v(x) in a (x) = u(x)v(x), and g (x) v(x) + Evaluating at neighboorhood of such k x = shows k+ () v(), which () 6=. We can therefore choose x = u(x)

9 FELIX KLEIN'S PAPER ON REAL FLEXES VINDICATED 9 as a new rst coordinate. In the new coordinates, again denoted by x ; : : : ; x N, the local equation of () near is x = and h h (;x 2;:::;x N ) = ; h k k+ () 6= k+ Therefore g(x ; : : : ; x N ) = x k+ g (x ; : : : ; x N ), with g () 6=. After changing perhaps the sign of the rst coordinate in the target, we may assume that g () >. If we take the new coordinate then has the desired form. x = x (g (x ; : : : ; x N )) =k proof of 3.. Let E = R 2 or C 2, ` E be a line and let ` be distinct from `, ` \ ` = fx g. Then the lines in a neighborhood of ` can be viewed as graphs of ane maps ` : `! `. Any vector x 2 E can be uniquely decomposed as x = x + x, where x 2 `, x 2 `. Denote by i` : `! E the natural inclusion and by L(`; `) the space of ane maps from ` to `; then the line corresponding to ` 2 L(`; `) is the image of the map i` + ` : `! E, and x 2 E is on that line if and only if x = `(x ). Let z = ([x ]; `; [f]) 2 W. Using the coordinates E L(`; `) U, where U is some ane coordinate system containing [f], we have the following equation for our various spaces : 9 = x `(x ) = f(x) = ; W df x (i` + `) = 9 >= >; d 2 f x (i` + `; i` + `) = 9 >= >; ; : Denoting tangent vectors with overlined symbols, we have the following equations for the corresponding tangent spaces at z : 9 = x `(x ) = () f(x ) + df x (x) = ; T W z df x (i`) + d 2 f x (x; i`) + df x (`) = 9 >= d 2 f x (i`; i`) + d 3 f x (x; i`; i`) + 2d 2 f x (i`; `) = >; T z and Ker(d z ) is thet set of tangent vectors to ; of the form (x; `; ), that is : Ker(d z ) = (x; `; ) j 9 >= >; T ; z x `(x ) = ; df x (x) = ; d 2 f x (x; i`) + df x (`) = ; d 3 f x (x; i`; i`) + 2d 2 f x (i`; `) = : If df x 6=, then df x (x) = =) x 2 ` =) x = =) `(x ) = and df x (`) = =) ` =. But then d 3 f(x; i`; i`) = with x 2 ` not zero is possible only if z 2 ;; (), that is d 3 f x (i`; i`; i`) =. If x is an ordinary double point on f, then df x =, and if ` is simply tangent to one of the branches through x, then d 2 f x (x; i`) = =) x 2 ` =) `(x ) = and d 3 f(i`; i`; i`) 6=. Thus : Ker(d z ) = (x; `; ) j x 2 ` ; `(x ) = ; d 3 f x (x; i`; i`) + 2d 2 f x (i`; `) = : and the projection (x; `) 7! x induces an isomorphism from Ker(d z ) to `. If f 2 Im(d z ), then it follows from equation () above that f(x ) =. But since dim(ker(d z )) =, we have exactly : Im(d z ) = f j f(x ) =

10 FELICE RONGA and so the intrinsic derivatives of at z identify with the ordinary derivatives of f at x restricted to ` : (x; `) 2 Ker(d z ) ; d h e z (x; `) = d 3 f x (x) : Therefore d 2e z =, d 3e z 6= and we can apply lemma 3.2. Let M 2 () = M 2 (; ; ; ; ) be the closure of M 2 (; ; ; ; ) in W. As "control map" for bitangents we take the restriction : M 2! P n of the natural projection S : W! P n. M 2 is not smooth, so in this case we must understand where it is not smooth and near which smooth points it might not be a local homeomorphism. 3.3 Proposition. There is a decompositon M 2 () = M 2 (; ; ; ; )[M 2 (; ;; ; ; )[M 2 (; ; ; ;; )[M 3 (; ; ; ; ; ; )[ ;;; ()[K 2 where K 2 is an algebraic subset of codimension 2 in M 2, and all the other strata, except M2 (; ; ; ; ), have codimension. M 2 is smooth at points of M2 (; ; ; ; ) [ ;;; [ M2 (; ;; ; ; ) and singular at points of M2 (; ; ; ;; ) [ M3 (; ; ; ; ; ; ) Let z = ([x]; `; [f]) 2 M2 (; ; ; ; ) [ ;;; [ M2 (; ;; ; ; ). Then d z is not bijective () z 2 ;;; () or x is singular on V (f) (i.e. (z) 2 ()) Proof. The expression of the closure of M2 (; ; ; ; ) and smoothness at the points stated are general facts for maps with the transversality properties proclaimed in 3.. One can check them on the explicit example.2, where K 2 is empty. If z 2 ;;; (), Ker(d z ) T M 2 () z. Since is the composition M 2 ()! W! T P n, Ker(d z ) Ker(d) z, and so d z is not bijective. Let z = ([x ]; `; [f]) 2 M2 (; ;; ; ; ), (z ) = (z ), z = ([x ]; `; [f]) 2 ; ), [x ] 6= [x ]). We will use the local coordinates on P 2 P 2 P n, notation and expressions of tangent spaces to the various strata introduced in the proof of 3.. Let z = (x; ; ) 2 Ker(d z ) \ Ker(d z ); since z 2 ;;, z 2 T ( ; ) z and d z (z) =, and since T (M 2 ) z = d( j ()) z dz T ( ; ()) z, we have that z 2 T (M 2 ) z, but z 2 Ker(d z ). If z = ([x ]; `; [f]) 2 M2 (; ; ; ; ), z = ([x ]; `; [f]) 2 ; ), [x ] 6= [x ], then (z ) = (z ) and and Ker(dS z ) \ T ( ) z = (x; `; ) j x 2 ` ; `(x ) = ; d 2 f x (x; i`) + df x (`) = Ker(dS z ) \ d( j ()) z dz T ( ; ()) z = n(x; `; ) x 2 ` ; `(x ) = `(x ) = ; d 2 f x (x; i`) + df x (`) = but `(x ) = `(x ) = =) ` = =) d 2 f x (x; i`) = =) x =, which proves that d z is injective. If z = ([x ]; `; [f]) 2, z = ([x ]; `; [f]) 2, [x ] 6= [x ]), and df x =, then z 2 M 2 ; assume that z 2 M 2 (; ; ; ; )[M 2 (; ;; ; ; ). Then (; `; ) j `(x ) = T ( ()) z \Ker(d z ) T (M 2 ) z, and so (; `; ) j `(x i ) = ; i = ; Ker(d z ). We prove now theorem I. The control map for real exes is R : ; (R)! P n;r, whereas the control map for bitangents of type t is the restriction t : M 2 (t )! P n;r of C, where M 2 (t ) = M 2 (C ) \ P 2 C n P2 R P 2 R P n;r. For [f] 2 n;r, we denote by i(f) the number of real exes on V (f) and by t (f) the number of its bitangents of type t.

11 FELIX KLEIN'S PAPER ON REAL FLEXES VINDICATED 3.4 Theorem. Let [f ]; [f ] 2 n;r. Then i(f) + t (f) = i(g) + t (g) Proof. We can nd a path in n;r that misses the ramication loci of codimension 2 of the control maps R and t, and is transversal to those of codimension, namely : () Both for real exes and bitangents of type t : (2) For bitangents of type t only : R ;;; (R) = t ;;; (R) and R( R) t M 2 (C ; ;; ; ; ) \ M 2 (t ) = t and t M 2 (C ; ; ; ;; ) \ M 2 (t ) M 3 (C ; ; ; ; ; ; ) \ M 2 (t ) : It follows from 3. 2) that R is a local homeomorphism near points of R, so i doesn't change when R( ) is crossed. If z = ([x]; `; [f]) 2 M2 ( ; ; ;; ) and (z) 2, then [x] is a real ordinary double point of f and ` is not tangent to the branches of f through [x], but it is simply tangent to a further real smooth point [x ] 2 V (f). Nearby bitangents will be tangent at points [x ] and [x ], near to [x] and [x ] respectively. Since [x] and [x ] aren't conjugate, neither are [x ] and [x ], and so t doesn't change in the neighborhood of [f]. If ([x ]; `; [f]) 2 M2 (R; ;; ; ; ), let [x 2 ] 6= [x ] be the second point on ` tangent to V (f). Since [x ] is a ex and [x 2 ] is not, they cannot be complex conjugated, and the same holds for nearby bitangents. Therefore t M 2 (C ; ;; ; ; ) \ M 2 (t ) is empty. If ([x ]; `; [f]) 2 M3 (C ; ; ; ; ; ; ), let further ` be simply tangent to V (f) at [x 2 ], [x 3 ], with [x ], [x 2 ] and [x 3 ] all distinct. If [x ] and [x 2 ] are complex conjugate, then bitangents near ` at points near [x ] and [x 2 ] will be of the same type, therefore t doesn't change when crossing t M 3 (C ; ; ; ; ; ; ) \ M 2 (t ). It remains to examine what happens when we cross R ;;; (R). For this we use the explicit example.2 : F (x ; x 2 ; x 3 ) = (x ; x 2 ; x x 2 3 x + x 3 x 2 ) : As explained in x, if z 2 ;;; (), in a suitable system of coordinates around z and (z), the map reads : (x ; x 2 ; x 3 ; u) 7! (F (x ; x 2 ; x 3 ); u) : Recall the parametrisations : and F M 2 (F; ; ) : x 3 7! (2x 2 3; ; x 4 3) : F ; (F ) : x 3 7! (6x 2 3; 8x 3 3; x 3 ) : If we call (X ; X 2 ; X 3 ) the three coordinates in the target of F, we see that : () for c <, the plane X = c cuts F ( ;; (F )) transversally in the 2 distinct real points whose p 3 p p 3 p preimages by F are c; 8 c=6 ; c=6 and c; 8 c=6 ; c=6 ; and it cuts F (M 2 ( ; ; ; )) transversally in one point, with 2 real preimages by F : c; ; p c=2 (2) for c >, the intersection of the plane X = c with F ( ;; (F )) has no real points, and it cuts F (M 2 ( ; ; ; )) transversally in one point, with the 2 imaginary conjugate preimages by F : c; ; p c=2

12 2 FELICE RONGA It suces now to show that the projection of the X -axis (of the target) on P n is transversal to the stratum R( ;;; (R)). We have seen in the proof of 3. that for z = ([x ]; `; [f]) 2 ; () n the tangent space T () z can be expressed as (x; `; f) j x `(x ) = ; f(x ) + df x (x) ; df x (i`) + d 2 f x (x; i`) + df x (`) = If (x; `; ) 2 T () z \ Ker(dS z ), where S : W! P n is the natural projection, then and df x (x) = =) x 2 ` =) x = =) `(x ) = z 2 ; =) d 2 f x (x; i`) = =) df x`) = =) ` = : Therefore T () z \Ker(dS z ) = ` = Ker(d z ). For the map F, a non-zero vector on the x -axis is contained in T ( ), but not in Ker(dF ), and it is not tangent to ; ; the image of the x -axis by df is the X -axis. Therefore the projection the X -axis on P n is transversal to R ;;; (). In conclusion, when crossing this stratum in the direction of increasing X, i decreases by 2 and t increases by, and so i + 2t remains unchanged. It follows from the proofs of propositions 3.2 and 3.3, and of theorem I, that the structure of the control maps R and t near generic points of strata of codimension does not depend on n, for n 3. Therefore, in order to see how i and t change when crossing a stratum of codimension, it is enough to work out explicit examples (a procedure used, or referred to, by Klein, sometimes implicitely). For R( R ), we can use the family of cubics f (x; y) = y 2 (x + ) 2 (x ), near. For R( ;;; (R)), we have to use quartics, because ;;; (R) = ; for cubics; we can take f (x; y) = y 2 x 4 + x 2, near. x 4. Constructing examples of curves 4. Remark. Let [f] 2 P n;r be a real curve, x and x 2 two real distinct ordinary double points on V (f). Assume that the line ` through x and x 2 is not tangent to the branches of f through x and x 2 and that it is not tangent to f elsewhere. Then there exist arbitrary small neighborhoods V of [f] in P n;r, and U ; U 2 of x ; x 2 respectively in P 2 C, with U \ U 2 = ; with the following property : if there exist [f ] 2 V, x 2 U, x 2 2 U 2 such that the line ` through x and x 2 is tangent to f at x and x 2, then ` is not of type t. Indeed, it suces to choose U and U 2 such that x and x 2 cannot be conjugate. In fact, it is possible to show, more precisely, that under the above conditions, the control map : M 2! P n can be written in suitable local coordinates near z = (x ; `; [f]) as : (x ; x 2 ; : : : ; x N ) 7! (x 2 ; x2 2 ; x 3; : : : ; x N ) but we don't need this here. Now we will follow closely Klein's paper to construct examples of curves with the maximal number of real exes. Let rst n be even, n = 2. In the ane real plane, take the ellipse x 2 + 2y 2 = and its images by rotations of an angle of = i, i = ; : : : ;. The union of these ellipses is a curve C n of degree n, with 4 2 = n(n 2)=2 real ordinary double points, and the lines joining 2 of them are not tangent to the other ellipses. When we deform the equation of C n slightly into Cn, double tangents can arise in three ways : () near lines joining pairs of distinct double points. They cannot be of type t by remark 4.. (2) near a line ` passing through a double point, intersection x of two ellipses, and tangent to a third ellipse. Then (x; `; C n ) is in M 2 (; ; ; ; ) and we have seen in the proof of 3.4, by an argument similar to remark 4., that there are no bitangents of type t nearby. (3) near a line ` tangent to distinct pairs of ellipses, at real points x, x 2. Then the derivative of at z = (x ; `; C n ) is bijective, and so nearby bitangents of nearby curves cannot be of type t.

13 FELIX KLEIN'S PAPER ON REAL FLEXES VINDICATED 3 As a consequence, t (Cn) =. According to 3. 2), applied to R, each ordinary double point x on C n gives rise to 2 real exes near x on Cn, one for each real branch through x. Therefore : i(cn) = 2 4 = n(n 2) : 2 This proves theorem I for n even. If n = 2 +, we start with the same ellipses, but add a non singular cubic which cuts each ellipse in 6 distinct points, not on the intersection of 2 ellipses. Such a cubic can be obtained as a small deformation of the cubic y 3 =. The union of the ellipses and the cubic form a curve C n of degree n. As before, a small deformation Cn of C n will have t =, and i(cn) will be equal to twice the number of double points of C n, plus the 3 real exes of the cubic. Hence : i(cn) = 2(4 + 6) + 3 = n(n 2) : 2 This completes the proof of theorem I. For theorem II, we rst construct curves with the minimal number of exes. f 2 = (x 2 + y 2 + ) has no real points, and if f 3 is the equation of a real cubic, f 2+ = f 3 f 2 has the same real points as the cubic. Small deformations of f 2 and f 2+ provide smooth curves gn min without real exes for n even and with 3 real exes for n odd. Denote by gn max the curves with maximal number of real exes constructed in the proof of theorem I and let gt n, t be a path in P n;r, joining gmin n to gn max, which avoids the ramication loci of codimension 2 of R, the control map for real exes, and transversal to those of codimension. Then, as in the proof of 3.4, when g crosses a stratum of codimension, i can change by 2, 2 or. Therefore, as t varies from to, i must take all the values of the form i = n(n 2) 2k, with i for n even and i 3 for n odd. References. S. Akbulut (ed.), Real Algebraic Geometry and Topology, Contemporary Mathematics 82 (996). 2. R. Benedetti and R. Silhol, Spin and Pin structures, immersed and embedded surfaces and a result of B. Segre on real cubic surfaces, Topology 34 (995), 65{ M. Golubitski and V. Guillemin, Stable Mappings and their Singularities, Springer-Verlag, F. Klein, Eine neue Relation zwischen den Singularitaten einer algebraischen Curve, math. Ann. (876), I.R. Porteous, preprint, Columbia Univ. (962); reprinted in : Proc. of the Liverpool Singularities Symp., Springer Lecture Notes in Math., vol. 92, 97, pp. 27{ F. Sottile, Enumerative Geometry for real Varieties, to appear. 7. R. Thom, Stabilite structurelle et morphogenese, W.A. Benjamin Inc., O. Ya. Viro, Some integral calculus based on Euler characteristic, Topology and geometry { Rohlin Seminar, Springer Lecture Notes in Mathematics, vol. 346, Springer, Berlin-New York, 988, pp. 27{ C.T.C. Wall, Duality of real projective plane curves : Klein's equation, Topology 35 (996), H.G. Zeuthen, Sur les dierentes formes des courbes planes du quatrieme ordre, Math. Ann. 7 (874), Section de Mathematiques, Universite de Geneve, C.P. 24, CH-2 Geneve 24, Switzerland. address: Ronga@math.unige.ch