Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XIV (2015), 881-906
On the class of caustics by reflection of planar curves
ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
Abstract. Given any light position S P2 and any algebraic curve of P2 (with any kind of singularities), we con2sider the incident lines comingCfrom S (i.e. the lines containing S) and their reflected lines after reflection on the mirror curve . The caustic by reflection 6S( ) is the Zariski closure of the envelope of these rCeflected lines. We introduce the Cnotion of reflected polar curve and express the class of 6S( ) in terms of intersection numbers of with the reflected polar curve, thanks to Ca fundamental lemma established in [16C]. This approach enables us to state an explicit formula for the class of 6S( ) in every case in terms of intersection numbers of the initial curve . C C Mathematics Subject Classification (2010): 14H50 (primary); 14E05, 14N05, 14N10 (secondary).
Introduction
Let V be a three dimensional complex vector space endowed with some fixed basis. 2 We consider a light point S x0 y0 z0 P P(V) and a mirror given by an [ : : ] 22 := d irreducible algebraic curve V (F) of P , with F Sym (V_) (F corresponds C = 2 to a polynomial of degree d in C x, y, z ). We denote by d_ the class of . We consider the caustic by reflection[6 ( )]of the mirror curve with sourceCpoint S C C S. Recall that 6S( ) is the Zariski closure of the envelope of the reflected lines associated to the incCident lines coming from S after reflection off . When S is not C at infinity, Quetelet and Dandelin [9,18] proved that the caustic by reflection 6S( ) is the evolute of the S-centered homothety (with ratio 2) of the pedal of fromCS (i.e. the evolute of the orthotomic of with respect to S). This decomposCition has also been used in a modern approach bCy [2–4] to study the source genericity (in the real case). In [16] we proved formulas for the degree of the caustic by reflection of planar algebraic curves. For a presentation of the classical notions of envelope, evolute, pedal, contrapedal, orthotomic, we refer to [4,8,10,20,23]. In [7], Chasles proved that the class of 6S( ) is equal to 2d_ d for a generic ( , S). In [1], Brocard and Lemoyne gave (wCithout any proof)+a more general C Franc¸oise Pe`ne is supported by the french ANR project GEODE (ANR-10-JCJC-0108). Received April 13, 2013; accepted in revised version September 23, 2013. 882 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE formula only when S is not at infinity. The Brocard and Lemoyne formula appears to be the direct composition of formulas got by Salmon and Cayley in [19, page 137, 154] for some geometric characteristics of evolute and pedal curves. The formula given by Brocard and Lemoyne is not satisfactory for the following reasons. The results of Salmon and Cayley apply only to curves having no singularities other than ordinary nodes and cusps [19, page 92], but the pedal of such a curve is not necessarily a curve satisfying the same properties. For example, the pedal curve of the rational cubic V (y2z x3) from 4 0 1 is a quartic curve with an ordinary triple point. Therefore it is not corre[ct:to :com] pose directly the formulas got by Salmon and Cayley as Brocard and Lemoyne apparently did (see also Section 5 for a counterexample of the Brocard and Lemoyne formula for the class of the caustic by reflection). Let us mention some works on the evolute and on its generalization in higher dimension [6, 11, 21]. In [11], Fantechi gave a necessary and sufficient condition for the birationality of the evolute of a curve and studied the number and type of the singularities of the general evolute. Let us insist on the fact that there exist irreducible algebraic curves (other than lines and circles) for which the evolute map is not birational. This study of evolute is generalized in higher dimension by Trifogli in [21] and by Catanese and Trifogli [6]. The aim of the present paper is to give a formula for the class (with multiplic- ity) of the caustic by reflection for any algebraic curve (without any restriction neither on the singularity points nor on the flex poinCts) and for any light posi- tion S (including the case when S is at infinity or when S is on the curve ). C In Section 1, we define the reflected lines m at a generic m and the 2 2 R 2 C (rational) “reflected map” R ,S P 99K P mapping a generic m to the equation of . C : 2 C Rm In Section 2, we define the caustic by reflection 6S( ), we give conditions ensuring that 6 ( ) is an irreducible curve and we prove thaCt its class is the degree S C of the image of by R ,S. In Section 3C, we gCive formulas for the class of caustics by reflection valid for any ( , S). These formulas describe precisely how the class of the caustic depends on geCometric invariants of and also on the relative positions of S and of the two cyclic points I, J with respCect to . As a consequence of this result, we obtain the C following formula for the class of 6S( ) valid for any of degree d 2 and for a generic source position S: C C
class 6S( ) 2d_ d , ` µI ( ) µJ ( ), C = + C 1 C C where ( , ` ) is the co ntact number of with t he line at infinity ` and where C 1 C 1 µI ( ) and µJ ( ) denote the multiplicity numbers of respectively I and J on . CIn SectionC4, our formulas are illustrated on two examples of curves (theClem- niscate of Bernoulli and the quintic considered in [16]). In Section 5, we compare our formula with the one given by Brocard and Lemoyne for a light position not at infinity. We also give an explicit counter- example to their formula. ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 883
In Section 6, we prove our main theorem. First we give a formula for the class of the caustic in terms of intersection numbers of with a generic “reflected polar” C at the base points of R ,S. We then compute these intersection numbers in terms of C the degree d and of the class d_ of but also in terms of intersection numbers of with each line of the triangle I J S. C C In appendix A, we prove a useful formula expressing the classical intersection number in terms of probranches.
ACKNOWLEDGEMENTS. The authors thank Jean Marot for stimulating discussions and for having indicated them the formula of Brocard and Lemoyne.
1. Reflected lines m and rational map R ,S R C 2 Recall that we consider a light position S x0 y0 z0 P and an irreducible [ : : ] 2 algebraic (mirror) curve V (F) of P2 given by a homogeneous polynomial F Symd (V) with d C2.=We write Sing( ) for the set of singular points of . 2 C C For any non singular point m, we write m for the tangent line to at m. We set T C 2 C S(x0, y0, z0) V 0 . For any m x y z P , we write m(x, y, z) V 0 . 2 \ { } [2 : : ] 2 2 \ { } We write as usual ` V (z) P for the line at infinity. For any P(x1, y1, z1) V 0 , we define 1 = ⇢ 2 \ { } d 1 1 F x F y F z F Sym V_ . P := 1 x + 1 y + 1 z 2 2 Recall that V (1P F) is the polar curve of with respect to P x1 y1 z1 P . C 2 [ : : ] 2 Since the initial problem is euclidean, we endow P with an angular structure for which I 1 i 0 P2 and J 1 i 0 P2 play a particular role. To this end, let[us:reca:ll t]he2definition o[ f t:he cro:ss-]ra2tio of 4 points of ` . Given 1 four points (Pi ai bi 0 )i 1,...,4 such that each point appears at most 2 times, we [ : : ] = define the cross-ratio (P1, P2, P3, P4) of these four points as follows:
b3a1 b1a3 b4a2 b2a4 P1, P2, P3, P4 , = b3a2 b2a3 b4a1 b1a4 1 with convention 0 . For any distinct lines 1 and 2 not equal to ` , con- taining neither I nor=J1, we define the oriented angAular meAasure between 1and A1 A2 by ✓ (modulo ⇡Z) such that
2i✓ a1 ib1 a2 ib2 e P1, P2, I, J + = = a1 ib1 a2 ib2 + 2 (where Pi ai bi 0 is the point at infinity of i ). Let Q Sym (V_) be defined by[ Q(:x, y, z:) ] x2 y2. It will be worthAnoting that Q2( F) F2 := + r = x + 884 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
2 Fy 1I F1J F. For every non singular point m of ` , we recall that tm Fy = 2 C \ 1 [ : Fx 0 P is the point at infinity of m and so tm I, J is equivalent to m V: (Q] (2 F)). T C 62 { } 62 r Now, for any m (` Q( F)) and any incident line ` containing m, we de- 2C\ 1[ r fine as follows the associated reflected line Rm(`) (for the reflexion on at m with respect to the Snell-Descartes reflection law Angle(`, ) Angle( C, R (`))). TmC = TmC m Definition 1.1. For every m (` V (Q( F))), we define rm ` ` 2 C \ 1 [ r : 1 ! 1 mapping P ` to the unique rm(P) such that (P, tm, I, J) (tm, rm(P),I,J). 2 1 2 = We define Rm m m with m ` G(1, P ), m ` by Rm(`) : F ! F F := { 2 2 } = (m rm(P`)) if P` is the point at infinity of `. We have (on coordinates) r x y 0 x F2 F2 2y F F y F2 F2 2x F F 0 . m 1 : 1 : = 1 x y + 1 x y : 1 x y + 1 x y : ⇥ ⇤ ⇥ ⇤ 1 Remark 1.2. Observe that rm is an involution on ` P with exactly two fixed 1 =⇠ points tm and nm Fx Fy 0 . As a consequence, Rm is an involution with two fixed points ( [) and: (: )] (mn ) the normal line to at m. m m := m MoreovTer rC(I) NJ anCd r (J) I. C m = m = Definition 1.3. For any m x y z ( S ` V (Q( F)) we define [ : : ] 2 C \ { } [ 1 [ r the reflected line m on at m (of the incident line coming from S) as the line R ((mS))R. C Rm := m For m x y z ( S ` V (Q( F)), the point at infinity of (Sm) is s [x z: z: x ] 2y zC \z {y } [0 . 1Du[e to therEuler identity, on , we have m[ 0 0 : 0 0 : ] C x Fx yFy zFz 0 and so (x0z z0x)Fx (y0z z0 y)Fy z1S F. Hence + + = + = 2 r(sm) vm um 0 and the reflected line m is the set of P X Y Z P such th=at[u X : v Y: ] w Z 0, with R [ : : ] 2 m + m + m = 2 2 2d 1 u z y zy F F 2zF 1 F Sym V_ m := 0 0 x + y + y S 2 2 2 2d 1 vm zx0 z0x F F 2zFx 1S F Sym V_ := x + y 2 xu yv w m m xy yx F2 F2 21 F x F yF m := z = 0 0 x + y S y x 2d 1 Sym V_ . 2 Definition 1.4. We call reflected map of from S the following rational map C 2 2 P 99K P R ,S . C : m um vm wm 7! [ : : ] 2 We also define the rational map T ,S (R ,S) 99K P . C := C |C : C ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 885
For any m V, it will be useful to define RF,S(m) (um, vm, wm) V and to notice that 2 := 2
R , (m) Q( F(m)) (m S) 21 F(m) (m n ) V, F S = r · ^ S · ^ m 2 with1 n (F (m), F (m), 0) V. m x y 2 Proposition 1.5. The base points of T ,S are the following: I, J, S (if these points are in ), the singular points of andCthe points of tangency of with some line of the triaCngle (I J S). C C
Proof. We have to prove that the set of base points of T ,S is the following set: C ( I,J,S V (1S F,Q( F)) V (Fx ,Fy)). We just prove that Base(T ,S) M,:=thCe\co{nverse}[being obviours (ob[serve that if m I, J , we automaCticall⇢y hMave Q( F(m)) 0 and n Vect(m)). Let m2x {y z } be such that r = m 2 [ ; ; ] 2 C RF,S(m) 0. Then m and Q( F(m)) S 21S F(m) nm are colinear. Due to the Eule=r identity, we have 0 r DF(m·) m (with DF(m· ) the differential of F = · at m) and so 0 1S F(m) Q( F(m)) since DF(m) S 1S F(m) and since DF(m) n =Q ( F(m)). H· encre 1 F(m) 0 or Q( ·F(=m)) 0. · m = r S = r = If 1S F(m) 0, then either Q( F(m)) 0 or m S. If 1 F(m) = 0 and Q( F(mr)) 0 ,=then F (m=) F (m) 0 or m S 6= r = x = y = = Fx (m) Fy(m) 0 . Assume that m Fx (m) Fy(m) 0 . Then, since [Q( F(m: )) 0, w:e c]onclude that m I,=J [. : : ] r = 2 { } We recall the following result established at the same period in [17] and by Catanese in [5], with two different proofs. Proposition 1.6 ([5,17]). Let be an irreducible curve of degree d 2. Then, for 3 C a generic S P , the map T ,S is birational. 2 C 2. Caustic by reflection Definition 2.1. The caustic by reflection 6 ( ) is the Zariski closure of the enve- S C lope of the reflected lines m m ( S ` V (Q( F)) . {R ; 2 C \ { } [ 1 [ r } Recall that, in [16], we have defined a rational map 8F,S called caustic map mapping a generic m to the point of tangency of 6 ( ) with and that 2 C S C Rm 6S( ) is the Zariski closure of 8F,S( ). CIn the present work, we will noCt consider the cases in which the caustic by reflection 6S( ) is a single point. We recall that these cases are easily characterized as follows. C Proposition 2.2. Assume that (i) S I, J , 62 { } z y z y 1 2 1 1 2 With V V V being given in coordinates by (x1, y1, z1) (x2, y2, z2) z1x2 z2x1 . ^ : ⇥ ! ^ = x y y x ✓ 1 2 1 2 ◆ 886 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
(ii) is not a line (i.e. d 1), (iii) iCf d 2, then S is not6=a focus of the conic . = C Then 6 ( ) is not reduced to a point and is an irreducible curve. S C Proof. Assume (i), (ii) and (iii) and that 6 ( ) S with S x y z . S C = { 0} 0 = [ 1 : 1 : 1] When S ` , we will use the fact that 6S( ) is the evolute of the orthotomic of with resp62ect1to S. Since is not a line, the oCrthotomic of with respect to S is notCreduced to a point but its eCvolute is a point. This implies thaCt the orthotomic of with respect to S is either a line (not equal to ` ) or a circle. But is the contrapedaCl (or orthocaustic) curve (from S) of the imag1e by the S-centereCd homothety (with ratio 1/2) of the orthotomic of . Therefore d 2 and S is a focal point of , which contradicts (iii). C = C When S ` but S0 ` , then, for symmetry reasons, we also have 6 ( ) S a2nd w1e conclude62ana1logously. S0 C = { } Suppose now that S, S0 ` . We have z0 z1 0. For every m x y 2 1 = = = [ : : 1 (` V (Q( F))), we have (S, tm, I, J) (tm, S0, I, J). Therefore w]e2haCve\ 1 [ r = ix y i F F i F F ix y 0 0 y + x y + x 1 1 i F F ix y = i F F ix y y + x 0 0 y + x 1 1 and so
ix y ix y i F F 2 i F F 2 ix y ix y . 0 0 1 1 y + x = y + x 0 0 1 1 Now, according to (i), ix0 y0 0, ix0 y0 0, ix1 y1 0, ix1 y1 0. 2 6= 2 6= 6= 6= Hence ( i Fy Fx ) a(i Fy Fx ) for some a 0. So, there exists (↵, ) 2 + = + 6= 2 C (0, 0) such that ↵Fx Fy 0 (setting (↵, ) (1 a0, i ia0) for \ { } 2 + = = some a0 C satisfying a0 a). This implies that F(x, y, z) G( x ↵y, z) for 2 =d 2 = some irreducible G Sym ((C )_). We conclude that deg G 1 and so d 1 which contradicts (ii)2. = = Hence we proved that 6 ( ) is not reduced to a point. Now the irreducibility S C of 6S( ) comes from the fact that 6S( ) 8F,S( ) and that is an irreducible curve. C C = C C
Proposition 2.3. Assume that 6 ( ) is not reduced to a point. Then we have S C
class 6S( ) deg T ,S( ) , (2.1) C = C C where T ,S( ) stands for the Zariski closure of T ,S( ). C C C C Proof. This comes from the fact that 6 ( ) is the Zariski closure of the envelope S C of m, m (Sing( ) S ` V (Q( F)) and can be precised as foll{oRws. For 2eveCry\algebraiCc c[urv{e}0[ 1V ([G) (withrG in} Symk(V ) for some k), = _ ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 887
2 2 we consider the Gauss map 0 P 99K P defined on coordinates by 0( x y z ) G G G , we obtain: immediately the commutative diagram: [ : : ] = [ x : y : z]
/ 6S( ) (2.2) C 99 C 99 9 6S( ) T ,S 9 C C 9 ✏ 9⌧ 6 ( )(6S( )) (6 ( ))_ S C C =⇠ S C with 8F,S the caustic map defined in [16] (see the begining of the present sec- tion).
Let us notice that, according to the proof of Proposition 2.3, the rational map T ,S has the same degree as the rational map (8F,S) (since 6S( ) is irreducible C |C C and since the Gauss map ( 6 ( )) 6 ( ) is birational [12]). S C | S C
3. Formulas for the class of the caustic
Since the map T ,S may be non birational, we introduce the notion of class with multiplicity of 6C ( ): S C mclass 6 ( ) (S, ) mclass 6 ( ) S C = 1 C ⇥ S C where class(6 ( )) is the class o f the algebraic curve 6 ( ) a nd where (S, ) S C S C 1 C is the degree of the rational map T ,S. We recall that 1(S, ) corresponds to the C C number of preimages on of a generic point of 6S( ) by T ,S. Before stating our mCain result, let us introducCe someC notations. For every 2 m1 P , we write µm1 µm1 ( ) for the multiplicity of m1 on and consider the 2 = C C set Branchm1 ( ) of branches of at m1. We denote by the set of couples point- branch (m , C) of with m Cand Branch ( ). EFor every (m , ) , we 1 B C 1 2 C B 2 m1 C 1 B 2 E write e for the multiplicity of and m ( ) the tangent line to at m1; we observe B B T 1 B B that µm1 Branch ( ) e . We write im1 (0, 00) the intersection number of two = m1 B B2 C 2 curves 0 and 00 at m1. For any algebraic curve 0 of , we also define the contact P 2 C P number m1 ( , 0) of and 0 at m1 P by C C C C 2
, 0 i , 0 µ ( )µ 0 if m 0 m1 C C := m1 C C m1 C m1 C 1 2 C \ C and , 0 0 if m 0. m1 C C := 1 62 C \ C Recall that ( , ) 0 mean s that m or that and intersect m1 C C0 = 1 62 C \ C0 C C0 transversally at m1.
Theorem 3.1. Assume that the hypotheses of Proposition 2.2 hold. 888 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
(1) If S ` , the class (with multiplicity) of 6S( ) is given by 62 1 C mclass(6 ( )) 2d_ d 2 f 0 g f g0 q0, (3.1) S C = + + where g is the contact number of with ` , i.e. g ( , ` ), m1 ` m1 • f is the multiplicity numbCer at a c1yclic poi:=nt of 2wCi\th1an isotCropi1c line • from S, i.e. PC f i ( , (I S)) i ( , (J S)), := I C + J C f 0 is the contact number of with an isotropic line from S outside I, J, S , • i.e. C { }
f 0 ( , (I S)) ( , (J S)), := m1 + m1 m ( (I S)) I,S C m ( (J S)) J,S C 12 C\X \{ } 12 C\X \{ } g0 given by g0 iS( , (I S)) iS( , (J S)) µS • q is given by := C + C ; • 0 q0 im ( , m ( )) 2e . := [ 1 B T 1 B B] (m1, ) m1 I,J,S ,Tm1 (I S) B 2E: 62{Xor } B= Tm (J S), im ( , m ( )) 2e 1 B= 1 B T 1 B B (2) If S ` , the class of 6S( ) is 2 1 C mclass 6 ( ) 2d_ d 2g µ µ µ c0(S), (3.2) S C = + I J S with
c0(S) e min(iS , OscS( ) 3e , 0 , := B + B B B BranchS( ) iS( ,` ) 2e B2 CX: B 1 = B where Osc ( ) is any smooth algebraic osculating curve to at S (i.e. any S B B smooth algebraic curve 0 such that iS( , 0) > 2e ). C B C B The notation introduced in this theorem is close to that of Salmon and Cayley [19] (see Section 5 for further explanations). Let us point out that, in this article, g is not the geometric genus of the curve. Remark 3.2. Observe that we also have
c0(S) (e min( 1(S, ) 3e , 0)), := B + B B BranchS( ) iS( ,` ) 2e B2 CX: B 1 = B where 1(S, ) is the first characteristic exponent of which is not a multiple of e (see [25]).B B B Observe that, when iS( , S( )) 2e , we have min(iS( , OscS( )) 3e , 0) 0 except if S is a sBingTularBpoin=t andBif the probranches ofB are givBen b y B 1= 2 1 B 1 Y x0 y0 ↵Z ↵1 Z in the chart X 1 if x0 0 (or X y0 x0 2 = + + · · · = 6= = ↵Z ↵1 Z 1 in the chart Y 1 otherwise), with ↵ 0, ↵1 0 and 2 < + < 3. He+nc·e· ·c (S) = e when6= admits6=no such 1 0 BranchS( ) iS( ,` ) 2e branch tangent at S to ` .= B2 C : B 1 = B B C 1 P ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 889
Combining Proposition 1.6 and Theorem 3.1, we obtain: Corollary 3.3 (A source-generic formula for the class). Let P2 be a fixed curve of degree d 2. For a generic source point S, we havCe ⇢(S, ) 1 and 1 C = class(6S( )) 2d_ d g µI µJ where g denotes the contact number of with ` .C = + C 1 2 Proof. Due to Proposition 1.6, 1(S, ) 1 for a generic S P . So class(6S( )) C = 2 C = mclass(6S( )). AssumeCmoreover, that S ` (so we apply the first formula of Theorem 3.1), 62 1 S (so g0 0), that (I S) and (J S) are not tangent to (so f 0 q0 0 and f 62 Cµ ( ) =µ ( )). We obtain the result. C = = = I C + J C
4. Examples
Let us now illustrate our result for two particular mirror curves.
4.1. Example of the lemniscate of Bernoulli We consider the case when V (F) is the lemniscate of Bernoulli given by 2 2 2 C2 = 2 2 2 F(x, y, z) (x y ) 2(x y )z and when S P I, J . The degree of is d 4. T=he sin+gular po ints of are : I 1 i 0 , J2 1 \ {i 0} and 0 0 1C. Thes=e three points are double poCints, eac[h o:ne:ha]ving[ tw: o di:ffe]rent taOn[gen:t li:nes]. Hence the class of is given by d d(d 1) 3 2 6 and so C _ = ⇥ = 2d_ d 16. + = The tangent lines to at I are ` , V (y iz ix) and ` , V (y iz ix) (the C 1 I := 2 I := + intersection number of with ` , or with ` , at I is equal to 4). The tangent lines C 1 I 2 I to at J are ` , V (y iz ix) and ` , V (y iz ix) (the intersection C 1 J := + 2 J := + + number of with ` , or with ` , at J is equal to 4). This ensures that we have C 1 J 2 J
f 2 2 1S ` , 1S ` , 1S ` , 1S ` , . = + 2 1 I + 2 2 I + 2 1 J + 2 2 J Observe that ` is not tangent to . Indeed I and J are the only points in ` 1 C C \ 1 and ` is not tangent to at these points. Therefore we have g 0 and c0(S) 0. S1ince I and J are alCso the only points at which is tangent=to an isotropic=line C (i.e. a line containing I or J), we have f 0 0, g0 µS, q0 0. In this case, one can check that (S, ) 1. Finally, we ge=t = = 1 C =
if S ` , class 6S( ) 12 2 1S ` , ` , 1S ` , ` , µS. (4.1) 62 1 C = 2 1 I [ 2 I + 2 1 J [ 2 J Moreover we have
if S ` I, J , class 6S( ) 16 2 2 12, (4.2) 2 1 \ { } C = = (since µ µ 2 and since µ 0). For example, for S 1 0 1 , we get I = J = S = [ : : ] class(6 ( )) 8, since S is in ` , ` , but not in (so µ 0). S C = 2 I \ 1 J C S = 890 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
4.2. Example of a quintic curve As in [16], we consider the quintic curve V (F) with F(x, y, z) y2z3 x5. C = 2 = We also consider a light point S x0 y0 z0 P I, J . This curve admits two [ : : ] 2 \ { } singular points: m1 0 0 1 and m2 0 1 0 , we have d 5. We recall that[ :adm: its] a single[ b:ran:ch]at m , which=has multiplicity 2 and C 1 which is tangent to V (y). We observe that im1 ( , V (y)) 5. Analogously, admits a single branch atCm , whi=ch has multiplicity 3 and C 2 which is tangent to ` . We observe that im ( , ` ) 5. 1 2 C 1 = We obtain that the class of is d_ 5 and that has no inflexion point (these two facts are proved in [16]). InCparticul=ar, we get thaCt 2d d 15. _ + = Since m2 is the only point of ` , we get that g m2 ( , ` ) 2 and f 0. C \ 1 = C 1 = = The curve admits six (pairwise distinct) isotropic tangent lines other than C ` : `1, `2 and `3 containing I defined by 1 3i k 3 2i⇡ ` V ix y ↵ p20z , where ↵ e 3 k = + 25 := ✓ ◆ for k 1, 2, 3 and ` , ` and ` containing J defined by = 4 5 6 3i k 3 `3 k V ix y ↵ p20z + = + + 25 ✓ ◆ for k 1, 2, 3. For every i 1, 2, 3, 4, 5, 6 , we write ai the point at which is tangen=t to ` (the points a co2rre{spond to the p}oints of V (F2 F2) m , mC ). i i C \ x + y \ { 1 2} Since contains no inflexion point and since m1 and m2 are the only singular points of , wCe get that, C f 0 # i 1, 2, 3, 4, 5, 6 S ` a and q0 0 = { 2 { } : 2 i \ { i }} = when S ` . Now62re1call that g i ( , (I S)) i ( , (J S)) µ . Again, in this case, one 0 = S C + S C S can check that 1(S, ) 1. If S ` , we have C = 62 1 class 6 ( ) 13 2 # i 1, 2, 3, 4, 5, 6 S ` a g0 (4.3) S C = ⇥ { 2 { } : 2 i \ { i }} and if S ` I, J , we have 2 1 \ { } class 6S( ) 11 3 1S m . (4.4) C = ⇥ = 2 2 We observe that the points of P I, J belonging to two dictinct `k are outside . The set of these points is \ { } C
3 3 3 3 p3 20↵k 0 1 , p3 20↵k p3p3 20↵k 1 , E := k 1 25 : : 50 : 50 : [= ⇢ 3 3 p3 20↵k p3p3 20↵k 1 50 : 50 : ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 891
2i⇡ with ↵ e 3 . Finally, the class of the caustic in the different cases is summarized in the fo=llowing table.
2 Condition on S P I, J class(6S( )) S m2 \ { } 8 C = = 2 S 9 62 E S `k ak 10 2 C \ k 1 \ S= 6 S ` m2 `k ( ) m1 a1, . . . , a6 11 2 1 \ [ k 1 \ E [ C ! [ [ S= 6 S ` m1 `k 12 2 C \ 1 [ { } [ k 1 ! = otherwise S 13
5. On the formulas by Brocard and Lemoyne and by Salmon and Cayley
5.1. Formulas given by Brocard and Lemoyne
Recall that, when S ` , 6S( ) is the evolute of an homothetic of the pedal of from S. 62 1 C C The work of Salmon and Cayley is under ordinary Plu¨cker conditions (no worse multiple tangents than ordinary double tangents, no singularities other than ordinary cusps and ordinary nodes). In [19, page 137], Salmon and Cayley gave the following formula for the class of the evolute :
n0 m n f g. = + Replace now m, n, f and g by M, N, F and G (respectively) given in [19, page 154] for the pedal. Doing so, one gets precisely (with the same notations) the formula of the class of caustics by reflection given by Brocard and Lemoyne in [1, page 114]. As explained in introduction, this composition of formulas of Salmon and Cay- ley is incorrect because of the non-conservation of the Plu¨cker conditions by the pedal transformation. Nevertheless, for completeness sake, let us present the Bro- card and Lemoyne formula and compare it with our formula. Brocard and Lemoyne gave the following formula for the class of the caustic by reflection 6S( ) when S ` : C 62 1 BL d 2 d_ f 0 g f g0 q0, (5.1) = + ˆ ˆ ˆ ˆ + ˆ for an algebraic curve of degr ee d, of c lass d_, g times tangent to ` , passing C ˆ 1 fˆ times through a cyclic point, fˆ0 times tangent to an isotropic line of S, passing g times through S, q being the coincidence number of contact points when an ˆ0 ˆ0 892 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
isotropic line is multiply tangent. In [19], q0 is defined as the coincidence number 2 ˆ of tangents at points ◆1, ◆2 of P _ (corresponding to (I S) and (J S)) if these points are multiple points of the image of by the polar reciprocal transformation with C center S; i.e. q0 represents the number of ordinary flexes of . When S ˆ ` , let us compare terms appearing in our foCrmula (3.1) with terms of (5.1) : 62 1 g seems to be equal to g; • ˆ it seems that f µ µ and so f f ( , (I S)) ( , (J S)) • ˆ = I + J = ˆ + I C + J C ; it seems that f m ( , (I S)) m ( , (J S)) and ˆ0 m1 (I S) 1 m1 (J S) 1 • so = 2C\ C + 2C\ C P P f 0 f 0 ( , (I S)) ( , (J S)) ( , (I S)) ( , (J S)) = ˆ I C J C S C S C ; it seems that g µ , therefore g g ( , (I S)) ( , (J S)) • ˆ0 = S 0 = ˆ0 + S C + S C ; our definition of q0 appears as an extension of q0 (except that we exclude the • points m I, J, S ). ˆ 1 2 { } Observe that these terms coincide with the definition of Brocard and Lemoyne if (I S) and (J S) are not tangent to at S, I, J. In particular, the first item of Theorem 3.1 states that, when S is not at inCfinity we have mclass(6 ( )) BL ( , (I S)) ( , (J S)) ( , (I S)) ( , (J S)), S C = + I C + J C + S C + S C with BL the Brocard and Lemoyne formula recalled in (5.1). This last formula point out the error in the Brocard and Lemoyne formula.
5.2. A counterexample to the formula of Brocard and Lemoyne
We consider an example in which I ( , (I S)) J ( , (J S)) 1, which means that (I S) is tangent to at I and (J S) iCs tangent=to atCJ. Let us=consider the non- singular quartic curve C V (2yz3 2z2 y2 2zy3 C2y4 2z3x 2zyx2 5y2x2 4 C = + + + + + + 3x ) and S 0 0 1 . This curve has degree d 4 and class d_ 4 3 12, is not tange[nt :to `: ,]is tangent to C(SI) at I and no=where else, is tan=gent⇥to (S=J) at J and nowhere els1e; these tangent points are ordinary. S is a non singular point of . Therefore, with our definitions, we have g 0, f 2 2 4, f 0, g C = = + = 0 = 0 = 1 1 1 1, q0 0, which gives class(6S( )) 4 2(12 0) 0 4 1 0 23, si+nce in t=his case= (S, ) 1. In comparisCon, =the +Brocar d and L emo yn e for=mula 1 C = would give g 0, fˆ 1 1 2, fˆ0 1 1 2, g0 1, q0 0 and so their formula giveˆs =BL 4= 2(+12 =2) 0 =2 1+ 0= 21ˆw=hich isˆ n=ot class(6 ( )). = + = S C
6. Proof of Theorem 3.1
To compute the degree of T ,S( ), we will use the Fundamental Lemma given in [16]. Let us first recall the dCefinCition of '-polar introduced in [16] and extending the notion of polar. ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 893
Definition 6.1. Let p 1, q 1 and let W be a complex vector space of di- p q mension p 1. Given ' P P(W) 99K P a rational map defined by + : := d q ' '0 'q (with '0, . . . , 'q Sym (W_)) and a a0 aq P , = [ : · · · : ] 2 = [ : · · · : ] 2 we define the '-polar at a, denoted by ',a, the hypersurface of degree d given by q p P ',a V j 0 a j ' j P . P := = ✓ ⇣ ⌘ 2 With thiPs definition, the “classical” polar of a curve V (F) of P (for some C = homogeneous polynomial F C x, y, z ) at a is the -polar curve at a, where 2 [ ] C x y z Fx Fy Fz . C : [ : : ] 7! [ : : ] Definition 6.2. We call reflected polar (or r-polar) of the plane curve with (r) C respect to S at a the R ,S-polar at a, i.e. the curve S,a( ) R ,S,a. C P C := P C From a geometric point of view, (r) ( ) is an algebraic curve such that, for PS,a C every m (r) ( ), contains a (if is well defined), this means that line 2 C \ PS,a C Rm Rm (am) is tangent to 6S( ) at the point m0 8F,S(m) 6S( ) associated to m (see picture). C = 2 C
Let us now recall the statement of the fundamental lemma proved in [16]. Lemma 6.3 (Fundamental lemma [16]). Let W be a complex vector space of di- mension p 1, let be an irreducible algebraic curve of Pp P(W) and p +q C := ' P 99K P be a rational map given by ' '0 'q with '0, . . . , 'q : = [ : · · · : ] 2 Sym (W_). Assume that Base(') and that ' has degree 1 N . C 6✓ q |C 2 [ {1} Then, for generic a a0 aq P , the following formula holds = [ : · · · : ] 2
. deg '( ) . deg( ) i , ', , 1 = p a C C p Base(' ) C P ⇣ ⌘ 2 X |C with convention 0. 0 and deg('( )) 0 if #'( ) < . 1 = C = C 1 894 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
Due to this lemma and to Proposition 2.3, we have
mclass 6 ( ) d(2d 1) i , (r) ( ) . (6.1) S C = m1 C PS,a C m1 Base(T ,S) 2 X C ⇣ ⌘ Now, we enter in the most technical stuff which is the computation of the inter- section numbers i ( , (r) ( )) of with its reflected polar at the base points of m1 C PS,a C C R ,S. To compute these intersection numbers, it will be useful to observe the form C of the image of R ,S by linear changes of variable. It is worth noting that RF,S can be rewritten C
R , id 1 F1 F S 1 F1 F J 1 F1 F I . F S = ^ I J · S I · S J · Proposition 6.4. Let M⇥ GL(V). We have ⇤ 2 1 1 R M Com(M) R(M (I),M (J)), F,S F M,M 1(S) = · with Com(M) det(M) t M 1 and := · (A,B) R id 1AG1BG S0 1S G1AG B 1S G1BG A . G,S0 := ^ · 0 · 0 · ⇥ ⇤ Proof. We use M(u) M(v) (Com(M))(u v) and 1M(u)(F)(M(P)) 1u(F M)(P). ^ = ^ =
2 2 We write 5 V 0 P for the canonical projection, P0 0 0 1 P : \ { } ! [ : : ] 2 and P0(0, 0, 1) V. Let m1 be a base point of and M GL(V) be such that 5(M(P )) m2and such that the tangent cone ofCV (F M)2at P does not contain 0 = 1 0 V (x). Let µm1 be the multiplicity of m1 in (m1 is a singular point of if and only 2 C 1 C if µm1 > 1). Then, for every a P , writing a0 M (a), we have 2 := (r) i , ( ) i , V a, R , ( ) m1 C PS,a C = m1 C F S · i V (F M), V a, R , M( ) = P0 ⌦ ↵ F S · 1 1 i V (F M), V ⌦ a , R(M (I),M↵ (J))( ) P0 0 F M,M 1(S) = · 1 1 ⇣ ⇣D (M (I),EM⌘⌘ (J)) i P0 , V a0, R 1 ( ) , = F M,M (S) · Branch (V (F M)) B B2 XP0 ⇣ ⇣D E⌘⌘ where BranchP0 (V (F M)) is the set of branches of V (F M) at P0. The last equality comes from Pr oposition A.1 proved in appendix (see formula (A.1)). Let b be the number of such branches. Of course, b 1 for non-singular points. Writing = e for the multiplicity of the branch , we have µm1 Branch (V (F M)) e . B = P0 B 1 1 B B2 1 Let us write C x N and C x N , y for the rings of convergePnt power series of x N , y. h i h i ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 895
1 1 Let C x⇤ N 1 C x N and C x⇤, y N 1 C x N , y . For every h h aix:=q x , whe deifine thehvaluaitio:=n of h a s folhlows: i = q Q q C ⇤ 2 + 2Sh i S P val(h) valx (h(x)) min q Q , aq 0 . := := { 2 + 6= } 2 Let be a branch of V (F M) at P0. Observe that 0 M( ) P is a B 1 B = B ⇢1 branch of at m1. Let A(x A, yA, z A) M (I), B(xB, yB, zB) M (J) and 1C := := S0 M (S). Let be the tangent line to at P0. The branch can be split in := TB B B e pro-branches with equations y gi, (x) in the chart z 1 (for i 1, . . . , e ) B = B = 2 { B} with gi C x⇤ having (rational) valuation larger than or equal to 1 (so gi0(0) 0). 2 , h. . . ,i ( ) = For j 1 e 0 , consider also the equations y g j, 0 x (in the chart z 1) 2 { B } = (B ( )) = of the pro-branches j, 0 for each branch 0 BranchP0 V F M . This notion of pro-branches comVesBfrom the combinaBtion2of the Weierstra ss and the Puiseux theorems. It has been used namely by Halphen in [14] and by Wall in [24]. One can also see [16]. There exists a unit U in C x, y such that the following equality h i holds in C x⇤, y h i e B0 F(M(x, y, 1)) U(x, y) (y g j, (x)). = B0 Branch (V (F M)) j 1 B02 YP0 Y= For a generic a (with a M 1(a)), using (A.2)), we obtain 0 := (A,B) (A,B) i P , V a0, R ( ) valx a0, R x, gi, (x), 1 0 F M,S0 F M,S0 B B · = i ⇣ ⇣D E⌘⌘ X ⇣D E⌘ (A,B) min valx R x, gi, (x), 1 . = j 1,2,3 F M,S0 B j i = ✓ ◆ X h i Hence formula (6.1) becomes mclass(6 ( )) d(2d 1) S := C e B (A,B) (6.2) min valx RF M,S x, gi, (x), 1 , j 1,2,3 0 B j m1 i 1 = ✓ ◆ X2C XB X= h i where, for every m , M depends on m and is as above, where the sum is over 1 2 C 1 BranchP0 (V (F M)). Due to Lemma 33 of [16], for every P(xP , yP , z P ) VB 2 0 , we have 2 \ { } 1M(P) F M x, gi, (x), 1 1P(F M) x, gi, (x), 1 Di, (x)WP,i, (x), B = B = B B w ith WP,i, (x) yP gi0, (x)xP z P xgi0, (x) gi, (x) B := B + B B and with
Di, (x) U(x, gi, (x)) (gi, (x) g j, (x)). B := B B B0 0 BranchP0 (V (F M)) j 1,...,e ( 0, j) ( ,i) B 2 Y = B0Y: B 6= B 896 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
Hence we have (A,B) 2 RF M,S x, gi, (x), 1 Di, Ri, (x) (6.3) 0 B := B · ˆ B with x Ri, (x) gi, (x) WA,i, (x)WB,i, (x) S0 WS ,i, (x)WA,i, (x) B ˆ B := 0 B 1 ^ B B · 0 B B · 1 h @ A WS ,i, (x)WB,i, (x) A . 0 B B · i First, with the notations of [16] (since U(0, 0) 0), we have 6= e B val(Di, ) Vm , B = 1 Branch (V (F M)) i 1 B2 XP0 X=
(which is zero if m1 is a nonsingular point of ). Second, writing hm ,i, C 1 B := min(val( Ri, j ), j 1, 2, 3), we observe that, due to Proposition 29 and to Re- [ ˆ B] = e mark 34 of [16], the quantity i B1 hm1,i, only depends on m1 and on the branch = B 0 M( ) of at m1 (it does not depend on the choice of M GL(V) such that 5B (M=(P )B) mC and such thaPt V (x) is not tangent to M 1( )2). Hence we write 0 = 1 B0 e B hm , hm ,i, . 1 B0 1 B := i 1 X= With these notations, due to (6.3), formula (6.2) becomes mclass(6S( )) 2d(d 1) d 2 Vm hm , . C = + 1 1 B0 m Sing( ) m Branchm ( ) 12X C X12C B02 X 1 C
Moreover, as noticed in [16], we have d(d 1) Vm d , where m1 Sing( ) 1 = _ d is the class of . Therefore, we get 2 C _ C P
mclass(6S( )) 2d_ d hm , . (6.4) C = + 1 B0 m Branchm ( ) X12C B02 X 1 C
Theorem 3.1 will come directly from the computation of hm1,i, given in the fol- lowing result. B
Lemma 6.5. Let m1 and 0 Branchm1 ( ). Writing m1 0 for the tangent line to at m , i (2 C, B) f2or the interseCction numbeTr ofB with at B0 1 m1 B0 Tm1 B0 B0 Tm1 B0 m1 and e for the multiplicity of 0, we have B0 B (1) hm , 0 if I, J, S m 0. 1 B0 = 62 T 1 B (2) hm , 0 if #( m 0 I, J, S ) 1 and m1 I, J, S . 1 B0 = T 1 B \ { } = 62 { } ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 897
(3) hm , e if #( m 0 I, J, S ) 1 and m1 I, J, S . 1 B0 = B0 T 1 B \ { } = 2 { } (4) hm1, 0 im1 ( 0, m1 0) min(im1 ( 0, m1 0) 2e 0 , 0) if m1 0 (I S), J B = anBd mT BI, S+. B T B B T B = 62 Tm1 B0 1 62 { } hm1, 0 im1 ( 0, m1 0) min(im1 ( 0, m1 0) 2e 0 , 0) if m1 0 (J S), I B = andBm T BJ, S+. B T B B T B = 62 Tm1 B0 1 62 { } (5) hm , im ( 0, m 0) if m 0 (I S), J m 0 and m1 I, S . 1 B0 = 1 B T 1 B T 1 B = 62 T 1 B 2 { } hm , im ( 0, m 0) if m 0 (J S), I m 0 and m1 J, S . 1 B0 = 1 B T 1 B T 1 B = 62 T 1 B 2 { } (6) hm1, 0 im1 ( 0, m1 0) e 0 if m1 0 (I J), that S m1 0 and m1 I, JB. = B T B B T B = 62 T B 62 { } (7) hm , im ( 0, m 0) if m 0 (I J), that S m 0 and m1 I, J . 1 B0 = 1 B T 1 B T 1 B = 62 T 1 B 2 { } (8) hm , 2im ( 0, m 0) 2e if I, J, S m 0 and m1 I, J, S . 1 B0 = 1 B T 1 B B0 2 T 1 B 62 { } (9) hm , 2im ( 0, m 0) e if I, J, S m 0 and m1 I, J . 1 B0 = 1 B T 1 B B0 2 T 1 B 2 { } (10) hm1, 0 2im1 ( 0, m1 0) e 0 if I,J,S m1 0, m1 Sand im1( 0, m1 0) 2e B. = B T B B 2T B = B T B 6= B0 (11) hm , e (1 min( 1, 3)) if I,J,S m 0, m1 S and im ( 0, m 0) 1 B0 = B0 + 2T 1 B = 1 B T 1 B = 2e , e 1 im ( 0, Oscm ( 0)), where Oscm ( 0) is any osculating B0 B0 = 1 B 1 B 1 B smooth algebraic curve to 0 at m1 (the last formula of hm , holds if we B 1 B0 replace e 0 1 by the first characteristic exponent of 0 non multiple of e 0 , see [25]).B B B
1 Proof. We take M such that V (y) (with M ( 0)). To simplify nota- TB = B = B tions, we omit indices in WP,i, and consider i 1, . . . , e . B B 2 { B}
Suppose that I, J, S . Then W , (0) y 0, W , (0) y 0 • 62 Tm1 B0 B i = B 6= A i = A 6= and W , (0) y 0 so S0 i = S0 6=
0 R (0) 0 y y S y y B y y A ˆi A B 0 A S0 B S0 = 0 1 1 ^ · · · ⇥ ⇤ @ A yA yB yS0 yA yB xS0 yA yS0 xB yB yS0 x A . = 0 0 1 @ A
Hence hm1,i, 0 and the sum over i 1, . . . , e of these quantities is equal to 0. B = = B Suppose I , J, and m I. Take M such that S (0, 1, 0), • 2 Tm1 B0 S 62 Tm1 B0 1 6= 0 A(1, 0, 0), y 0. We have W , (0) y , W , (0) 0 and W , (0) 1 and B 6= B i = B A i = S0 i = 0 yB 0 so R (0) 0 0 y . Hence h 0 and the sum ˆi B m1,i, = 0 1 1 ^ 0 0 1 = 0 0 1 B = over i 1, .@. . , eA of@these quAantiti@es is equAal to 0. Suppos=e I B , J, S and m I. Take M such that S (0, 1, 0), • 2 Tm1 B0 62 Tm1 B0 1 = 0 A(0, 0, 1), y 0. We have W , (x) y g (x)x z (xg (x) g (x)), B 6= B i = B i0 B + B i0 i 898 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
W , (x) xg (x) g (x) and W , (x) 1 and so A i = i0 i S0 i =
x (xgi0(x) gi )xB ( ) g (x) (xg (x) g )( g (x)x z (xg (x) g (x))) Rˆi x i i0 i i0 B B i0 i = 0 1 1 ^ 0 y g ( x)x 2z+(xg (x) g (x)) 1 B + i0 B B i0 i @ A @ 2 2 2 A yB gi (x) x(gi0(x)) xB zB((xgi0(x)) (gi (x)) ) x (2xg (+x) g (x)) x y 2xz (xg ( x) g (x)) , B i0 i + B + B i0 i = 0 x (xg (x) g (x))2 z (xg (x) g (x)) 1 B i0 i + B i0 i @ A the valuation of the coordinates of which are larger than or equal to 1 and the valuation of the second coordinate is 1. Hence hm ,i, 1 and the sum over 1 B = i 1, . . . , e e 0 of these quantities is equal to e 0 . Su=ppose S B = B , I, J and m S. TaBke M such that A(0, 1, 0), • 2 Tm1 B0 62 Tm1 B0 1 6= S (1, 0, 0), y 0. We have W , (0) y 0, W , (0) 0 and W , (0) 1 0 B 6= B i = B 6= S0 i = A i = 0 yB 0 and so R (0) 0 0 y . Hence h 0 and the sum ˆi B m1,i, = 0 1 1 ^ 0 0 1 = 0 0 1 B = over i 1, . . . , e@ ofAthes@e quanAtities@is equAal to 0. Suppos=e m SBand I, J . Take M such that S (0, 0, 1), A(0, 1, 0), • 1 = 62 Tm1 B0 0 y 0. We have W , (x) y g (x)x z (xg (x) g (x)), W , (x) B 6= B i = B i0 B + B i0 i S0 i = xg (x) g (x) and W , (x) 1 and so i0 i A i = x (xg (x) g )x i0 i B R (x) g (x) (xg (x) g (x))(2y g (x)x z (xg (x) g (x))) ˆi i i0 i B i0 B + B i0 i =0 1 1 ^ 0 y g (x)x 1 B i0 B g@(x)(y A g@(x)x ) (xg (x) g (x))(2y g (x)x z (xg (x) g (xA))) i B i0 B + i0 i B i0 B + B i0 i (xg (x) g (x))x x(y g (x)x ) , i0 i B B i0 B =0 (xg (x) g (x))(2xy g (x)xx z x(xg (x) g (x)) g (x)(xg (x) g (x))x )1 i0 i B i0 B + B i0 i + i i0 i B @ A the valuation of the coordinates of which are larger than or equal to 1 and the valuation of the second coordinate is 1. Hence hm ,i, 1 and the sum over 1 B = i 1, . . . , e of these quantities is equal to e 0 . Su=ppose B (I S), J and Bm I, S . Take M such that • Tm1 B0 = 62 Tm1 B0 1 62 { } S (1, 0, 0), B(0, 1, 0), y 0, x 0, z 0. We have W , (x) g (x), 0 A = A 6= A 6= S0 i = i0 W , (x) g (x)x z (xg (x) g (x)) and W , (x) 1 and so A i = i0 A + A i0 i B i =
x z A(xgi0(x) gi (x)) 2 Ri (x) gi (x) (g (x)) x g (x)(xg (x) g (x))z ˆ = 0 1 ^ 0 i0 A + i0 i0 i A 1 1 gi0(x)z A @ A @ 2 A gi (x)gi0(x)z A (gi0(x)) x A gi0(x)(xgi0(x) gi (x))z A + g (x )z , i A = 0 x(g (x))2x (xg (x) g (x))2z 1 i0 A + i0 i A @ A the valuation of the coordinates of which are respectively 2val(g ) 2, val(g ) i i and 2val(gi ) 1. Hence hm ,i, val(gi ) min(val(gi ) 2, 0) and the 1 B = + ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 899
sum over i 1, . . . , e of these quantities is equal to im ( 0, m 0) = B 1 B T 1 B + min(im1 ( 0, m1 0) 2e 0 , 0). Suppose B T B (I S), JB and m I. Take M such that S (1, 0, 0), • Tm1 B0 = 62 Tm1 B0 1 = 0 B(0, 1, 0), A(0, 0, 1). We have W , (x) g (x), W , (x) xg (x) g (x) S0 i = i0 A i = i0 i and W , (x) 1 and so B i = x xg (x) g (x) g0(x)(2gi (x) xg0(x)) i0 i i i Ri (x) gi (x) gi0(x)(xgi0(x) gi (x)) gi (x) , ˆ =0 1 ^ 0 1=0 2 1 1 g0(x) (xg (x) g (x)) i i0 i @ A @ A @ A the valuation of the coordinates of which are larger than or equal to val(gi ), the second coordinate has valuation val(gi ). Hence hm ,i, val(gi ) and the sum 1 B = over i 1, . . . , e of these quantities is equal to im1 ( 0, m1 0). Suppos=e B (I S), J and m S. TakBe MT suBch that A(1, 0, 0), • Tm1 B0 = 62 Tm1 B0 1 = B(0, 1, 0), S (0, 0, 1). We have W , (x) xg (x) g (x), W , (x) g (x) 0 S0 i = i0 i A i = i0 and W , (x) 1 and so B i = x (xg (x) g (x)) x(g (x))2 i0 i i0 Ri (x) gi (x) gi0(xgi0(x) gi (x)) gi (x) , ˆ = 0 1 ^ 0 1 = 0 2 2 1 1 g0(x) (xg (x)) (g (x)) i i0 i @ A @ A @ A the valuation of the coordinates of which being larger than or equal to val(gi ) and the valuation of the second coordinate is equal to val(gi ). Hence hm ,i, val(gi ) and the sum over i 1, . . . , e of these quantities is equal to 1 B = = B im1 ( 0, m1 0). SuppBoseT B (I J), S and m I, J . Take M such that • Tm1 B0 = 62 Tm1 B0 1 62 { } S (0, 1, 0), B(1, 0, 0), y 0, x 0, z 0. We have W , (x) g (x), 0 A = A 6= A 6= B i = i0 W , (x) g (x)x z (xg (x) g (x)) and W , (x) 1 and so A i = i0 A + A i0 i S0 i =
x 2gi0(x)x A z A(xgi0(x) gi (x)) 2 Ri (x) gi (x) (g (x)) x g (x)(xg (x) g (x))z ˆ = 0 1 ^ 0 i0 A i0 i0 i A 1 1 gi0(x)z A @ A @ 2 A (gi0(x)) (xz A x A) 2g (x)x z (2xg (x) g (x)) , i0 A A i0 i = 0 z (xg (x) g (x))2 x g (x)(xg (x) 2g (x)) 1 A i0 i + A i0 i0 i @ A the valuation of the coordinates of which are respectively 2val(g ) 2, val(g ) 1 i i and larger than val(gi ). Hence hm ,i, val(gi ) 1 and the sum over i 1 B = = 1, . . . , e of these quantities is equal to im1 ( 0, m1 0) e 0 . SupposeBthat (I J), that S B TanBd m B I. Take M such • Tm1 B0 = 62 Tm1 B0 1 = that S (0, 1, 0), B(1, 0, 0), A(0, 0, 1). We have W , (x) g (x), W , (x) 0 B i = i0 A i = xg (x) g (x) and W , (x) 1 and so i0 i S0 i = x (xg (x) g (x)) x(g (x))2 i0 i i0 Ri (x) gi (x) gi0(x)(xgi0(x) gi (x)) (2xgi0(x) gi (x)) , ˆ = 0 1^0 1=0 2 1 1 g0(x) (xg (x) g (x)) i i0 i @ A @ A @ A 900 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
the valuation of the coordinates of which being larger than or equal to val(gi ) and the valuation of the second coordinate is equal to val(gi ). Hence hm ,i, val(gi ) and the sum over i 1, . . . , e of these quantities is equal to 1 B = = B im1 ( 0, m1 0). SuppBoseTthaBt I, J, S and m I, J, S . Take M such that S (1, 0, 0), • 2 Tm1 B0 1 62 { } 0 yA yB 0, x A 0, z A 0, xB 0, zB 0, x AzB xB z A. We have =( ) = ( ) 6= ( ) 6= ( )6= (6= ( ) (6=)) ( ) WS0,i x gi0 x , WA,i x gi0 x x A z A xgi0 x gi x and WB,i x g (x)x = z (xg (x) g (=x) ) and so + = i0 B + B i0 i x x (g (x))2x z z (xg (x) g (x))2 A i0 B + A B i0 i R (x) gi (x) 0 ˆi = 0 1 ^ 0 1 1 (g (x))2(xz x z ) 2z z g (x)(xg (x) g (x)) i0 B + B A A + A B i0 i0 i @ A @ A g (x)(g (x))2(xz x z ) 2z z g (x)g (x)(xg (x) g (x)) i i0 B + B A A + A B i i0 i0 i 2 2 , 0 x A(gi0(x)) xB z AzB(xgi0(x) gi (x)) x . . . . 1 = 2+ [ 2] x Agi (x)(g0(x)) xB z AzB gi (x)(xg0(x) gi (x)) B i i C the valuati@on of the coordinates of which are larger than or equal to 2val(Ag ) i 2, the valuation of the second coodinate is 2val(gi ) 2. Hence hm ,i, 1 B = 2val(gi ) 2 and the sum over i 1, . . . , e of these quantities is equal to = B 2im1 ( 0, m1 0) 2e 0 . SuppoBse TthatBI, J , S B and m J. Take M such that B(0, 0, 1), • 2 Tm1 B0 1 = S (1, 0, 0), y 0, x 0 and z 0. We have W , (x) g (x), 0 A = A 6= A 6= S0 i = i0 WA,i (x) gi0(x)x A z A(xgi0(x) gi (x)) and WB,i (x) xgi0(x) gi (x) and so = + = 2 x z A(xgi0(x) gi (x)) ( ) g (x) 0 Rˆi x i =0 1 1 ^ 0 x (g (x))2 2z (xg (x) g (x))g (x) 1 A i0 + A i0 i i0 @ A g@(x)g (x)( g (x)x 2z (xg (x) g (x)))A i i0 i0 A + A i0 i z (xg (x) g (x))2 xg (x)( g (x)x 2z (xg (x) g (x))) , A i0 i i0 i0 A + A i0 i =0 g (x)z (xg (x) g (x))2 1 i A i0 i the valua@tion of the coordinates of which are larger than or equal to 2val(g ) A1 i and the valuation of the second coordinate is 2val(gi ) 1. Hence hm ,i, 1 B = 2val(gi ) 1 and the sum over i 1, . . . , e of these quantities is equal to = B 2im1 ( 0, m1 0) e 0 . SuppoBse TthatBI, J , S B and m S. Take M such that S (0, 0, 1), • 2 Tm1 B0 1 = 0 B(1, 0, 0), yA 0, x A 0 and z A 0. We have WB,i (x) gi0(x), ( ) =( ) 6=( ( ) 6=( )) ( ) (=) ( ) WA,i x gi0 x x A z A xgi0 x gi x and WS0,i x xgi0 x gi x and so = + = x 2x g (x)(xg (x) g (x)) z (xg (x) g (x))2 A i0 i0 i A i0 i Ri (x) gi (x) 0 ˆ = 0 1 ^ 0 2 1 1 (gi0(x)) x A @ A @ 2 A gi (x)(gi0(x)) x A x g (x)(xg (x) 2g (x)) z (xg (x) g (x))2 . = 0 A i0 i0 i A i0 i 1 2x g (x)g (x)(xg (x) g (x)) z g (x)(xg (x) g (x))2 A i i0 i0 i + A i i0 i @ A ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 901
The valuation of the first coordinate is 3val(gi ) 2 is smaller than or equal to the valuation of the third coordinate. If val(g ) 2, the valuation of the second coordinate is 2val(g ) 1; hence i 6= i hm ,i, 2val(gi ) 1 and the sum over i 1, . . . , e of these quantities is 1 B = = B equal to 2im ( 0, m 0) e . 1 B T 1 B B0 Suppose now that val(gi ) 2, then 3val(gi ) 2 4 and there exist ↵, ↵1 C = 2 = 2 and 1 > 2 such that gi (x) ↵x ↵1x 1 . . .. Then, the second coordinate = + 1 + 4 has the following form (x A2↵( 1 2)x 1+ . . .) x (. . .). Therefore hm ,i, + + 1 B = min( 1 1, 4) and the sum over i 1, . . . , e of these quantities is equal to + = B e (1 min( 1, 3)). B0 + Proof of Theorem 3.1. Recall that (6.4) says
mclass(6S( )) 2d_ d hm , C = + 1 B0 m Branchm ( ) X12C B02 X 1 C and that the values of hm , have been given in Lemma 6.5. 1 B0
Assume first S ` . Then we have to sum the hm1, 0 coming from Items 3, 4, • 5, 6 and 7 of Le62mm1a 6.5. B The sum of the hm , coming from Items 3 and 5 applied with m1 S gives 1 B0 = directly g0. The sum of the hm , coming from Items 3, 5 and 7 applied with m1 I, J 1 B0 2 { } gives f I ( , ` ) J ( , ` ). + C 1 + C 1 The sum of the hm , coming from Item 6 gives g I ( , ` ) J ( , ` ). 1 B0 C 1 C 1 The sum of the hm , coming from Item 4 gives 2 f 0 q0 (notice that hm , 1 B0 1 B0 = 2(im1 ( 0, m1 0) e 0 ) (im1 ( 0, m1 0) 2e 0 )1im ( 0, m 0) 2e ). B T B B B T B B 1 B T 1 B B0 Assume first S ` . Then we have to sum the hm1, 0 coming from Items 3, 8, • 9, 10 and 11 of 62Lem1ma 6.5. B The sum of the hm , coming from Items 3 (with m1 S), 10 and 11 gives 1 B0 = 2S( , ` ) µS c0(S). C 1 + + The sum of the hm , coming from Items 3 and 9 applied with m1 I, J 1 B0 2 { } gives 2(I ( , ` ) J ( , ` )) µI µJ . C 1 + C 1 + + The sum of the hm , coming from Item 8 gives 2(g I ( ,` ) J ( ,` ) 1 B0 C 1 C 1 S( ,` )). C 1
Appendix
A. Intersection numbers of curves and pro-branches
The following result expresses the classical intersection number im1 ( , 0) defined in [15, page 54] thanks to the use of probranches. C C 2 Proposition A.1. Let m P . Let V (F) and 0 V (F0) be two algebraic 2 C = C = plane curves containing m, with homogeneous polynomials F, F0 C X, Y, Z . 2 [ ] 902 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE
3 Let M GL(C ) be such that 5(M(P0)) m and such that the tangent cones of V (F 2M) and of V (F M) do not contai=n V (X). 0 Assume that V (F M) admits b branches at P and that its -th branch 0 B has multiplicity e . Assume that V (F M) admits b branches at P and that its 0 0 0 0-th branch 0 has multiplicity e0 . B 0 0 Then we have
e 1 b e 1 b 0 1 0 0 1 j e j0 e0 im( , 0) valx h ⇣ x h0 ⇣ 0 x 0 , C C = 0 1 j 0 0 1 j0 0 " ✓ ◆ !# X= X= X= X= 1 j e with y h (⇣ x ) C x⇤ an equation of the j-th probranch of at P0, = 1 2 h i B e j0 0 y h0 (⇣ 0 x 0 ) x⇤ an equation of the k0-th probranch of 0 at P0, with 0 C 0 = 2 2i⇡h i B 2i⇡ e e0 ⇣ e and ⇣ e 0 . := 0 := With the notations of Proposition A.1, we get
b
im( , 0) i P0 , V (F0) , (A.1) C C = 1 B X= with the usual definition given in [24] of intersection number of a branch with a curve
e 1 1 j e i , V (F0 M) val F0 M x, h , ⇣ x . (A.2) P0 B = x j j 0 ✓ ✓ ✓ ◆◆◆ X= Proof of Proposition A.1. By definition, the intersection number is defined by
C X, Y, Z i ( , 0) i (V (F M, F0 M) length [ ] m C C = P0 = (F M, F M) ✓ 0 ◆(X,Y,Z)! C X,Y,Z where ( [ ] )(X,Y,Z) is the local ring in the maximal ideal (X, Y, Z) of P0 (F M,F0 M) [15, page 5 3]. A ccording to [13], we have
C X, Y, Z i ( , 0) dim [ ] . m C C = C (F M, F M) ✓ 0 ◆(X,Y,Z)!
Let f, f 0 be defined by f (x, y) F M(x, y, 1), f 0(x, y) F0 M(x, y, 1). We get = = C x, y C x, y i ( , 0) dim [ ] dim h i. m C C = C ( f, f ) = C ( f, f ) ✓ 0 ◆(x,y)! 0 ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 903
Recall that, according to the Weierstrass preparation theorem, there exist two units U and U 0 of C x, y and f1, . . . , fb, f10, . . . , fb0 C x y monic irreducible such that h i 0 2 h i[ ] b b0 f U f and f 0 U 0 f 0 , = = 0 1 1 Y= Y0= f 0 being an equation of and f 0 0 being an equation of 0 . According = B 0 = B 0 to the Puiseux theorem, (respectively 0 ) admits a parametrization B B 0
e e0 x t x t 0 = respectively = . x h (t) C t x h0 (t) C t ⇢ = 2 h i ( = 0 2 h i ! 1 j e We know that, for every 1, .., b and every j 0, .., e , h (⇣ x ) 2 { } 1 2 { 1 } 2 1 e e e j0 0 0 C x are the y-roots of f (respectively h (⇣ 0 x 0 ) C x 0 are the y-roots h i 0 2 h i of f 0 ). In particular, we have 0
e 1 e 0 1 1 1 0 j e j0 e0 f (x, y) y h ⇣ x and f 0 (x, y) y h0 ⇣ 0 x 0 . = 0 = 0 j 0 ✓ ✓ ◆◆ j0 0 !! Y= Y= Therefore we have the following sequence of C-algebra-isomorphisms: b C x, y C x, y h i h i A , ( , ) b ⇠ f f 0 = = 1 f (x, y), f 0(x, y) Y= 1 ! Q= C x,y where A h i . Let 1, . . . , b . We observe that we have := ( f (x,y), f 0(x,y)) 2 { }
e 1 C x A h i . = 1 j 0 j e Y= f 0 x, h ⇠ x ✓ ✓ ✓ ◆◆◆ On another hand, we have
1 1 C x e , y C x e , y D ⌧ ⌧ e 1 := f (x, y), f 0(x, y) = 1 j e y h ⇣ x , f 0(x, y) j 0 ! = ✓ ✓ ◆◆ 1 Q e e 1 x , y e 1 C ⌧ D ⇠ 1 ⇠ , j = j e = j 0 y h ⇣ x , f (x, y) j 0 Y= 0 Y= ✓ ✓ ◆ ◆ 904 ALFREDERIC JOSSE AND FRANC¸ OISE PE` NE with 1 C x e ⌧ . D , j 1 := j e f 0 x, h ⇣ x ✓ ✓ ✓ ◆◆◆ We consider now the natural extension of rings i A , , D , such that : j ! j g A , val 1/e i (g) (x) e val (g(x)). 8 2 x = x We have 1 e e 1 x C D ⌧ , ⇠ v = j 0 x Y= 1 1 e j e where v is the valuation in x of ( f 0(x, h (⇣ x ))), i.e.
1 e j j e v val f 0 t , h ⇣ t e val f 0 x, h ⇣ x . := t = x We get
b b e 1 1 e j im( , 0) dimC A valt f 0 t , h ⇣ t C C = 1 = 1 j 0 e X= X= X= b e 1 1 j e valx f 0 x, h ⇣ x = 1 j 0 X= X= ✓ ✓ ✓ ◆◆◆ b e 1 b 0 1 j e valx f 0 x, h ⇣ x . = 0 1 j 0 0 1 ✓ ✓ ✓ ◆◆◆ X= X= X= Observe now that 1 j e 1 valx f 0 x, h ⇣ x N 0 2 e ✓ ✓ ✓ ◆◆◆ and that
e0 1 1 1 0 1 j e j0 e0 j e f 0 x, h ⇣ x Res f 0 , f y h0 ⇣ 0 x 0 h ⇣ x , 0 ⌘ 0 ; ⌘ 0 ✓ ✓ ◆◆ j0 0 ! ✓ ◆! Y= where Res denotes the resultant and where means “up to a non zero scalar”. Finally, we get ⌘
e 1 b e 1 b 0 1 0 0 1 j0 e0 j e im( , 0) valx h0 ⇣ 0 x 0 h ⇣ x . C C = 0 1 j 0 0 1 j0 0 " ! ✓ ◆# X= X= X= X= ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 905
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Universite´ de Brest UMR CNRS 6205 Laboratoire de Mathe´matique de Bretagne Atlantique 6 avenue Le Gorgeu 29238 Brest cedex, France [email protected]
Universite´ de Brest and Institut Universitaire de France UMR CNRS 6205 Laboratoire de Mathe´matique de Bretagne Atlantique 6 avenue Le Gorgeu 29238 Brest cedex, France [email protected]