On the class of caustics by reflection
aa r X i v : . [ m a t h . AG ] A p r ON THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES
ALFREDERIC JOSSE AND FRANC¸ OISE P`ENE
Abstract.
Given any light position S ∈ P and any algebraic curve C of P (with any kindof singularities), we consider the incident lines coming from S (i.e. the lines containing S ) andtheir reflected lines after reflection on the mirror curve C . The caustic by reflection Σ S ( C ) is theZariski closure of the envelope of these reflected lines. We introduce the notion of reflected polarcurve and express the class of Σ S ( C ) in terms of intersection numbers of C with the reflectedpolar curve, thanks to a fundamental lemma established in [15]. This approach enables us tostate an explicit formula for the class of Σ S ( C ) in every case in terms of intersection numbersof the initial curve C . Introduction
Let V be a three dimensional complex vector space endowed with some fixed basis. Weconsider a light point S [ x : y : z ] ∈ P := P ( V ) and a mirror given by an irreducible algebraiccurve C = V ( F ) of P , with F ∈ Sym d ( V ∨ ) ( F corresponds to a polynomial of degree d in C [ x, y, z ]). We denote by d ∨ the class of C . We consider the caustic by reflection Σ S ( C ) of themirror curve C with source point S . Recall that Σ S ( C ) is the Zariski closure of the envelope ofthe reflected lines associated to the incident lines coming from S after reflection off C . When S is not at infinity, Quetelet and Dandelin [17, 9] proved that the caustic by reflection Σ S ( C ) isthe evolute of the S -centered homothety (with ratio 2) of the pedal of C from S (i.e. the evoluteof the orthotomic of C with respect to S ). This decomposition has also been used in a modernapproach by [2, 3, 4] to study the source genericity (in the real case). In [15] we stated formulasfor the degree of the caustic by reflection of planar algebraic curves.In [7], Chasles proved that the class of Σ S ( C ) is equal to 2 d ∨ + d for a generic ( C , S ). In [1],Brocard and Lemoyne gave (without any proof) a more general formula only when S is not atinfinity. The Brocard and Lemoyne formula appears to be the direct composition of formulasgot by Salmon and Cayley in [18, p. 137, 154] for some geometric characteristics of evolute andpedal curves. The formula given by Brocard and Lemoyne is not satisfactory for the followingreasons. The results of Salmon and Cayley apply only to curves having no singularities otherthan ordinary nodes and cusps [18, p. 82], but the pedal of such a curve is not necessarily a curvesatisfying the same properties. For example, the pedal curve of the rational cubic V ( y z − x )from [4 : 0 : 1] is a quartic curve with a triple ordinary point. Therefore it is not correct tocompose directly the formulas got by Salmon and Cayley as Brocard and Lemoyne apparentlydid (see also Section 5 for a counterexample of the Brocard and Lemoyne formula for the classof the caustic by reflection).Let us mention some works on the evolute and on its generalization in higher dimension[10, 19, 6]. In [10], Fantechi gave a necessary and sufficient condition for the birationality of theevolute of a curve and studied the number and type of the singularities of the general evolute. Let Date : January 10, 2018.2000
Mathematics Subject Classification.
Key words and phrases. caustic, class, polar, intersection number, pro-branch, Pl¨ucker formulaFran¸coise P`ene is supported by the french ANR project GEODE (ANR-10-JCJC-0108).
1N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 2 us insist on the fact that there exist irreducible algebraic curves (other than lines and circles) forwhich the evolute map is not birational. This study of evolute is generalized in higher dimensionby Trifogli in [19] and by Catanese and Trifogli [6].The aim of the present paper is to give a formula for the class (with multiplicity) of the causticby reflection for any algebraic curve C ( without any restriction neither on the singularitypoints nor on the flex points ) and for any light position S (including the case when S isat infinity or when S is on the curve C ).In Section 1, we define the reflected lines R m at a generic m ∈ C and the (rational) “reflectedmap” R C ,S : P → P mapping a generic m ∈ C to the equation of R m .In section 2, we define the caustic by reflection Σ S ( C ), we give conditions ensuring that Σ S ( C )is an irreducible curve and we prove that its class is the degree of the image of C by R C ,S .In section 3, we give formulas for the class of caustics by reflection valid for any ( C , S ). Theseformulas describe precisely how the class of the caustic depends on geometric invariants of C and also on the relative positions of S and of the two cyclic points I, J with respect to C . As aconsequence of this result, we obtain the following formula for the class of Σ S ( C ) valid for any C of degree d ≥ S : class (Σ S ( C )) = 2 d ∨ + d − Ω( C , ℓ ∞ ) − µ I ( C ) − µ J ( C ) , where Ω( C , ℓ ∞ ) is the contact number of C with the line at infinity ℓ ∞ and with µ I ( C ) and µ J ( C )are the multiplicities number of respectively I and J on C .In Section 4, our formulas are illustrated on two examples of curves (the lemniscate of Bernoulliand the quintic considered in [15]).In section 5, we compare our formula with the one given by Brocard and Lemoyne for a lightposition not at infinity. We also give an explicit counter-example to their formula.In Section 6, we prove our main theorem. In a first time, we give a formula for the class ofthe caustic in terms of intersection numbers of C with a generic “reflected polar” at the basepoints of R C ,S . In a second time, we compute these intersection numbers in terms of the degree d and of the class d ∨ of C but also in terms of intersection numbers of C with each line of thetriangle IJ S .In appendix A, we prove a useful formula expressing the classical intersection number in termsof probranches. 1.
Reflected lines R m and rational map R C ,S Recall that we consider a light position S [ x : y : z ] ∈ P and an irreducible algebraic(mirror) curve C = V ( F ) of P given by a homogeneous polynomial F ∈ Sym d ( V ) with d ≥ Sing ( C ) for the set of singular points of C . For any non singular point m , we write T m C for the tangent line to C at m . We set S ( x , y , z ) ∈ V \ { } . For any m [ x : y : z ] ∈ P ,we write m ( x, y, z ) ∈ V \ { } . We write as usual ℓ ∞ = V ( z ) ⊂ P for the line at infinity. Forany P ( x , y , z ) ∈ V \ { } , we define∆ P F := x F x + y F y + z F z ∈ Sym d − ( V ∨ ) . Recall that V (∆ P F ) is the polar curve of C with respect to P [ x : y : z ] ∈ P .Since the initial problem is euclidean, we endow P with an angular structure for which I [1 : i : 0] ∈ P and J [1 : − i : 0] ∈ P play a particular role. To this end, let us recall thedefinition of the cross-ratio β of 4 points of ℓ ∞ . Given four points ( P i [ a i : b i : 0]) i =1 ,..., such N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 3 that each point appears at most 2 times, we define the cross-ratio β ( P , P , P , P ) of these fourpoints as follows : β ( P , P , P , P ) = ( b a − b a )( b a − b a )( b a − b a )( b a − b a ) , with convention = ∞ . For any distinct lines A and A not equal to ℓ ∞ , containing neither I nor J , we define the oriented angular measure between A and A by θ (modulo π Z ) such that e − iθ = β ( P , P , I, J ) = ( a + ib )( a − ib )( a − ib )( a + ib )(where P i [ a i : b i : 0] is the point at infinity of A i ). Let Q ∈ Sym ( V ∨ ) be defined by Q ( x, y ) := x + y . It will be worth noting that Q ( ∇ F ) = F x + F y = ∆ I F ∆ J F . For every non singularpoint m of C \ ℓ ∞ , we recall that t m [ F y : − F x : 0] ∈ P is the point at infinity of T m C and so t m
6∈ {
I, J } is equivalent to m V ( Q ( ∇ F )).Now, for any m ∈ C \ ( ℓ ∞ ∪ Q ( ∇ F )) and any incident line ℓ containing m , we define asfollows the associated reflected line R m ( ℓ ) (for the reflexion on C at m with respect to theSnell-Descartes reflection law Angle ( ℓ, T m ) = Angle ( T m , R m )). Definition 1.
For every m ∈ C \ ( ℓ ∞ ∪ V ( Q ( ∇ F ))) , we define r m : ℓ ∞ → ℓ ∞ mapping P ∈ ℓ ∞ to the unique r m ( P ) such that β ( P, t m , I, J ) = β ( t m , r m ( P ) , I, J ) .We define R m : F m → F m with F m := { ℓ ∈ G (1 , P ) , m ∈ ℓ } by R m ( ℓ ) = ( m r m ( P ℓ )) if P ℓ is the point at infinity of ℓ . We have (on coordinates) r m ([ x : y : 0]) = [ x ( F x − F y ) + 2 y F x F y : − y ( F x − F y ) + 2 x F x F y : 0] Remark 2.
Observe that r m is an involution on ℓ ∞ ∼ = P with exactly two fixed points t m and n m [ F x : F y : 0] . As a consequence, R m is an involution with two fixed points T m ( C ) and N m ( C ) := ( mn m ) the normal line to C at m .Moreover r m ( I ) = J and r m ( J ) = I . Definition 3.
For any m [ x : y : z ] ∈ C \ ( { S } ∪ ℓ ∞ ∪ V ( Q ( ∇ F )) we define the reflected line R m on C at m (of the incident line coming from S ) as the line R m := R m (( mS )) . For m [ x : y : z ] ∈ C \ ( { S } ∪ ℓ ∞ ∪ V ( Q ( ∇ F )), the point at infinity of ( Sm ) is s m [ x z − z x : y z − z y : 0]. Due to the Euler identity, on C , we have xF x + yF y + zF z = 0 and so( x z − z x ) F x + ( y z − z y ) F y = z ∆ S F . Hence r ( s m ) = [ − v m : u m : 0] and the reflected line R m is the set of P [ X : Y : Z ] ∈ P such that u m X + v m Y + w m Z = 0, with u m := ( z y − zy )( F x + F y ) + 2 z ∆ S F.F y ∈ Sym d − ( V ∨ ) v m := ( zx − z x )( F x + F y ) − z ∆ S F.F x ∈ Sym d − ( V ∨ ) w m := − xu m − yv m z = ( xy − yx )( F x + F y ) − S F ( xF y − yF x ) ∈ Sym d − ( V ∨ ) . Definition 4.
We call reflected map of C from S the following rational map R C ,S : P → P m [ u m : v m : w m ] . We also define the rational map T C ,S := ( R C ,S ) |C : C → P . N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 4
For any m ∈ V , it will be useful to define R F, S ( m ) := ( u m , v m , w m ) ∈ V and to notice that R F, S ( m ) = Q ( ∇ F ( m )) · ( m ∧ S ) − S F ( m ) · ( m ∧ n m ) ∈ V , with n m ( F x ( m ) , F y ( m ) , ∈ V . Proposition 5.
The base points of T C ,S are the following: I , J , S (if these points are in C ), the singular points of C and the points of tangency of C withsome line of the triangle ( IJ S ) .Proof. We have to prove that the set of base points of T C ,S is the following set: M := C ∩ ( { I, J, S } ∪ V (∆ S F, Q ( ∇ F )) ∪ V ( F x , F y )). We just prove that Base ( T C ,S ) ⊂ M , the conversebeing obvious (observe that if m ∈ { I, J } , we automatically have Q ( ∇ F ( m )) = 0 and n m ∈ V ect ( m )). Let m [ x ; y ; z ] ∈ C be such that R F, S ( m ) = . Then m and Q ( ∇ F ( m )) · S − S F ( m ) · n m are colinear. Due to the Euler identity, we have 0 = DF ( m ) · m (with DF ( m )the differential of F at m ) and so 0 = − ∆ S F ( m ) · Q ( ∇ F ( m )) since DF ( m ) · S = ∆ S F ( m ) andsince DF ( m ) · n m = Q ( ∇ F ( m )). Hence ∆ S F ( m ) = 0 or Q ( ∇ F ( m )) = 0.If ∆ S F ( m ) = 0, then either Q ( ∇ F ( m )) = 0 or m = S .If ∆ S F ( m ) = 0 and Q ( ∇ F ( m )) = 0 , then F x ( m ) = F y ( m ) = 0 or m = [ F x ( m ) : F y ( m ) : 0].Assume that m = [ F x ( m ) : F y ( m ) : 0]. Then, since Q ( ∇ F ( m )) = 0, we conclude that m ∈{ I, J } . (cid:3) In the following result, we state the S -generic birationality of T C ,S . We give a short versionof the proof of [16]. Let us indicate that another proof of the same result has been establishedat the same period by Catanese in [5]. Proposition 6 (see also [16, 5]) . Let C be an irreducible curve of degree d ≥ . Then, for ageneric S ∈ P , the map T C ,S is birational.Proof. For every m ∈ C := C \ ( ℓ ∞ ∪ V ( Q ( ∇ F ))) and every S ∈ P \ { m } , we write R m,S forthe reflected line R m (( mS )). For every m ∈ C , we consider the set K m := { S ∈ P \ C : ∃ m ′ ∈C \ { m } , R m,S = R m ′ ,S } . • Let us prove that, for any m ∈ C , K m is contained in an algebraic curve ¯ K m of degreeless than 2 d + 2.Let m ∈ C . Consider S ∈ P \ C and m ′ ∈ C \ { m } such that R m,S = R m ′ ,S . Thenwe have R m,S = R m ′ ,S = ( mm ′ ) and so S ∈ R m,m ′ ∩ R m ′ ,m .Assume first that R m,m ′ = R m ′ ,m . Then this line is ( mm ′ ) and it is its own reflectedline both at m and at m ′ . This implies that ( mm ′ ) is either T m C or N m C , so that S ∈ T m C ∪ N m C .Assume now that R m,m ′ = R m ′ ,m . Then S = τ m ( m ′ ) with τ m : P → P the rationalmap associated to τ m : V → V with τ m ( m ′ ) = R F, m ′ ( m ) ∧ R F, m ( m ′ ). Hence K m ⊆ ¯ K m := T m C ∪ N m C ∪ τ m ( C ), where A is the Zariski closure of a set A . Since the degree(in m ′ ) of the coordinates of τ m is 2 d , we conclude that deg ¯ K m ≤ d + 2. • The set K of points S ∈ P \ C such that R C ,S is not birational is contained in¯ K := [ E ⊂C : E< ∞ \ m ∈C \ E ¯ K m . with ∧ : V × V → V being given in coordinates by ( x , y , z ) ∧ ( x , y , z ) = z y − z y z x − z x x y − y x . N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 5
To conclude we will apply the Zorn lemma. We have to prove that { T m ∈C \ E ¯ K m , E < ∞} is inductive for the inclusion. Let ( F j := T m ∈C \ E j ¯ K m ) j ≥ be an increasing sequenceof sets (with E j finite subsets of C ). Write Z for the union of these sets. Observe that Z ⊆ ¯ K m for some fixed m ∈ C \ S i ≥ E i . The set ¯ K m is the union of irreduciblealgebraic curves C , ..., C p . We write d i for the degree of C i . If C i ⊆ Z , we write N i := min { j ≥ C i ⊂ F j } . If C i Z , then ( C i ∩ F j ) j ≥ is an increasing sequence offinite sets containing at most d i (2 d + 2) points and we set N i := min { j : ( C i ∩ Z ) ⊆ F j } .We obtain Z = F max( N ,...,N p ) . Due to the Zorn lemma, there exists a finite set E suchthat K ⊂ T m ∈C \ E ¯ K m , from which the result follows. (cid:3) Caustic by reflection
Definition 7.
The caustic by reflection Σ S ( C ) is the Zariski closure of the envelope of thereflected lines {R m ; m ∈ C \ ( { S } ∪ ℓ ∞ ∪ V ( Q ( ∇ F )) } . Recall that, in [15], we have defined a rational map Φ F, S called caustic map mapping ageneric m ∈ C to the point of tangency of Σ S ( C ) with R m and that Σ S ( C ) is the Zariski closureof Φ F, S ( C ).In the present work, we will not consider the cases in which the caustic by reflection Σ S ( C ) isa single point. We recall that these cases are easily characterized as follows. Proposition 8.
Assume that (i) S
6∈ {
I, J } , (ii) C is not a line (i.e. d = 1 ), (iii) if d = 2 , then S is not a focus of the conic C .Then Σ S ( C ) is not reduced to a point and is an irreducible curve.Proof. Assume (i), (ii) and (iii) and that Σ S ( C ) = { S ′ } with S ′ = [ x : y : z ].When S ℓ ∞ , we will use the fact that Σ S ( C ) is the evolute of the orthotomic of C withrespect to S . Since C is not a line, the orthotomic of C with respect to S is not reduced to apoint but its evolute is a point. This implies that the orthotomic of C with respect to S is eithera line (not equal to ℓ ∞ ) or a circle. But C is the contrapedal (or orthocaustic) curve (from S )of the image by the S -centered homothety (with ratio 1 /
2) of the orthotomic of C . Therefore d = 2 and S is a focal point of C , which contradicts (iii).When S ∈ ℓ ∞ but S ′ ℓ ∞ , then, for symetry reasons, we also have Σ S ′ ( C ) = { S } and weconclude analogously.Suppose now that S, S ′ ∈ ℓ ∞ . We have z = z = 0. For every m = [ x : y : 1] ∈ C \ ( ℓ ∞ ∪ V ( Q ( ∇ F ))), we have β ( S, t m , I, J ) = β ( t m , S ′ , I, J ) Therefore we have( ix − y )( − iF y + F x )( iF y + F x )( − ix − y ) = ( iF y + F x )( − ix − y )( − iF y + F x )( ix − y )and so ( ix − y )( ix − y )( − iF y + F x ) = ( iF y + F x ) ( − ix − y )( − ix − y ) . Now, according to (i), ix − y = 0, − ix − y = 0, ix − y = 0, − ix − y = 0. Hence( − iF y + F x ) = a ( iF y + F x ) for some a = 0, which implies that d = 1 and contradicts (ii). N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 6
Hence we proved that Σ S ( C ) is not reduced to a point. Now the irreducibility of Σ S ( C ) comesfrom the fact that Σ S ( C ) = Φ F, S ( C ) and that C is an irreducible curve. (cid:3) Proposition 9.
Assume that Σ S ( C ) is not reduced to a point. Then we haveclass (Σ S ( C )) = deg ( T C ,S ( C )) , (1) where T C , S ( C ) stands for the Zariski closure of T C , S ( C ) .Proof. This comes from the fact that Σ S ( C ) is the Zariski closure of the envelope of {R m , m ∈C \ ( Sing ( C ) ∪ { S } ∪ ℓ ∞ ∪ V ( Q ( ∇ F )) } and can be precised as follows. For every algebraic curveΓ = V ( G ) (with G in Sym k ( V ∨ ) for some k ), we consider the Gauss map δ Γ : P −→ P definedon coordinates by δ Γ ([ x : y : z ]) = [ G x : G y : G z ], we obtain immediately the commutativediagram : C ( Φ F, S ) |C −→ Σ S ( C ) T C ,S ց ↓ δ Σ S ( C ) δ Σ S ( C ) (Σ S ( C )) ∼ = (Σ S ( C )) ∨ , (2)with Φ F, S the caustic map defined in [15] (see the begining of the present section). (cid:3) Let us notice that, according to the proof of Proposition 9, the rational map T C ,S as thesame degree as the rational map (Φ F, S ) |C (since Σ S ( C ) is irreducible and since the Gauss map( δ Σ S ( C ) ) | Σ S ( C ) is birational [11]).3. Formulas for the class of the caustic
Since the map T C ,S may be non birational, we introduce the notion of class with multiplicityof Σ S ( C ): mclass(Σ S ( C )) = δ ( S, C ) × class(Σ S ( C ))where class(Σ S ( C )) is the class of the algebraic curve Σ S ( C ) and where δ ( S, C ) is the degree ofthe rational map T C ,S . We recall that δ ( S, C ) corresponds to the number of preimages on C ofa generic point of Σ S ( C ) by T C ,S .Before stating our main result, let us introduce some notations. For every m ∈ P , we write µ m = µ m ( C ) for the multiplicity of m on C and consider the set Branch m ( C ) of branchesof C at m . We denote by E the set of couples point-branch ( m , B ) of C with m ∈ C and B ∈
Branch m ( C ). For every ( m , B ) ∈ E , we write e B for the multiplicity of B and T m ( B ) thetangent line to B at m ; we observe that µ m = P B∈ Branch m ( C ) e B . We write i m (Γ , Γ ′ ) theintersection number of two curves Γ and Γ ′ at m . For any algebraic curve C ′ of P , we alsodefine the contact number Ω m ( C , C ′ ) of C and C ′ at m ∈ P byΩ m ( C , C ′ ) := i m ( C , C ′ ) − µ m ( C ) µ m ( C ′ ) if m ∈ C ∩ C ′ and Ω m ( C , C ′ ) := 0 if m
6∈ C ∩ C ′ . Recall that Ω m ( C , C ′ ) = 0 means that m
6∈ C ∩ C ′ or that C and C ′ intersect transversally at m . Theorem 10.
Assume that the hypotheses of Proposition 8 hold true. (1) If S ℓ ∞ , the class (with multiplicity) of Σ S ( C ) is given bymclass (Σ S ( C )) = 2 d ∨ + d − f ′ − g − f − g ′ + q ′ , (3) where N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 7 • g is the contact number of C with ℓ ∞ , i.e. g := P m ∈C∩ ℓ ∞ Ω m ( C , ℓ ∞ ) , • f is the multiplicity number at a cyclic point of C with an isotropic line from S , i.e. f := i I ( C , ( IS )) + i J ( C , ( J S )) , • f ′ is the contact number of C with an isotropic line from S outside { I, J, S } , i.e. f ′ := X m ∈ ( C∩ ( IS )) \{ I,S } Ω m ( C , ( IS )) + X m ∈ ( C∩ ( JS )) \{ J,S } Ω m ( C , ( J S )) , • g ′ given by g ′ := i S ( C , ( IS )) + i S ( C , ( J S )) − µ S ; • q ′ is given by q ′ := X ( m , B ) ∈E : m I,J,S } ,T m B =( IS ) or T m B =( JS ) , i m ( B , T m ( B )) ≥ e B [ i m ( B , T m ( B )) − e B ] . (2) If S ∈ ℓ ∞ , the class of Σ S ( C ) ismclass (Σ S ( C )) = 2 d ∨ + d − g − µ I − µ J − µ S − c ′ ( S ) , (4) with c ′ ( S ) := X B∈ Branch S ( C ): i S ( B ,ℓ ∞ )=2 e B ( e B + min( i S ( B , Osc S ( B )) − e B , , where Osc S ( B ) is any smooth algebraic osculating curve to B at S (i.e. any smoothalgebraic curve C ′ such that i S ( B , C ′ ) > e B ). The notations introduced in this theorem are directly inspired by those of Salmon and Cayley[18] (see Section 5). Let us point out that, in this article, g is not the geometric genus of thecurve. Remark 11.
Observe that we also have c ′ ( S ) := X B∈ Branch S ( C ): i S ( B ,ℓ ∞ )=2 e B ( e B + min( β ( S, B ) − e B , , where β ( S, B ) is the first characteristic exponent of B non multiple of e B (see [22] ).Observe that, when i S ( B , T S ( B )) = 2 e B , we have min( i S ( B , Osc S ( B )) − e B ,
0) = 0 except if S is a singular point and if the probranches of B are given by Y − x − y = αZ + α Z β + ... in the chart X = 1 if x = 0 (or X − y − x = αZ + α Z β + ... in the chart Y = 1 otherwise),with α = 0 , α = 0 and < β < . Hence c ′ ( S ) = P B∈ Branch S ( C ): i S ( B ,ℓ ∞ )=2 e B e B when C admitsno such branch tangent at S to ℓ ∞ . Combining Proposition 6 and Theorem 10, we obtain
Corollary 12 (A source-generic formula for the class) . Let
C ⊂ P be a fixed curve of degree d ≥ . For a generic source point S , we have δ ( S, C ) = 1 and class (Σ S ( C )) = 2 d ∨ + d − g − µ I − µ J with g the contact number of C with ℓ ∞ .Proof. Due to Proposition 6, δ ( S, C ) = 1 for a generic S ∈ P . So class (Σ S ( C )) = mclass (Σ S ( C )).Assume moreover, that S ℓ ∞ (so we apply the first formula of Theorem 10), S
6∈ C (so g ′ = 0), that ( IS ) and ( J S ) are not tangent to C (so f ′ = q ′ = 0 and f = µ I ( C ) + µ J ( C )). Weobtain the result. (cid:3) Examples
Let us now illustrate our result for two particular mirror curves.
N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 8
Example of the lemniscate of Bernoulli.
We consider the case when C = V ( F ) is thelemniscate of Bernoulli given by F ( x, y, z ) = ( x + y ) − x − y ) z and when S ∈ P \ { I, J } .The degree of C is d = 4. The singular points of C are : I [1 : i : 0], J [1 : − i : 0] and O [0 : 0 : 1].These three points are double points, each one having two different tangent lines. Hence theclass of C is given by d ∨ = d ( d − − × d ∨ + d = 16 . The tangent lines to C at I are ℓ ,I := V ( y − iz − ix ) and ℓ ,I := V ( y − iz + ix ) (the intersectionnumber of C with ℓ ,I or with ℓ ,I at I is equal to 4). The tangent lines to C at J are ℓ ,J := V ( y + iz − ix ) and ℓ ,J := V ( y + iz + ix ) (the intersection number of C with ℓ ,J or with ℓ ,J at J is equal to 4). This ensures that we have f = 2(2 + S ∈ ℓ ,I + S ∈ ℓ ,I + S ∈ ℓ ,J + S ∈ ℓ ,J ) . Observe that ℓ ∞ is not tangent to C . Indeed I and J are the only points in C ∩ ℓ ∞ and ℓ ∞ isnot tangent to C at these points. Therefore we have g = 0 and c ′ ( S ) = 0.Since I and J are also the only points at which C is tangent to an isotropic line (i.e. a linecontaining I or J ), we have f ′ = 0, g ′ = µ S , q ′ = 0. In this case, one can check that δ ( S, C ) = 1.Finally, we getif S ℓ ∞ , class(Σ S ( C )) = 12 − S ∈ ℓ ,I ∪ ℓ ,I + S ∈ ℓ ,J ∪ ℓ ,J ) − µ S . (5)Moreover, since µ I = µ J = 2, we haveif S ∈ ℓ ∞ \ { I, J } , class(Σ S ( C )) = 16 − − , (6)(since µ I = µ J = 2 and since µ S = 0). For example, for S [1 : 0 : 1], we get class(Σ S ( C )) = 8,since S is in ℓ ,I ∩ ℓ , J but not in C (so µ S = 0).4.2. Example of a quintic curve.
As in [15], we consider the quintic curve C = V ( F ) with F ( x, y, z ) = y z − x . We also consider a light point S [ x : y : z ] ∈ P \ { I, J } . This curveadmits two singular points: A [0 : 0 : 1] and A [0 : 1 : 0], we have d = 5.We recall that C admits a single branch at A , which has multiplicity 2 and which is tangentto V ( y ). We observe that i A ( C , V ( y )) = 5.Analogously, C admits a single branch at A , which has multiplicity 3 and which is tangentto ℓ ∞ . We observe that i A ( C , ℓ ∞ ) = 5.We obtain that the class of C is d ∨ = 5 and that C has no inflexion point (these two facts areproved in [15]). In particular, we get that 2 d ∨ + d = 15.Since A is the only point of C ∩ ℓ ∞ , we get that g = Ω A ( C , ℓ ∞ ) = 2 and f = 0.The curve C admits six (pairwise distinct) isotropic tangent lines other than ℓ ∞ : ℓ , ℓ and ℓ containing I ∀ k ∈ { , , } , ℓ k = V (cid:18) ix − y + 3 i α k √ z (cid:19) , with α := e iπ and ℓ , ℓ and ℓ containing J : ∀ k ∈ { , , } , ℓ k = V (cid:18) ix + y + 3 i α k √ z (cid:19) . For every i ∈ { , , , , , } , we write a i the point at which C is tangent to ℓ i (the points a i correspond to the points of C ∩ V ( F x + F y ) \ { A , A } ). Since C contains no inflexion point andsince A and A are the only singular points of C , we get that, f ′ = { i ∈ { , , , , , } : S ∈ ℓ i \ { a i }} and q ′ = 0 N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 9 when S ℓ ∞ .Now recall that g ′ = i S ( C , ( IS )) + i S ( C , ( J S )) − µ S . Again, in this case, one can check that δ ( S, C ) = 1. If S ℓ ∞ , we haveclass(Σ S ( C )) = 13 − × { i ∈ { , , , , , } : S ∈ ℓ i \ { a i }} − g ′ (7)and if S ∈ ℓ ∞ \ { I, J } , we have class(Σ S ( C )) = 11 − × S = A . (8)We observe that the points of P \ { I, J } belonging to two dictinct ℓ k are outside C . The set ofthese points is E := [ k =1 (cid:26)(cid:20) − √ α k : 0 : 1 (cid:21) , (cid:20) √ α k : 350 √ √ α k : 1 (cid:21) , (cid:20) √ α k : − √ √ α k : 1 (cid:21)(cid:27) with α = e iπ . Finally, the class of the caustic in the different cases is summarized in thefollowing table. Condition on S ∈ P \ { I, J } class(Σ S ( C )) = S = A S ∈ E S ∈ C ∩ S k =1 ( ℓ k \ { a k } ) 10 S ∈ ( ℓ ∞ \ { A } ) ∪ (cid:16)S k =1 ℓ k \ ( E ∪ C ) (cid:17) ∪ { A } ∪ { a , ..., a } S ∈ C \ (cid:16) ℓ ∞ ∪ { A } ∪ S k =1 ℓ k (cid:17) On the formulas by Brocard and Lemoyne and by Salmon and Cayley
Formulas given by Brocard and Lemoyne.
Recall that, when S ℓ ∞ , Σ S ( C ) is theevolute of an homothetic of the pedal of C from S .The work of Salmon and Cayley is under ordinary Pl¨ucker conditions (no hyper-flex, nosingularities other than ordinary cups and ordinary nodes). In [18, p.137], Salmon and Cayleygave the following formula for the class of the evolute : n ′ = m + n − f − g. Replace now m , n , f and g by M , N , F and G (respectively) given in [18, p. 154] for thepedal. Doing so, one exactly get (with the same notations) the formula of the class of causticsby reflection given by Brocard and Lemoyne in [1, p. 114].As explained in introduction, this composition of formulas of Salmon and Cayley is incorrectbecause of the non-conservation of the Pl¨ucker conditions by the pedal transformation. Never-theless, for completeness sake, let us present the Brocard and Lemoyne formula and compare itwith our formula. Brocard and Lemoyne gave the following formula for the class of the causticby reflection Σ S ( C ) when S ℓ ∞ : class (Σ S ( C )) = d + 2( d ∨ − ˆ f ′ ) − ˆ g − ˆ f − ˆ g ′ + ˆ q ′ , (9)for an algebraic curve C of degree d , of class d ∨ , ˆ g times tangent to ℓ ∞ , passing ˆ f times througha cyclic point, ˆ f ′ times tangent to an isotropic line of S , passing ˆ g ′ times through S , ˆ q ′ being thecoincidence number of contact points when an isotropic line is multiply tangent. In [18], ˆ q ′ isdefined as the coincidence number of tangents at points ι , ι of P ∨ (corresponding to ( IS ) and N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 10 ( J S )) if these points are multiple points of the image of C by the polar reciprocal transformationwith center S ; i.e. ˆ q ′ represents the number of ordinary flexes of C .When S ℓ ∞ , let us compare terms appearing in our formula (3) with terms of (9) : • ˆ g seems to be equal to g ; • it seems that ˆ f = µ I + µ J and so f = ˆ f + Ω I ( C , ( IS )) + Ω J ( C , ( J S )); • it seems that ˆ f ′ = P m ∈C∩ ( IS ) Ω m ( C , ( IS )) + P m ∈C∩ ( JS ) Ω m ( C , ( J S )) and so f ′ := ˆ f ′ − Ω I ( C , ( IS )) − Ω J ( C , ( J S )) − Ω S ( C , ( IS )) − Ω S ( C , ( J S )); • it seems that ˆ g ′ = µ S , therefore g ′ := ˆ g ′ + Ω S ( C , ( IS )) + Ω S ( C , ( J S )); • our definition of q ′ appears as an extension of ˆ q ′ (except that we exclude the points m ∈ { I, J, S } ).Observe that these terms coincide with the definition of Brocard and Lemoyne if ( IS ) and ( J S )are not tangent to C at S , I , J . In particular, if we call BL the right hand side of (9), the firstitem Theorem 10 states that, when S is not at infinity we have mclass (Σ S ( C )) = BL + Ω I ( C , ( IS )) + Ω J ( C , ( J S )) + Ω S ( C , ( IS )) + Ω S ( C , ( J S )) . A counterexample to the formula of Brocard and Lemoyne.
We consider anexample in which Ω I ( C , ( IS )) = Ω J ( C , ( J S )) = 1, which means that ( IS ) is tangent to C at I and ( J S ) is tangent to C at J . Let us consider the non-singular quartic curve C = V (2 yz +2 z y +2 zy +2 y − z x +2 zyx +5 y x +3 x ) and S [0 : 0 : 1]. This curve C has degree d = 4 and class d ∨ = 4 × ℓ ∞ , is tangent to ( SI ) at I and nowhere else,is tangent to ( SJ ) at J and nowhere else; these tangent points are ordinary. S is a non singularpoint of C . Therefore, with our definitions, we have g = 0, f = 2+2 = 4, f ′ = 0, g ′ = 1+1 − q ′ = 0, which gives class(Σ S ( C )) = 4+2(12 − − − − − δ ( S, C ) = 1.In comparison, the Brocard and Lemoyne formula would give ˆ g = 0, ˆ f = 1+1 = 2, ˆ f ′ = 1+1 = 2,ˆ g ′ = 1, ˆ q ′ = 0 and so their formula gives class(Σ S ( C )) = 4 + 2(12 − − − − − Proof of Theorem 10
To compute the degree of T C ,S ( C ), we will use the Fundamental Lemma given in [15]. Let usfirst recall the definition of ϕ -polar introduced in [15] and extending the notion of polar. Definition 13.
Let p ≥ , q ≥ and let W be a complex vector space of dimension p + 1 .Given ϕ : P p := P ( W ) → P q a rational map defined by ϕ = [ ϕ : · · · : ϕ q ] (with ϕ , . . . , ϕ q ∈ Sym d ( W ∨ ) ) and a = [ a : · · · : a q ] ∈ P q , we define the ϕ -polar at a , denoted by P ϕ,a , thehypersurface of degree d given by P ϕ,a := V (cid:16)P qj =0 a j ϕ j (cid:17) ⊆ P p . With this definition, the “classical” polar of a curve C = V ( F ) of P (for some homogeneouspolynomial F ∈ C [ x, y, z ]) at a is the δ C -polar curve at a , where δ C : [ x : y : z ] [ F x : F y : F z ]. Definition 14.
We call reflected polar (or r -polar) of the plane curve C with respectto S at a the R C ,S -polar at a , i.e. the curve P ( r ) S,a ( C ) := P R C ,S ,a . N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 11
From a geometric point of view, P ( r ) S,a ( C ) is an algebraic curve such that, for every m ∈C ∩ P ( r ) S,a ( C ), R m contains a (if R m is well defined), this means that line ( am ) is tangent to Σ S ( C )at the point m ′ = Φ F, S ( m ) ∈ Σ S ( C ) associated to m (see picture).Let us now recall the statement of the fundamental lemma proved in [15]. Lemma 15 (Fundamental lemma [15]) . Let W be a complex vector space of dimension p + 1 ,let C be an irreducible algebraic curve of P p := P ( W ) and ϕ : P p → P q be a rational map givenby ϕ = [ ϕ : · · · : ϕ q ] with ϕ , ..., ϕ q ∈ Sym δ ( W ∨ ) . Assume that C 6⊆
Base ( ϕ ) and that ϕ |C hasdegree δ ∈ N ∪ {∞} . Then, for generic a = [ a : · · · : a q ] ∈ P q , the following formula holds true δ . deg (cid:16) ϕ ( C ) (cid:17) = δ. deg ( C ) − X p ∈ Base ( ϕ |C ) i p ( C , P ϕ,a ) , with convention . ∞ = 0 and deg ( ϕ ( C )) = 0 if ϕ ( C ) < ∞ . Due to this lemma and to Proposition 9, we havemclass(Σ S ( C )) = d (2 d − − X m ∈ Base ( T C ,S ) i m ( C , P ( r ) S,a ( C )) . (10)Now, we enter in the most technical stuff which is the computation of the intersection numbers i m ( C , P ( r ) S,a ( C )) of C with its reflected polar at the base points of R C ,S . To compute theseintersection numbers, it will be useful to observe the form of the image of R C ,S by linear changesof variable. It is worth noting that R F, S can be rewritten R F, S = id ∧ [∆ I F ∆ J F · S − ∆ S F ∆ I F · J − ∆ S F ∆ J F · I ] . N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 12
Proposition 16.
Let M ∈ GL ( V ) . We have R F, S ◦ M = Com ( M ) · R ( M − ( I ) ,M − ( J )) F ◦ M,M − ( S ) , with Com ( M ) := det ( M ) · t M − and R ( A , B ) G, S ′ := id ∧ (cid:2) ∆ A G ∆ B G · S ′ − ∆ S ′ G ∆ A G · B − ∆ S ′ G ∆ B G · A (cid:3) . Proof.
We use M ( u ) ∧ M ( v ) = ( Com ( M ))( u ∧ v ) and ∆ M ( u ) ( F )( M ( P )) = ∆ u ( F ◦ M )( P ). (cid:3) We write Π : V \ { } → P for the canonical projection, P [0 : 0 : 1] ∈ P and P (0 , , ∈ V .Let m be a base point of C and M ∈ GL ( V ) be such that Π( M ( P )) = m and such thatthe tangent cone of V ( F ◦ M ) at P does not contain V ( x ). Let µ m be the multiplicity of m in C ( m is a singular point of C if and only if µ m > a ∈ P , writing a ′ := M − ( a ), we have i m ( C , P ( r ) S,a ( C )) = i m ( C , V ( h a , R F, S ( · ) i ))= i P ( V ( F ◦ M ) , V ( h a , R F, S ◦ M ( · ) i ))= i P ( V ( F ◦ M ) , V ( h a ′ , R ( M − ( I ) ,M − ( J )) F ◦ M,M − ( S ) ( · ) i ))= X B∈ Branch P ( V ( F ◦ M )) i P ( B , V ( h a ′ , R ( M − ( I ) ,M − ( J )) F ◦ M,M − ( S ) ( · ) i )) , where Branch P ( V ( F ◦ M )) is the set of branches of V ( F ◦ M ) at P . The last equality comesfrom Proposition 18 proved in appendix (see formula (14)). Let b be the number of such branches.Of course, b = 1 for non-singular points. Writing e B for the multiplicity of the branch B , we have µ m = P B∈ Branch P ( V ( F ◦ M )) e B . Let us write C h x N i and C h x N , y i for the rings of convergentpower series of x N , y . Let C h x ∗ i := S N ≥ C h x N i and C h x ∗ , y i := S N ≥ C h x N , y i . For every h = P q ∈ Q + a q x q ∈ C h x ∗ i , we define the valuation of h as follows: val ( h ) := val x ( h ( x )) := min { q ∈ Q + , a q = 0 } . Let B be a branch of V ( F ◦ M ) at P . We precise that B = M ( B ) ⊂ P is a branch of C at m . Let A ( x A , y A , z A ) := M − ( I ), B ( x B , y B , z B ) := M − ( J ) and S ′ := M − ( S ). Let T B be the tangentline to B at P . The branch B can be splitted in e B pro-branches with equations y = g i, B ( x ) inthe chart z = 1 (for i ∈ { , ..., e B } ) with g i ∈ C h x ∗ i having (rational) valuation larger than orequal to 1 (so g ′ i (0) = 0). For j ∈ { , ..., e B ′ } , consider also the equations y = g j, B ′ ( x ) (in thechart z = 1) of the pro-branches V j, B ′ for each branch B ′ ∈ Branch P ( V ( F ◦ M )). This notionof pro-branches comes from the combination of the Weierstrass and of the Puiseux theorems.It has been used namely by Halphen in [13] and by Wall in [21]. One can also see [15]. Thereexists a unit U of C h x, y i such that the following equality holds true in C h x ∗ , y i F ( M ( x, y, U ( x, y ) Y B ′ ∈ Branch P ( V ( F ◦ M )) e B′ Y j =1 ( y − g j, B ′ ( x )) . For a generic a (with a ′ := M − ( a )), using (15)), we obtain i P ( B , V ( h a ′ , R ( A , B ) F ◦ M, S ′ ( · ) i )) = X i val x (cid:16) h a ′ , R ( A , B ) F ◦ M, S ′ ( x, g i, B ( x ) , i (cid:17) = X i min j =1 , , val x (cid:18)h R ( A , B ) F ◦ M, S ′ ( x, g i, B ( x ) , i j (cid:19) . N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 13
Hence Formula (10) becomesmclass(Σ S ( C )) := d (2 d − − X m ∈C X B e B X i =1 min j =1 , , val x (cid:18)h R ( A , B ) F ◦ M , S ′ ( x , g i , B ( x ) , ) i j (cid:19) , (11)where, for every m ∈ C , M depends on m and is as above, where the sum is over B ∈
Branch P ( V ( F ◦ M )). Due to Lemma 33 of [15], for every P ( x P , y P , z P ) ∈ V \ { } , we have(∆ M ( P ) F ) ◦ M ( x, g i, B ( x ) ,
1) = ∆ P ( F ◦ M )( x, g i, B ( x ) ,
1) = D i, B ( x ) W P ,i, B ( x ) , with W P ,i, B ( x ) := y P − g ′ i, B ( x ) x P + z P ( xg ′ i, B ( x ) − g i, B ( x ))and with D i, B ( x ) := U ( x, g i, B ( x )) Q B ′ ∈ Branch P ( V ( F ◦ M )) Q j =1 ,...,e B′ :( B ′ ,j ) =( B ,i ) ( g i, B ( x ) − g j, B ′ ( x )).Hence we have R ( A , B ) F ◦ M, S ′ ( x, g i, B ( x ) ,
1) := ( D i, B ) · ˆ R i, B ( x ) (12)withˆ R i, B ( x ) := xg i, B ( x )1 ∧ (cid:2) W A ,i, B ( x ) W B ,i, B ( x ) · S ′ − W S ′ ,i, B ( x ) W A ,i, B ( x ) · B − W S ′ ,i, B ( x ) W B ,i, B ( x ) · A (cid:3) . First, with the notations of [15] (since U (0 , = 0), we have X B∈ Branch P ( V ( F ◦ M )) e B X i =1 val ( D i, B ) = V m , (which is null if m is a nonsingular point of C ). Second, writing h m ,i, B := min( val ([ ˆ R i, B ] j ) , j =1 , , P e B i =1 h m ,i, B only depends on m and on the branch B = M ( B ) of C at m (it does not depend on the choiceof M ∈ GL ( V ) such that Π( M ( P )) = m and such that V ( x ) is not tangent to M − ( B )).Hence we write h m , B := e B X i =1 h m ,i, B . With these notations, due to (12), formula (11) becomes mclass (Σ S ( C )) = 2 d ( d −
1) + d − X m ∈ Sing ( C ) V m − X m ∈C X B ∈ Branch m ( C ) h m , B . Moreover, as noticed in [15], we have d ( d − − P m ∈ Sing ( C ) V m = d ∨ , where d ∨ is the class of C . Therefore, we get mclass (Σ S ( C )) = 2 d ∨ + d − X m ∈C X B ∈ Branch m ( C ) h m , B . (13)Theorem 10 will come directly from the computation of h m ,i, B given in following result. Lemma 17.
Let m ∈ C and B ∈ Branch m ( C ) . Writing T m B for the tangent line to B at m , i m ( B , T m B ) for the intersection number of B with T m B at m and e B for themultiplicity of B , we have (1) h m , B = 0 if I, J, S
6∈ T m B . (2) h m , B = 0 if T m B ∩ { I, J, S } ) = 1 and m
6∈ {
I, J, S } . (3) h m , B = e B if T m B ∩ { I, J, S } ) = 1 and m ∈ { I, J, S } . N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 14 (4) h m , B = i m ( B , T m B ) + min( i m ( B , T m B ) − e B , if T m B = ( IS ) , J
6∈ T m B and m
6∈ {
I, S } . h m , B = i m ( B , T m B ) + min( i m ( B , T m B ) − e B , if T m B = ( J S ) , I
6∈ T m B and m
6∈ {
J, S } . (5) h m , B = i m ( B , T m B ) if T m B = ( IS ) , J
6∈ T m B and m ∈ { I, S } . h m , B = i m ( B , T m B ) if T m B = ( J S ) , I
6∈ T m B and m ∈ { J, S } . (6) h m , B = i m ( B , T m B ) − e B if T m B = ( IJ ) , that S
6∈ T m B and m
6∈ {
I, J } . (7) h m , B = i m ( B , T m B ) if T m B = ( IJ ) , that S
6∈ T m B and m ∈ { I, J } . (8) h m , B = 2 i m ( B , T m B ) − e B if I, J, S ∈ T m B and m
6∈ {
I, J, S } . (9) h m , B = 2 i m ( B , T m B ) − e B if I, J, S ∈ T m B and m ∈ { I, J } . (10) h m , B = 2 i m ( B , T m B ) − e B if I, J, S ∈ T m B , m = S and i m ( B , T m B ) = 2 e B . (11) h m , B = e B (1 + min( β , if I, J, S ∈ T m B , m = S and i m ( B , T m B ) = 2 e B , e B β = i m ( B , Osc m ( B )) , where Osc m ( B ) is any osculating smooth algebraic curveto B at m (the last formula of h m , B holds true if we replace e B β by the first char-acteristic exponent of B non multiple of e B , see [22] ).Proof. We take M such that T B = V ( y ) (with B = M − ( B )). To simplify notations, we ommitindices B in W P ,i, B and consider i ∈ { , ..., e B } . • Suppose that
I, J, S
6∈ T m B . Then W B ,i (0) = y B = 0, W A ,i (0) = y A = 0 and W S ′ ,i (0) = y S ′ = 0 so ˆ R i (0) = ∧ [ y A y B · S ′ − y A y S ′ · B − y B y S ′ · A ]= y A y B y S ′ y A y B x S ′ − y A y S ′ x B − y B y S ′ x A . Hence h m ,i, B = 0 and the sum over i = 1 , ..., e B of these quantities is equal to 0. • Suppose I ∈ T m B , J, S 6∈ T m B and m = I . Take M such that S ′ (0 , , A (1 , , y B = 0. We have W B ,i (0) = y B , W A ,i (0) = 0 and W S ′ ,i (0) = 1 and so ˆ R i (0) = ∧ − y B = − y B . Hence h m ,i, B = 0 and the sum over i = 1 , ..., e B ofthese quantities is equal to 0. • Suppose I ∈ T m B , J, S
6∈ T m B and m = I . Take M such that S ′ (0 , , A (0 , , y B = 0. We have W B ,i ( x ) = y B − g ′ i ( x ) x B + z B ( xg ′ i ( x ) − g i ( x )), W A ,i ( x ) = xg ′ i ( x ) − g i ( x )and W S ′ ,i ( x ) = 1 and soˆ R i ( x ) = xg i ( x )1 ∧ − ( xg ′ i ( x ) − g i ) x B ( xg ′ i ( x ) − g i )( − g ′ i ( x ) x B + z B ( xg ′ i ( x ) − g i ( x ))) − y B + g ′ i ( x ) x B − z B ( xg ′ i ( x ) − g i ( x )) = − y B g i ( x ) + x ( g ′ i ( x )) x B − z B (( xg ′ i ( x )) − ( g i ( x )) ) − x B (2 xg ′ i ( x ) − g i ( x )) + xy B + 2 xz B ( xg ′ i ( x ) − g i ( x )) − x B ( xg ′ i ( x ) − g i ( x )) + z B ( xg ′ i ( x ) − g i ( x )) , the valuation of the coordinates of which are larger than or equal to 1 and the valuationof the second coordinate is 1. Hence h m ,i, B = 1 and the sum over i = 1 , ..., e B = e B ofthese quantities is equal to e B . • Suppose S ∈ T m B , I, J
6∈ T m B and m = S . Take M such that A (0 , , S ′ (1 , , y B = 0. We have W B ,i (0) = y B = 0, W S ′ ,i (0) = 0 and W A ,i (0) = 1 and so ˆ R i (0) = N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 15 ∧ y B = y B . Hence h m ,i, B = 0 and the sum over i = 1 , ..., e B of thesequantities is equal to 0. • Suppose m = S and I, J
6∈ T m B . Take M such that S ′ (0 , , A (0 , , y B = 0.We have W B ,i ( x ) = y B − g ′ i ( x ) x B + z B ( xg ′ i ( x ) − g i ( x )), W S ′ ,i ( x ) = xg ′ i ( x ) − g i ( x ) and W A ,i ( x ) = 1 and soˆ R i ( x ) = xg i ( x )1 ∧ − ( xg ′ i ( x ) − g i ) x B − ( xg ′ i ( x ) − g i ( x ))(2 y B − g ′ i ( x ) x B + z B ( xg ′ i ( x ) − g i ( x ))) y B − g ′ i ( x ) x B = g i ( x )( y B − g ′ i ( x ) x B ) + ( xg ′ i ( x ) − g i ( x ))(2 y B − g ′ i ( x ) x B + z B ( xg ′ i ( x ) − g i ( x ))) − ( xg ′ i ( x ) − g i ( x )) x B − x ( y B − g ′ i ( x ) x B ) − ( xg ′ i ( x ) − g i ( x ))(2 xy B − g ′ i ( x ) xx B + z B x ( xg ′ i ( x ) − g i ( x )) + g i ( x )( xg ′ i ( x ) − g i ( x )) x B ) , the valuation of the coordinates of which are larger than or equal to 1 and the valuationof the second coordinate is 1. Hence h m ,i, B = 1 and the sum over i = 1 , ..., e B of thesequantities is equal to e B . • Suppose T m B = ( IS ), J
6∈ T m B and m
6∈ {
I, S } . Take M such that S ′ (1 , , B (0 , , y A = 0, x A = 0, z A = 0. We have W S ′ ,i ( x ) = − g ′ i ( x ), W A ,i ( x ) = − g ′ i ( x ) x A + z A ( xg ′ i ( x ) − g i ( x )) and W B ,i ( x ) = 1 and soˆ R i ( x ) = xg i ( x )1 ∧ z A ( xg ′ i ( x ) − g i ( x )) − ( g ′ i ( x )) x A + g ′ i ( x )( xg ′ i ( x ) − g i ( x )) z A g ′ i ( x ) z A = g i ( x ) g ′ i ( x ) z A + ( g ′ i ( x )) x A − g ′ i ( x )( xg ′ i ( x ) − g i ( x )) z A − g i ( x ) z A − x ( g ′ i ( x )) x A + ( xg ′ i ( x ) − g i ( x )) z A , the valuation of the coordinates of which are respectively 2 val ( g i ) − val ( g i ) and2 val ( g i ) −
1. Hence h m ,i, B = val ( g i ) + min( val ( g i ) − ,
0) and the sum over i = 1 , ..., e B of these quantities is equal to i m ( B , T m B ) + min( i m ( B , T m B ) − e B , • Suppose T m B = ( IS ), J
6∈ T m B and m = I . Take M such that S ′ (1 , , B (0 , , A (0 , , W S ′ ,i ( x ) = − g ′ i ( x ), W A ,i ( x ) = xg ′ i ( x ) − g i ( x ) and W B ,i ( x ) = 1 andsoˆ R i ( x ) = xg i ( x )1 ∧ xg ′ i ( x ) − g i ( x ) g ′ i ( x )( xg ′ i ( x ) − g i ( x )) g ′ i ( x ) = g ′ i ( x )(2 g i ( x ) − xg ′ i ( x )) − g i ( x )( xg ′ i ( x ) − g i ( x )) , the valuation of the coordinates of which are larger than or equal to val ( g i ), the secondcoordinate has valuation val ( g i ). Hence h m ,i, B = val ( g i ) and the sum over i = 1 , ..., e B of these quantities is equal to i m ( B , T m B ). • Suppose T m B = ( IS ), J
6∈ T m B and m = S . Take M such that A (1 , , B (0 , , S ′ (0 , , W S ′ ,i ( x ) = xg ′ i ( x ) − g i ( x ), W A ,i ( x ) = − g ′ i ( x ) and W B ,i ( x ) = 1 andso ˆ R i ( x ) = xg i ( x )1 ∧ − ( xg ′ i ( x ) − g i ( x )) g ′ i ( xg ′ i ( x ) − g i ( x )) − g ′ i ( x ) = − x ( g ′ i ( x )) g i ( x )( xg ′ i ( x )) − ( g i ( x )) , the valuation of the coordinates of which being larger than or equal to val ( g i ) and thevaluation of the second coordinate is equal to val ( g i ). Hence h m ,i, B = val ( g i ) and thesum over i = 1 , ..., e B of these quantities is equal to i m ( B , T m B ). N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 16 • Suppose T m B = ( IJ ), S
6∈ T m B and m
6∈ {
I, J } . Take M such that S ′ (0 , , B (1 , , y A = 0, x A = 0, z A = 0. We have W B ,i ( x ) = − g ′ i ( x ), W A ,i ( x ) = − g ′ i ( x ) x A + z A ( xg ′ i ( x ) − g i ( x )) and W S ′ ,i ( x ) = 1 and soˆ R i ( x ) = xg i ( x )1 ∧ g ′ i ( x ) x A − z A ( xg ′ i ( x ) − g i ( x ))( g ′ i ( x )) x A − g ′ i ( x )( xg ′ i ( x ) − g i ( x )) z A g ′ i ( x ) z A = ( g ′ i ( x )) ( xz A − x A )2 g ′ i ( x ) x A − z A (2 xg ′ i ( x ) − g i ( x )) − z A ( xg ′ i ( x ) − g i ( x )) + x A g ′ i ( x )( xg ′ i ( x ) − g i ( x )) , the valuation of the coordinates of which are respectively 2 val ( g i ) − val ( g i ) − val ( g i ). Hence h m ,i, B = val ( g i ) − i = 1 , ..., e B of thesequantities is equal to i m ( B , T m B ) − e B . • Suppose that T m B = ( IJ ), that S
6∈ T m B and m = I . Take M such that S ′ (0 , , B (1 , , A (0 , , W B ,i ( x ) = − g ′ i ( x ), W A ,i ( x ) = xg ′ i ( x ) − g i ( x )and W S ′ ,i ( x ) = 1 and soˆ R i ( x ) = xg i ( x )1 ∧ − ( xg ′ i ( x ) − g i ( x )) − g ′ i ( x )( xg ′ i ( x ) − g i ( x )) g ′ i ( x ) = x ( g ′ i ( x )) − (2 xg ′ i ( x ) − g i ( x )) − ( xg ′ i ( x ) − g i ( x )) , the valuation of the coordinates of which being larger than or equal to val ( g i ) and thevaluation of the second coordinate is equal to val ( g i ). Hence h m ,i, B = val ( g i ) and thesum over i = 1 , ..., e B of these quantities is equal to i m ( B , T m B ). • Suppose that
I, J, S ∈ T m B and m
6∈ {
I, J, S } . Take M such that S ′ (1 , , y A = y B = 0, x A = 0, z A = 0, x B = 0, z B = 0, x A z B = x B z A . We have W S ′ ,i ( x ) = − g ′ i ( x ), W A ,i ( x ) = − g ′ i ( x ) x A + z A ( xg ′ i ( x ) − g i ( x )) and W B ,i ( x ) = − g ′ i ( x ) x B + z B ( xg ′ i ( x ) − g i ( x ))and soˆ R i ( x ) = xg i ( x )1 ∧ − x A ( g ′ i ( x )) x B + z A z B ( xg ′ i ( x ) − g i ( x )) − ( g ′ i ( x )) ( xz B + x B z A ) A + 2 z A z B g ′ i ( x )( xg ′ i ( x ) − g i ( x )) = − g i ( x )( g ′ i ( x )) ( xz B + x B z A ) A + 2 z A z B g i ( x ) g ′ i ( x )( xg ′ i ( x ) − g i ( x )) − x A ( g ′ i ( x )) x B + z A z B ( xg ′ i ( x ) − g i ( x )) − x [ .... ] x A g i ( x )( g ′ i ( x )) x B − z A z B g i ( x )( xg ′ i ( x ) − g i ( x )) , the valuation of the coordinates of which are larger than or equal to 2 val ( g i ) −
2, thevaluation of the second coodinate is 2 val ( g i ) −
2. Hence h m ,i, B = 2 val ( g i ) − i = 1 , ..., e B of these quantities is equal to 2 i m ( B , T m B ) − e B . • Suppose that
I, J, S ∈ T m B and m = J . Take M such that B (0 , , S ′ (1 , , y A = 0, x A = 0 and z A = 0. We have W S ′ ,i ( x ) = − g ′ i ( x ), W A ,i ( x ) = − g ′ i ( x ) x A + z A ( xg ′ i ( x ) − g i ( x )) and W B ,i ( x ) = xg ′ i ( x ) − g i ( x ) and soˆ R i ( x ) = xg i ( x )1 ∧ z A ( xg ′ i ( x ) − g i ( x )) − x A ( g ′ i ( x )) + 2 z A ( xg ′ i ( x ) − g i ( x )) g ′ i ( x ) = g i ( x ) g ′ i ( x )( − g ′ i ( x ) x A + 2 z A ( xg ′ i ( x ) − g i ( x ))) z A ( xg ′ i ( x ) − g i ( x )) − xg ′ i ( x )( − g ′ i ( x ) x A + 2 z A ( xg ′ i ( x ) − g i ( x ))) − g i ( x ) z A ( xg ′ i ( x ) − g i ( x )) , the valuation of the coordinates of which are larger than or equal to 2 val ( g i ) − val ( g i ) −
1. Hence h m ,i, B = 2 val ( g i ) − i = 1 , ..., e B of these quantities is equal to 2 i m ( B , T m B ) − e B . N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 17 • Suppose that
I, J, S ∈ T m B and m = S . Take M such that S ′ (0 , , B (1 , , y A = 0, x A = 0 and z A = 0. We have W B ,i ( x ) = − g ′ i ( x ), W A ,i ( x ) = − g ′ i ( x ) x A + z A ( xg ′ i ( x ) − g i ( x )) and W S ′ ,i ( x ) = xg ′ i ( x ) − g i ( x ) and soˆ R i ( x ) = xg i ( x )1 ∧ x A g ′ i ( x )( xg ′ i ( x ) − g i ( x )) − z A ( xg ′ i ( x ) − g i ( x )) g ′ i ( x )) x A = g i ( x )( g ′ i ( x )) x A x A g ′ i ( x )( xg ′ i ( x ) − g i ( x )) − z A ( xg ′ i ( x ) − g i ( x )) − x A g i ( x ) g ′ i ( x )( xg ′ i ( x ) − g i ( x )) + z A g i ( x )( xg ′ i ( x ) − g i ( x )) . The valuation of the first coordinate is 3 val ( g i ) − val ( g i ) = 2, the valuation of the second coordinate is 2 val ( g i ) −
1; hence h m ,i, B =2 val ( g i ) − i = 1 , ..., e B of these quantities is equal to 2 i m ( B , T m B ) − e B .Suppose now that val ( g i ) = 2, then 3 val ( g i ) − α, α ∈ C and β > g i ( x ) = αx + α x β + ... . Then, the second coordinate has the followingform ( x A α ( β − x β +1 + ... ) + x ( ... ) . Therefore h m ,i, B = min( β + 1 ,
4) and the sumover i = 1 , ..., e B of these quantities is equal to e B (1 + min( β , (cid:3) Proof of Theorem 10.
Recall that (13) says mclass (Σ S ( C )) = 2 d ∨ + d − X m ∈C X B ∈ Branch m ( C ) h m , B and that the values of h m , B have been given in Lemma 17. • Assume first S ℓ ∞ . Then we have to sum the h m , B coming from Items 3, 4, 5, 6 and7 of Lemma 17.The sum of the h m , B coming from Items 3 and 5 applied with m = S gives directly g ′ .The sum of the h m , B coming from Items 3, 5 and 7 applied with m ∈ { I, J } gives f + Ω I ( C , ℓ ∞ ) + Ω J ( C , ℓ ∞ ).The sum of the h m , B coming from Item 6 gives g − Ω I ( C , ℓ ∞ ) − Ω J ( C , ℓ ∞ ).The sum of the h m , B coming from Item 4 gives 2 f ′ − q ′ (notice that h m , B =2( i m ( B , T m B ) − e B ) − ( i m ( B , T m B ) − e B ) i m ( B , T m B ) ≥ e B ). • Assume first S ℓ ∞ . Then we have to sum the h m , B coming from Items 3, 8, 9, 10and 11 of Lemma 17.The sum of the h m , B coming from Items 3 (with m = S ), 10 and 11 gives 2Ω S ( C , ℓ ∞ )+ µ S + c ′ ( S ).The sum of the h m , B coming from Items 3 and 9 applied with m ∈ { I, J } gives2(Ω I ( C , ℓ ∞ ) + Ω J ( C , ℓ ∞ )) + µ I + µ J .The sum of the h m , B coming from Item 8 gives 2( g − Ω I ( C , ℓ ∞ ) − Ω J ( C , ℓ ∞ ) − Ω S ( C , ℓ ∞ )). (cid:3) N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 18
Appendix A. Intersection numbers of curves and pro-branches
The following result expresses the classical intersection number i m ( C , C ′ ) defined in [14, p.54] thanks to the use of probranches. Proposition 18.
Let m ∈ P . Let C = V ( F ) and C ′ = V ( F ′ ) be two algebraic plane curvescontaining m , with homogeneous polynomials F, F ′ ∈ C [ X, Y, Z ] . Let M ∈ GL ( C ) be such that Π( M ( P )) = m and such that the tangent cones of V ( F ◦ M ) and of V ( F ′ ◦ M ) do not contain X = 0 .Assume that V ( F ◦ M ) admits b branches at P and that its β -th branch B β has multiplicity e β .Assume that V ( F ′ ◦ M ) admits b ′ branches at P and that its β ′ -th branch B ′ β ′ has multiplicity e ′ β ′ .Then we have i m ( C , C ′ ) = b X β =1 e β − X j =0 b ′ X β ′ =1 e ′ β ′ − X j ′ =0 val x [ h β ( ζ j x eβ ) − h ′ β ′ ( ζ ′ j ′ x e ′ β ′ )] , with y = h β ( ζ j x eβ ) ∈ C h x ∗ i an equation of the j -th probranch of B β at P , y = h ′ β ′ ( ζ ′ j ′ x e ′ β ′ ) ∈ C h x ∗ i an equation of the k ′ -th probranch of B ′ β ′ at P , with ζ := e iπeβ and ζ ′ := e iπe ′ β ′ . With the notations of Proposition 18, we get i m ( C , C ′ ) = b X β =1 i P ( B β , V ( F ′ )) , (14)with the usual definition given in [21] of intersection number of a branch with a curve i P ( B β , V ( F ′ ◦ M )) = e β − X j =0 val x ( F ′ ◦ M ( x, h j,β ( ζ j x eβ ))) . (15) Proof of Proposition 18.
By definition, the intersection number is defined by i m ( C , C ′ ) = i P ( V ( F ◦ M, F ′ ◦ M ) = length (cid:18) C [ X, Y, Z ]( F ◦ M, F ′ ◦ M ) (cid:19) ( X,Y,Z ) ! where ( C [ X,Y,Z ]( F ◦ M,F ′ ◦ M ) ) ( X,Y,Z ) is the local ring in the maximal ideal ( X, Y, Z ) of P [14, p. 53].According to [12], we have i m ( C , C ′ ) = dim C (cid:18) C [ X, Y, Z ]( F ◦ M, F ′ ◦ M ) (cid:19) ( X,Y,Z ) ! Let f, f ′ be defined by f ( x, y ) = F ◦ M ( x, y, f ′ ( x, y ) = F ′ ◦ M ( x, y, i m ( C , C ′ ) = dim C (cid:18) C [ x, y ]( f, f ′ ) (cid:19) ( x,y ) ! = dim C C h x, y i ( f, f ′ ) . Recall that, according to the Weierstrass preparation theorem, there exist two units U and U ′ of C h x, y i and f , ..., f b , f ′ , ..., f ′ b ′ ∈ C h x i [ y ] monic irreducible such that f = U b Y β =1 f β and f ′ = U ′ b ′ Y β ′ =1 f ′ β ′ , N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 19 f β = 0 being an equation of B β and f ′ β ′ = 0 being an equation of B ′ β ′ . According to the Puiseuxtheorem, B β (resp. B ′ β ′ ) admits a parametrization (cid:26) x = t e β x = h β ( t ) ∈ C h t i (resp. ( x = t e ′ β ′ x = h ′ β ′ ( t ) ∈ C h t i ) . We know that, for every β ∈ { , .., b } and every j ∈ { , .., e β } , h β ( ζ j x eβ ) ∈ C h x eβ i are the y -roots of f β (resp. h β ′ ( ζ ′ j ′ x e ′ β ′ ) ∈ C h x e ′ β ′ i are the y -roots of f ′ β ′ ). In particular, we have f β ( x, y ) = e β − Y j =0 ( y − h β ( ζ j x eβ )) and f ′ β ′ ( x, y ) = e ′ β ′ − Y j ′ =0 ( y − h ′ β ′ ( ζ ′ j ′ x e ′ β ′ )) . Therefore we have the following sequence of C -algebra-isomorphisms: C h x, y i ( f, f ′ ) = C h x, y i ( Q bβ =1 f β ( x, y ) , f ′ ( x, y )) ∼ = b Y β =1 A β , where A β := C h x,y i ( f β ( x,y ) ,f ′ ( x,y )) . Let β ∈ { , ..., b } . We observe that we have A β = e β − Y j =0 C h x i ( f ′ ( x, h β ( ξ j x e β ))) . On another hand, we have D β := C h x eβ , y i ( f β ( x, y ) , f ′ ( x, y )) = C h x eβ , y i ( Q e β − j =0 ( y − h β ( ζ j x eβ )) , f ′ ( x, y )) ∼ = e β − Y j =0 C h x eβ , y i ( y − h β ( ζ j x eβ ) , f ′ ( x, y )) ∼ = e β − Y j =0 D β,j with D β,j := C h x eβ i ( f ′ ( x, h β ( ζ j x eβ ))) . We consider now the natural extension of rings i β : A β,j ֒ → D β,j such that ∀ g ∈ A β , val x /eβ (( i β ( g ))( x )) = e β val x ( g ( x )) . We have D β ∼ = e β − Y j =0 C h x eβ i ( x v β ) , where v β is the valuation in x eβ of ( f ′ ( x, h β ( ζ j x eβ ))), i.e. v β := val t ( f ′ ( t e β , h β ( ζ j t ))) = e β val x ( f ′ ( x, h β ( ζ j x eβ ))) . N THE CLASS OF CAUSTICS BY REFLECTION OF PLANAR CURVES 20
We get i m ( C , C ′ ) = b X β =1 dim C A β = b X β =1 e β − X j =0 e β val t ( f ′ ( t e β , h β ( ζ j t )))= b X β =1 e β − X j =0 val x ( f ′ ( x, h β ( ζ j x eβ ))) = b X β =1 e β − X j =0 b ′ X β ′ =1 val x ( f ′ β ′ ( x, h β ( ζ j x eβ ))) . Observe now that val x ( f ′ β ′ ( x, h β ( ζ j x eβ ))) ∈ e β N and that f ′ β ′ ( x, h β ( ζ j x eβ )) ≡ Res ( f ′ β ′ , f β ; y ) ≡ e ′ β ′ − Y j ′ =0 ( h ′ β ′ ( ζ ′ j ′ x e ′ β ′ ) − h β ( ζ j x eβ )) , where Res denotes the resultant and where ≡ means ”up to a non zero scalar”. Finally, we get i m ( C , C ′ ) = b X β =1 e β − X j =0 b ′ X β ′ =1 e ′ β ′ − X j ′ =0 val x [ h ′ β ′ ( ζ ′ j ′ x e ′ β ′ ) − h β ( ζ j x eβ )] . (cid:3) Acknowledgements :
The authors thank Jean Marot for stimulating discussions and for having indicated them theformula of Brocard and Lemoyne.
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