Caustics of plane curves, their birationality and matrix projections
aa r X i v : . [ m a t h . AG ] J un CAUSTICS OF PLANE CURVES, THEIR BIRATIONALITY ANDMATRIX PROJECTIONS.
FABRIZIO CATANESE
Dedicated to Klaus Hulek on the occasion of his -th birthday. Introduction and setup
Given a plane curve C and a point S , a source of light (which could also lie at infinity,as the sun), the light rays L P originating in S , and hitting the curve C in a point P , arereflected by the curve, and the caustic C of C is the envelope of the family of reflectedrays Λ P .Our first Theorem 3.1 says that the correspondence between the curve C and thecaustic curve C is birational, i.e., it is generically one to one, if the light source point S is chosen to be a general point.We learnt about this problem in [JP12a], to which we refer for an account of thehistory of the theory of caustics and for references to the earlier works of von Tschirn-hausen, Quetelet, Dandelin, Chasles, and more modern ones (as [BGG81], [BGG82]).Our methods are from algebraic geometry, so we got interested in a generalization ofthis result, in which the special form of a certain curve D plays no role: we achieve thisgoal in Theorem 5.1.Let us now describe the mathematical set up for the description of caustics.Let P = P C and let C ⊂ P be a plane irreducible algebraic curve, whose normaliza-tion shall be denoted by C ′ .Choose an orthogonality structure in the plane, i.e. two points, called classically thecyclic points, and let P ∞ be the line (‘at infinity’) joining them. The two cyclic pointsdetermine a unique involution ι on P ∞ for which the cyclic points are fixed, hence aninvolution, called orthogonality, on the pencils of lines passing through a given point ofthe affine plane P \ P ∞ .Without loss of generality, we choose appropriate projective coordinates such that ι : ( x, y, ( − y, x, , F ix ( ι ) = { (1 , ±√− , } . Let S ∈ P be a light source point, and to each point P ∈ P \ { S } associate theline L P := P S . In the case where P ∈ C , we define Λ P , the reflected light ray, as theelement of the pencil of lines through P determined by the condition that the crossratio CR ( N P , T P , L P , Λ P ) = − , ensuring the existence of a symmetry with centre P leaving the tangent line T P to C at P and the normal line N P := ι ( T P ) fixed, and exchanging the incoming light ray L P with the reflected light ray Λ P .We thus obtain a rational map of the algebraic curve C to the dual projective plane: AMS Classification: 14H50, 14E05, 14M12, 14N05.
Date : July 31, 2018.The present work took place in the realm of the DFG Forschergruppe 790 ”Classification of algebraicsurfaces and compact complex manifolds”.
Λ : C ( P ) ∨ . Definition 1.1.
The Caustic C of C is defined as the envelope of the family of lines { Λ P } : in other words, setting Γ := Λ( C ) , C = Γ ∨ . Remark 1.2. since the biduality map Γ Γ ∨ is birational (cf. [Wal50], pages 151-152), the map C C is birational iff Λ : C Γ is birational. Moreover, by thebiduality theorem, the class of the caustic C is the degree of Γ, and the degree of C isthe class of Γ.We shall quickly see in the next section the basic calculations which give the classof C , i.e. the degree of Γ, in the case where C and S are general (more precise Pl¨uckertype formulae which show how the singularities of the curve C and the special positionof S make these numbers decrease are to be found in [JP12a] and [JP12b]).In section 3 we show our fist result, that Λ is birational onto its image for generalchoice of the source point S , if C is not a line (in this case Γ is a line, and the causticis a point). The next section recalls a well known lemma about lines contained in thedeterminantal variety ∆ which is the secant variety of the Veronese surface V .This lemma plays a crucial role in the proof of our main result, which says thefollowing (see Theorem 5.1 for more details): Theorem 1.3.
Let D ⊂ P := P ( Sym ( C )) be a curve.Then, for general S ∈ P , the projection π S : P = P ( Sym ( C )) P given by π S ( B ) := BS has the property that its restriction to D , π S | D , is birational onto itsimage, unless (and this is indeed an exception) D is a curve contained in a plane ∆( S ′ ) = { B | BS ′ = 0 } (contained in the determinantal hypersurface ∆ = { B | det( B ) =0 } ) and D is not a line. This result suggests the investigation of a more general situation concerning thebirationality of linear projections given by matrix multiplications.
Problem 1.4.
Given a linear space P of matrices B , and a linear space P ′ of matrices S , consider the matrix multiplication π S ( B ) = BS . For which algebraic subvarieties D ⊂ P is the restriction of the projection π S | D birational onto its image for a generalchoice of S ∈ P ′ ? Equations in coordinates
Let f ( x , x , x ) = 0 be the equation of C in the appropriate system of homogeneouscoordinates, let d := deg ( f ), and let F := (( f ( x ) , f ( x )) be the first part of the gradientof f . For a point x = ( x , x , x ) we define( F, x ) := f ( x ) x + f ( x ) x , { F ∧ x } := f ( x ) x − f ( x ) x . Then the tangent line T P at a point P with coordinates x is the transpose of the rowvector ( f ( x ) , f ( x ) , f ( x )).The normal line N P is orthogonal to the tangent line, hence it has the form N P = t ( − f ( x ) , f ( x ) , f ( x )), and the condition that P ∈ N P forces the unknown rationalfunction f ( x ) to fulfill − f ( x ) x + f ( x ) x + f ( x ) x ≡
0, thus t N P is the row vector t N P = ( − x f ( x ) , x f ( x ) , −{ F ∧ x } ) . We find now the line L P as the line in the pencil spanned by T P and N P passingthrough S : as such the line L P is a column vector which is a linear combination λT P + AUSTICS 3 µN P ; the condition that S ∈ L P then determines λ = − t N P · S, µ = t T P · S , where S isthe transpose of the vector ( s , s , s ).Hence we get L P ( S ) = A ( P ) S, A ( P ) := − T P t N P + N P t T P , in particular the matrix A ( P ) is skew symmetric.To obtain the reflected ray Λ( P ) it is sufficient, by definition, to change the sign of λ , and we get therefore:Λ P ( S ) = B ( P ) S, B ( P ) := T P t N P + N P t T P . Remark 2.1.
1) The matrices A ( P ) and B ( P ) are functions which are defined for allgeneral points P of the plane.2) The matrix B ( P ) is symmetric and has rank at most two, since its image isgenerated by N P and T P ; moreover we have B ( P ) P = 0 , A ( P ) P = 0 , ∀ P ∈ C.
3) Assume that C is not a line passing through a cyclic point: then the matrix B ( P ) has precisely rank two on the non empty open set where f + f = 0 and x = 0 ; theformer condition clearly holds for a general point P ∈ C , otherwise the dual curve of C would be contained in a line y = ±√− y .4) The entries of the matrix B ( x ) are given by polynomials of degree d − . By the preceding remark follows easily the classical theorem asserting that
Theorem 2.1.
The class of the caustic, i.e., the degree of Γ , equals d (2 d − , for ageneral curve C and a general choice of S . In fact C has degree d , and B ( x ) S is given by polynomials of degree 2 d − x ,which have no base points on a general curve C .3. Birationality of the caustic map
Theorem 3.1. If C is not a line, then the caustic map C C is birational, forgeneral choice of S .Proof. As already remarked, the caustic map is birational iff the map Λ : C Γ isbirational. Observe that Λ defines a morhism C ′ → Γ which we also denote by Λ.The matrix B , whose entries are polynomials of degree 2 d −
1, yields a map B : C ′ → D ⊂ P = P ( Sym ( C )) . Lemma 3.2. B : C ′ → D := Φ( C ) is birational.Proof. It suffices to recall remark 2.1: for a general point P ∈ C , B ( P ) has rankexactly two, and B ( P ) P = 0. Hence P = ker( B ( P )), and the matrix B ( P ) determinesthe point P ∈ P . (cid:3) We have now a projection P ( Sym ( C )) P given by π S ( B ) := BS.
Consider the linear subspace W := { B | B , + B , = 0 } . FABRIZIO CATANESE
We observe preliminarily that the curve D is contained in the linear subspace W since, setting for convenience f i := f i ( x ), the matrix B ( x ) has the following entries: B , = − x f f , B , = 2 x f f . Then our main result follows from the next assertion, that, for a general choice of S ∈ P , the projection π S yields a birational map of D onto Γ := π S ( D ).In order to prove this, we set up the following notation:∆ S := { B | BS = 0 } , ∆ := { B | det( B ) = 0 } = ∪ S ∆ S . Observe that ∆ is the secant variety of the Veronese surface V := { B | rank ( B ) = 1 } . Observe that the curve D is contained in the linear subspace W, is contained in ∆but not contained in the Veronese surface V .We are working inside the subspace W , and we observe first of all that the centre ofthe projection π S restricted to W is the linear space W S := ∆ S ∩ W. Observe moreover that ∆ ∩ W = ∪ S W S . Now, the projection π S is not birational on D if and only if, for a general B ∈ D ,there exists another B ′ ∈ D, B = B ′ , such that the chord (i.e., secant line) B ∗ B ′ intersects W S in a point B ′′ (observe that the general point B ∈ D does not lie in theline W S ).There are two possible cases: Case I : B ′′ is independent of the point B ∈ D . Case II : B ′′ moves as a rational function of the point B ∈ D , hence the points B ′′ sweep theline W S . Lemma 3.3.
The assumption that case I holds for each S ∈ P leads to a contradiction.Proof of the Lemma. Under our assumption, for each S there is a point B ′′ ( S ) suchthat infinitely many chords of D meet W S in B ′′ ( S ).Let us see what happens if we specialize S to be a general point P ∈ C .The first alternative isI-1) B ′′ ( P ) = B ( P ): in this case, for each point B ∈ D there is B ∈ D such that B ( P ) , B , B are collinear. Since this happens for each choice of B ( P ) , B , every secantis a trisecant, hence , by the well known trisecant lemma (cf. [ACGH85], page 110), D is a plane curve of order at least three.Take now a general S ∈ P : since B ′′ ( S ) is on a secant to D , B ′′ ( S ) belongs to thesecant variety Σ of D (here a plane Π), but we claim that it is not in D . In fact, ifthere were a point P ∈ C ′ such that B ′′ ( S ) = B ( P ), then B ( P ) S = 0 contradictingthat S is a general point. Hence we obtain that the plane Π intersects ∆ in a biggerlocus than D : since ∆ is a cubic hypersurface, it follows that Π ⊂ ∆.By proposition 4.1 it follows that either there is a point S ′ such that S ′ ∈ ker( B ) , ∀ B ∈ Π, or there is a line L ∈ P such that ker( B ) ∈ L, ∀ B ∈ Π: both cases imply that thecurve C must be contained in a line, a contradiction.The second alternative isI-2 ) B ′′ := B ′′ ( P ) = B ( P ). Then there is a point B ′ ∈ D (possibly infinitely near)such that B ′ is a linear combination of B ′′ and B := B ( P ). AUSTICS 5
However, since BP = 0 , B ′′ P = 0 , and B = B ′′ , then also for their linear combination B ′ we have B ′ P = 0. The consequence is, since B ′ P = B ′ P ′ = 0, that B ′ has rank one.Therefore, if B ′ is not infinitely near, B ′ cannot be a general point of D , hence B ′ isindependent of P : but then C ⊂ ker( B ′ ), and since we assume that C is not a line, weobtain B ′ = 0, a contradiction.If P ′ is infinitely near to the point P ∈ C , i.e., P, P ′ span the tangent line to C at P ,and B, B ′ span the tangent line to D at B = B ( P ), we work over the ring of tangentvectors C [ ǫ ] / ( ǫ ), and we observe that( B + ǫB ′ )( P + ǫP ′ ) = 0 ⇒ BP ′ = 0 . For P ∈ C general this is a contradiction, since BP ′ = 0 , BP = 0 imply that B = B ( P ) has rank one. (cid:3) Lemma 3.4.
The assumption that case II holds for general S ∈ P leads to a contra-diction.Proof of the Lemma. As we already observed, for general S , B ′′ moves as a rationalfunction of the point B ∈ D , hence the points B ′′ sweep the line W S . Therefore the line W S is contained in the secant variety Σ of the curve D . As this happens for general S , and ∆ ∩ W = ∪ S W S , it follows that the threefold ∆ ∩ W is contained in the secantvariety Σ.Since Σ is irreducible, and has dimension at most three, it follows that we haveequality ∆ ∩ W = Σ . We conclude that, for P , P general points of C , the line joining B ( P ) and B ( P ) iscontained in ∆.By proposition 4.1, and since ker ( B ( P )) = P , ker ( B ( P )) = P , we have that thematrices in the pencil λ B ( P ) + λ B ( P ) send the span of P , P to its orthogonalsubspace.This condition is equivalent to t P ( B ( P )) P = 0 ∀ P , P ∈ C ( t P ( B ( P )) P = 0 follows in fact since P , P are general).Fix now a general point P : then we have a quadratic equation for C , hence C iscontained in a conic.A little bit more of attention: the matrix B ( P ) has rank two, hence the quadraticequation defines a reducible conic, and, C being irreducible, C is a line, a contradiction. (cid:3) Linear subspaces contained in the determinantal cubic ∆ := { B | det( B ) = 0 } Proposition 4.1.
Let λB + µB be a line contained in the determinantal hypersurface ∆ of the projective space of symmetric × matrices.Then the line is contained in a maximal projective subspace contained in ∆ , which iseither of the type ∆ S := { B | BS = 0 } , for some S ∈ P , or of the type FABRIZIO CATANESE ∆( L ) := { B | BL ⊂ L ⊥ } = { B | B | L ≡ } , for some line L ⊂ P .Proof. A pencil of reducible conics either has at most one (non infinitely near) base point S ∈ P , or it has a line L as fixed component.In the first case the pencil is ⊂ ∆ S , in the second case it is contained in the subspace∆( L ) consisting of the conics of the form L + L ′ , where L ′ is an arbitrary line in theplane. (cid:3) Remark 4.2.
Even if the result above follows right away from the classification ofpencils of conics, it is useful to recall the arguments which will be used in the sequel.For instance, we observe that the hyperplane sections of the Veronese surface V aresmooth conics, hence no line is contained in V .5. Birationality of certain matrix projections of curves
In this final section we want to show the validity of a much more general statement:
Theorem 5.1.
Let D ⊂ P := P ( Sym ( C )) be a curve and B : C ′ → D ⊂ P be itsnormalization.Then, for general S ∈ P , the projection π S : P = P ( Sym ( C )) P given by π S ( B ) := BS has the property that its restriction to D , π S | D is birational onto itsimage, unless D is a curve contained in a plane ∆( S ′ ) and is not a line.In the latter case, each projection π S | D has as image the line ( S ′ ) ⊥ and is not bira-tional.Proof. Let G := Gr (1 , P ) be the Grassmann variety of lines Λ ⊂ P : G has dimension 8.Define, for S ∈ P , G S := { Λ ∈ G| Λ ∩ ∆ S = ∅} . Indeed, these 6-dimensional subman-ifolds of G are the fibres of the second projection of the incidence correspondence I ⊂ G × P , I := { (Λ , S ) | Λ ∩ ∆ S = ∅} . In turn I is the projection of the correspondence J ⊂ G × ∆ × P , J := { (Λ , B, S ) | B ∈ Λ , BS = 0 } . Recall further that ∆ \ V has a fibre bundle structure K : ∆ \ V → P such that K ( B ) := ker ( B ), and with fibre over S equal to ∆ S \ V . Remark 5.2. (1) Observe that for matrices B ∈ V we can write them in the form B = x t x , for a suitable vector x , and in this case ker( B ) = x ⊥ , Im ( B ) = hh x ii .(2) In any case, since the matrices B are symmetric, we have always Im ( B ) = ker( B ) ⊥ . Consider now the fibres of I → G : for a general line Λ, its fibre S (Λ) is(1) if Λ ∩ ∆ = Λ , Λ ∩ ∆ ⊂ ∆ \ V , then S (Λ) consists of at most three points;(2) if Λ ∩ ∆ = Λ , (Λ ∩ V ) = ∅ , then S (Λ) consists of a line x ⊥ and at most onefurther point; AUSTICS 7 (3) if Λ ⊂ ∆ is of the form Λ ⊂ ∆ S , then S (Λ) consists of one or two lines containing S ;(4) if Λ ⊂ ∆ is of the form Λ ⊂ ∆( L ), S (Λ) consists of the line L .Since, if Λ ⊂ ∆( L ), the conics in Λ consist of L plus a line L ′ moving in thepencil of lines through a given point P .We let U ⊂ G × P , U := { (Λ , B ) | B ∈ Λ } be the universal tautological P -bundle, and we denote by p : U → P the secondprojection.Recall now that the secant variety Σ of D is defined as follows: we have a rationalmap ψ : C ′ × C ′ G associating to the pair ( s, t ) the line B ( s ) ∗ B ( t ) joining the twoimage points B ( s ) , B ( t ).Then one denotes by U ′ the pull back of the universal bundle, and defines Σ as theclosure of the image p ( U ′ ).The condition that for each S ∈ P the projection π S is not birational on D meansthat, if Y is the closure of the image of ψ , then Y ∩ G S has positive dimension.This implies that the correspondence I D := { (Λ y , S ) | y ∈ Y, Λ y ∩ ∆ S = ∅} ⊂ Y × P has dimension at least three and surjects onto P .Projecting I D on the irreducible surface Y , we obtain that all the fibres have positivedimension, and we infer that each secant line Λ y has a fibre S (Λ y ) of positive dimension.There are two alternatives:(i) a general secant Λ y is not contained in ∆, but intersects the Veronese surface V .(ii) each secant line Λ y ⊂ ∆.Step I) : the theorem holds true if D ⊂ V .Proof of step I.In this case any element of D is of the form B ( t ) = x ( t ) t x ( t ), and π S ( B ( t )) = x ( t )[ t x ( t ) S ] = ( x ( t ) , S ) x ( t ) = x ( t ) . Hence, for each S , the projection π S is the inverse of the isomorphism φ : x ∈ P → V, φ ( x ) = x t x. (cid:3) We may therefore assume in the sequel that D is not contained in V .Step II) : the theorem holds in case (i).Proof of step II.Observe preliminarly that, in case (i), D ∆; else we could take two smooth points B , B ∈ D ∩ (∆ \ V ), and the secant line B ∗ B could not fulfill (i).Choose then a point B ∈ D, B ∈ P \ ∆, hence w.l.o.g. we may assume that B isthe identity matrix I .Since any other point B ( t ) ∈ D is on the line joining B with a point x ( t ) t x ( t ) ∈ V ,we may write locally around a point of C ′ B ( t ) = I + ξ ( t ) t ξ ( t ) , where ξ ( t ) is a vector valued holomorphic function.Now, for each s, t, the secant line B ( t ) ∗ B ( s ) meets the Veronese surface V .Since B ( t ) cannot have rank equal to 1, there exists λ such that FABRIZIO CATANESE λB ( t ) + B ( s ) = λ ( I + ξ ( t ) t ξ ( t )) + ( I + ξ ( s ) t ξ ( s ))has rank equal to 1, i.e., K λ := ker( λB ( t ) + B ( s )) = { v | [ λ ( I + ξ ( t ) t ξ ( t )) + ( I + ξ ( s ) t ξ ( s ))] v = 0 } = { v | ( λ + 1) v + λξ ( t )( ξ ( t ) , v ) + ξ ( s )( ξ ( s ) , v ) = 0 } has dimension 2.Let us now make the assumption:(**) two general points ξ ( t ) , ξ ( s ) are linearly independent.The above formula shows however that, under assumption (**), it must be that v isa linear combination of ξ ( t ) , ξ ( s ). This is clear if λ + 1 = 0, otherwise v is orthogonalto the span of ξ ( t ) , ξ ( s ), contradicting that the kernel has dimension 2.Hence K λ = hh ξ ( t ) , ξ ( s ) ii and the condition that ξ ( t ) ∈ K λ yields( λ + 1) ξ ( t ) + λξ ( t )( ξ ( t ) , ξ ( t )) + ξ ( s )( ξ ( s ) , ξ ( t )) = 0and implies ( ∗ ∗ ∗ ) ∀ s, t ( ξ ( s ) , ξ ( t )) = 0 . (***) says that K λ = hh ξ ( t ) , ξ ( s ) ii is an isotropic subspace, which can have at mostdimension 1.Hence assumption (**) is contradicted, and we conclude that it must be:( ∗ ∗ ∗∗ ) ξ ( t ) = f ( t ) u, where u is an isotropic vector and f ( t ) is a scalar function.Even if this situation can indeed occur, we are done since in this case the matrix in V is unique, u t u , each secant Λ y contains u t u , hence S (Λ y ) = u ⊥ ∪ T y where T y is afinite set. Therefore, for general S , the fibre { y | Λ y ∩ ∆ S = ∅} is a finite set. (cid:3) Step III: the theorem holds true in case (ii).Proof of Step III.Consider the general secant line Λ y . We have two treat two distinct cases.Case (3) : Λ y ⊂ ∆ is of the form Λ y ⊂ ∆ S (then S (Λ y ) consists of one or two linescontaining S ).Case (4): Λ y ⊂ ∆ is of the form Λ y ⊂ ∆( L ) ( then S (Λ y ) consists of the line L ).In case (3), this means that two general matrices B ( s ) , B ( t ) have a common kernel S ( s, t ). Since the general matrix B ( t ) is in ∆ \ V , its rank equals 2 and S ( s, t ) = S ( t ) ∀ s .Hence the curve D is contained in a plane ∆ S . In this case however ImB ( t ) ⊂ S ⊥ and every projection π S ′ ( B ( t ) = B ( t ) S ′ lands in the line S ⊥ , so that the projectioncannot be birational, unless our curve D is a line.In case (4) for two general matrices B ( s ) , B ( t ) there exists a line L = L ( s, t ) suchthat B ( s ) , B ( t ) ∈ ∆( L ).Since two such general matrices have rank equal to 2, and B ( t ) L ⊂ L ⊥ , B ( s ) L ⊂ L ⊥ , if v ( t ) ∈ ker B ( t ) it follows that v ( t ) ∈ L ( since ker B ( t ) ∩ L = ∅ ). Therefore, if B ( t ) = B ( s ), then L ( t, s ) = hh v ( t ) , v ( s ) ii .However, the above conditions B ( t ) L ⊂ L ⊥ , B ( s ) L ⊂ L ⊥ are then equivalent to( B ( t ) v ( s ) , v ( s )) = t v ( s ) B ( t ) v ( s ) = 0 , ∀ t, s. Fixing t this is a quadratic equation in v ( s ), but, since the curve D is irreducible,and B ( t ) has rank equal to 2, we see that the vectors v ( s ) belong to a line. Therefore AUSTICS 9 the line L = L ( s, t ) is independent of s, t and the conclusion is that the curve D iscontained in the plane ∆( L ).In suitable coordinates for P , we may assume that L = hh e , e ii and L ⊥ = hh e ii .Choosing then S = e , we obtain an isomorphic projection, since for a matrix B = a b cb c we have B ( e ) = abc . (cid:3)(cid:3) Remark 5.3.
The referee suggested some arguments to simplify the proofs.For Theorem 3.1, this is the proposal:a) Firstly, in the case of the caustic, the curve D parametrizes the reducible conics ofthe form T P + N P , where T P is the tangent to the curve C at P , and N P is the normal.If S is a general point in P , then the degree of D equals the number of such conicspassing through S , hence, if ν is the degree of the curve N of normal lines, µ is thedegree of C ′ → N , then deg ( D ) = deg ( C ∨ ) + µν. The above formula shows that deg ( D ) ≥ C is not a line.In fact, then deg ( C ∨ ) ≥
2, while in general ν ≥ N P contains P ). But,if ν = 1, then the dual curve of N , the evolute, is a point, so C is a circle, but in thiscase µ = 2.b) Therefore, if one shows that D is contained in a plane π , then the plane π iscontained in the cubic hypersurface ∆, hence we can apply Proposition 4.1.c) In turn, to show that D is a plane curve, it is necessary and sufficient to show thattwo general tangent lines to D meet, which follows if one proves that:d) for each secant line there is a cone over D and with vertex a point B ′′ , such thatthe secant line passes through B ′′ (since then the two tangent lines are coplanar).In case I), d) follows since then, for each general S , there is a point B ′′ ( S ) such thata curve of secants passes through B ′′ ( S ), and we get a cone over D with vertex B ′′ ( S ).Varying S , the point B ′′ ( S ) must vary, since B ′′ ( S ) S = 0; hence the cone varies, andwe get that for each secant d) holds true.In case II), as we have shown, the secant variety of D equals W ∩ ∆, which is thesecant variety of the the rational normal quartic W ∩ V : but the singular locus ofthe secant variety of W ∩ V equals W ∩ V and contains D , hence W ∩ V = D , acontradiction.The argument suggested for Theorem 5.1 requires some delicate verification, so wedo not sketch it here. Acknowledgement:
I would like to thank Alfrederic Josse and Francoise P`ene for stimulating email cor-respondence and for spotting a mistake in my first naive attempt to prove birationalityof the caustic map for general source S , thus pushing me to find the proof of Theorem3.1, which I announced to them in an e-mail on December 20, 2012. At the moment of writing up the references for the present article, I became aware,by searching on the arXiv, that they have written an independent and different proofof birationality of the caustic map for general source, in [JP13].Thanks to the referee for helpful comments, and for sketching alternative arguments,which are reproduced in the remark above.
References [ACGH85]
Arbarello, E.; Cornalba, M.; Griffiths, P.A.; Harris, J. :
Geometry of al-gebraic curves. Volume I.
Grundlehren der mathematischen Wissenschaften, 267. NewYork etc.: Springer-Verlag. XVI, 386 p. (1985).[BGG81]
Bruce, J.W.; Giblin, P.J.; Gibson, C.G. :
On caustics of plane curves.
Am. Math.Mon. 88, 651–667 (1981).[BGG82]
Bruce, J.W.; Giblin, P.J.; Gibson, C.G. :
Source genericity of caustics by reflexionin the plane.
Q. J. Math., Oxf. II. Ser. 33, 169–190 (1982).[JP12a]
Josse, A., P`ene F. :
On the degree of caustics by reflection. arXiv:1201.0621, toappear in Comm. in Alg..[JP12b]
Josse, A., P`ene F. :
On the class of caustics by reflection. arXiv:1210.6551.[JP13]
Josse, A., P`ene F. :
Degree and class of caustics by reflection for a generic source. arXiv:1301.1846.[Wal50]
Walker, R. J. :
Algebraic curves. (Princeton Mathematical Series. 13) Princeton,N. J.: Princeton University Press; London: Oxford University Press X, 201 p. (1950),reprinted also by Dover and Springer.