Degree and class of caustics by reflection for a generic source

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by  Alfrederic Josse

DEGREE AND CLASS OF CAUSTICS BY REFLECTION
FOR A GENERIC SOURCE
ALFREDERICJOSSEAND FRANÇOISEPÈNE
Abstract. Given any irreducible algebraic (mirror) curve C ⊆ P2 :=
3 P2(C) and any (light position) S ∈ P2, the caustic by reflection ΣS(C)
1 of C from S is the Zariski closure of the envelope of the reflected lines
0 got from the lines coming from S after reflection on C. In [7, 8], we
2 establishedformulasforthedegreeandclass(withmultiplicity)ofΣS(C)
for any C and any S. In this paper, we prove the birationality of the
n
caustic map for a generic S in P2. Moreover, we give simple formulas
a
J for the degree and class (without multiplicity) of ΣS(C) for any C and
for a generic S in P2.
9

]
G
A
1. Introduction
.
h
We are interested in the study of caustics by reflection in the projective
t
a complex plane P2. Given an irreducible algebraic curve C = V(F) ⊂ P2 of
m
degree d ≥ 2 and given S = [x : y : z ] ∈ P2, the caustic by reflection
0 0 0
[
ΣS(C) of C from S is the Zariski closure of the envelope of the reflected lines

1 on C of the lines coming from S.
v
Form ∈ C,thereflectedlineR isdefinedastheorthogonalsymmetric
6 m,S,C
4 of the (incident) line (mS) with respect to the tangent line to C at m. In [7,
8 8], we detail theconstruction ofthe reflected lines andwe define two rational
1 maps ρ and Φ from P2 into itself satisfying the following property: For
F,S F,S
.
1 a generic m in C, ρ (m) corresponds to an equation of the reflected line
F,S
0
R and this line is tangent to Φ (C) at Φ (m). Hence the caustic
3 m,S,C F,S F,S
Σ (C) is the Zariski closure of Φ (C) and Φ is called the caustic map
1 S F,S F,S
: of C from S. Observe that the Zariski closure of ρ (C) is then the dual
v F,S
i curve of the caustic ΣS(C). In [7, 8], we used this approach to establish
X
precise formulas for the degree anf class (both with multiplicity) of Σ (C)
S
r for any C and any S. The degree with multiplicity of Σ (C) means its
a S
degree multiplied by the degree of the rational map Φ restricted to C
F,S
. The class with multiplicity of Σ (C) means its class multiplied by the
S
degree of the rational map ρ restricted to C. Our formulas complete the
F,S
formula obtained by Chasles in [3] for the class of a caustic by reflection
(for a generic C and a generic S). Let us indicate that, in [1], Brocard
and Lemoyne gave, without any proof, formulas for the degree and class of
caustics by reflection (for a Plücker curve C and for S not at infinity). It
seems that their formulas come from an incorrect composition of formulas
Date: January 10, 2013.
2000 Mathematics Subject Classification. 14H50,14E05,14N05,14N10.
Key words and phrases. caustic, degree, class, birationality
Françoise Pèneis supported by thefrench ANRproject GEODE(ANR-10-JCJC-0108).
1
DEGREE AND CLASS OF CAUSTICS BY REFLECTION FOR A GENERIC SOURCE 2
by Salmon and Cayley [10] for some caracteristic invariants of pedal and
evolute curves (using the representation of caustics by reflection given by
Quetelet and Dandelin). This is discussed in [8]. Let us also mention the
work of Catanese and Trifogli on focal loci, which generalize evolutes to
higher dimension [11, 2].
The question of the birationality of the rational maps ρ and Φ on
F,S F,S
C is not evident even if S is not at infinity. Indeed, according to results of
Quetelet and Dandelin [9, 4], when S is not at infinity, the caustic Σ (C) is
S
theevolute oftheS-centered homothety (with ratio2)ofthepedalof C from
S (i.e. the evolute of the orthotomic of C with respect to S). But we just
know that the evolute map is birational for a generic algebraic curve (see [5]
by Fantechi).
In this note, we prove the birationality on C of the maps ρ and Φ
F,S F,S
for any irreducible algebraic curve C ⊂ P2 of degree d ≥ 2 and for a generic
S in P2. This result enables us to establish simple formulas for the degree
and class of caustics by reflection valid for any irreducible algebraic curves
C ⊂ P2 of degree d ≥ 2 and for a generic S in P2. In this study, the cyclic
points I = [1 : i : 0] and J = [1 : −i : 0] play a particular role. We will also
use the canonical projection π : C3\{0} → P2.
2. Birationality
Theorem 2.1. Let C = V(F) ⊂ P2 be any irreducible algebraic curve of
degree d ≥ 2. For a generic S ∈ P2, the maps ρ and Φ are birational
F,S F,S
on C.
Before going into the proof of our Theorem, let us introduce some nota-
tions and recall some facts (see [7]). For any line D = V(ax+by+cz) ∈ P2
such that a2 +b2 6= 0, we define the orthogonal symmetric with respect to
D as the rational map σ :P2 → P2 (which is an involution) given by
D
σ [x :y : z] = π (a2+b2)·(x,y,z)+(ax+by+cz)·(a,b,0) .
D
(cid:0) (cid:1)
Let C = V(F) ⊂ P2 be an irreducible algebraic curve of degree d ≥ 2 and let
S ∈ P2\{I,J}bea(light)position. WedefineC := C\V(F2+F2). Observe
0 x y
that this set corresponds to the complement in C of the cyclic apparent
contour of C (the cyclic apparent contour of C from the cyclic points). We
recall that the reflected line R at m ∈ C \{S} is the line (mσ (S)),
m,S,C 0 TmC
where T C is the tangent to C at m. For any m ∈ C , we define the normal
m 0
line N C to C at m as the line containing m and [F (m): F (m) :0].
m x y
For any m ∈ C , we consider the set K of points S ∈ P2 such that
0 m
there exists m′ ∈ C \{m} satisfying ρ (m′) = ρ (m)6= 0. Observe that
0 F,S F,S
the set A of S ∈ P2 such that ρ is not birational can be written A =
F,S
K . To prove that ρ is birational for a generic S
SE⊂C0:#E<∞Tm∈C0\E m F,S
in P2, we prove that A is contained in a subvariety of codimension at least
1 in P2. Our proof is based on the following lemma.
Lemma 2.2. For any m ∈ C , the set K is contained in a (possibly non
0 m
irreducible) algebraic curve K¯ of degree at most 2d2+2.
m
Proof. Letusconsideranym ∈ C . LetS ∈ K andm′ ∈ C \{m}satisfying
0 m 0
ρF,S(m′) = ρF,S(m) 6= 0. Then (mm′) = Rm,S,C = Rm′,S,C and so S is in
DEGREE AND CLASS OF CAUSTICS BY REFLECTION FOR A GENERIC SOURCE 3
Am,m′ := σTmC((mm′))∩σTm′C((mm′)). Observe that, if σTmC((mm′)) =
σ ((mm′)), then these lines are (mm′) and so (mm′) is stable by σ
Tm′C TmC
and by σ . But T C and N C are the only lines containing m which
Tm′C m m
are stable by σ . Therefore, σ ((mm′)) = σ ((mm′)), implies that
TmC TmC Tm′C
(mm′) ∈ {TmC,NmC}∩{Tm′C,Nm′C}. If {TmC,NmC}∩{Tm′C,Nm′C} = ∅,
then S is the only point of Am,m′, so S is equal to
τ (m′) := π (m∧σ (m′))∧(m′∧σ (m)) .
m (cid:0) TmC Tm′C (cid:1)
Notice that τ is a rational map with coordinates of degree 2d. We ob-
m
tain that S belongs to the Zariski closure of τ (C), which (according to
m
[6, Proposition 4.4]) is contained in an algebraic curve of degree at most
C · τ∗(H) ≤ 2d2 (where H is the hyperplane class in P2). Otherwise,
m
S ∈ A = (mm′) ∈ {T C,N C}. Finally, we have K ⊆ K¯ :=
m,m′ m m m m
τ (C)∪T C ∪N C which is an algebraic curve of degree at most 2d2 +2
m m m
(use for example the fundamental lemma of [7] applied with τ ). (cid:3)
m
Proof of Theorem 2.1. Let us prove that ρ is birational on C for a generic
F,S
S in P2. The birationality of Φ will follow. Indeed, for a generic S in
F,S
P2, the caustic Σ (C) is a curve (see for example [8]). Therefore, for generic
S
m,m′ ∈ C, Φ (m) = Φ (m′) implies that ρ (m) = ρ (m′). With the
F,S F,S F,S F,S
notations of Lemma 2.2, we define A′ := K¯ . We
SE⊂C0:#E<∞Tm∈C0\E m
prove that the set F := K¯ , E ⊂ C , #E < ∞ is inductive for
nTm∈C0\E m 0 o
the inclusion. Let F := K¯ be an increasing sequence of
(cid:16) j Tm∈C0\Ej m(cid:17)j≥1
sets belonging to F. Let us show that the union Z of these sets is also in F.
First Z ⊆ K¯ ⊆ K¯ for some fixed m ∈ C \ E . Now
Tm∈C0\Si≥1Ei m m0 0 0 Si≥1 i
K¯ istheunionofafinitenumberofirreduciblealgebraiccurvesC ,...,C .
m0 1 p
Let i ∈ {1,...,p} and let d be the degree of C . If C ⊆ Z, then there exists
i i i
N ≥ 1 such that C ⊆ F . Assume now that C 6⊆ Z. Then (C ∩F ) is
i i Ni i i j j≥1
an increasing sequence of finite sets containing at most d ×(2d2+2) points.
i
Therefore, there exists N ≥ 1 such that (C ∩Z) ⊆ F . We conclude that
i i Ni
Z = F and so Z is in F. So F is inductive.
max(N1,...,Np)
From the Zorn lemma, either F is empty or it admits a maximal element
(for the inclusion). If it is empty, then A = A′ = ∅. If it is not empty
and if F := K¯ (with E ⊂ C and #E < ∞) is a maximal
0 Tm∈C0\E0 m 0 0 0
element of F, then A′ = F . Indeed, A′ contains F by definition of A′.
0 0
Conversely, let S ∈ A′, there exists E ⊂ C such that #E < ∞ and such
0
that S ∈ K¯ . Hence S ∈ K¯ . Since we also have
Tm∈C0\E m Tm∈C0\(E∪E0) m
K¯ ⊆ K¯ , we conclude that S ∈ K¯ .
Tm∈C0\E0 m Tm∈C0\(E∪E0) m Tm∈C0\E0 m
Therefore, in any case, A is contained in an algebraic curve, this gives the
S-genericity of the birationality of ρ and so the statement of Theorem
F,S
2.1. (cid:3)
3. Light generic formulas for the degree and the class of
caustics
Let C = V(F) ⊂ P2 be any irreducible algebraic curve of degree d ≥ 2.
We call isotropic tangent to C any tangent to C containing I or J. Before
stating our formulas, let us introduce some notations.
DEGREE AND CLASS OF CAUSTICS BY REFLECTION FOR A GENERIC SOURCE 4
For any P ∈ P2, we write µ (C) for the multiplicity of C at P. We recall
P
thatµ (C) = 1means thatP isanonsingularpointofC. Forany P ∈ C, we
P
writeBranch (C)forthesetofbranchesofC atP. LetuswriteE fortheset
P C
of couples (P,B) withP ∈C and with B ∈ Branch (C). Forany (P,B) ∈ E ,
P C
we write T B for the tangent line to B at P and e for the multiplicity of
P B
B. We recall that e = µ (C). For any (P,B) ∈ E and any
PB∈BranchP(C) B P C
algebraic curve C′, we denote by i (C,C′) (resp. i (B,C′)) the intersection
P P
number of C (resp. B) with C′ at P. We recall that the contact number
Ω (C,C′) of C with C′ is given by Ω (C,C′) = i (C,C′) − µ (C)µ (C′).
m1 m1 P P P
The line at infinity of P2 is written ℓ . Combining Theorem 2.1 with the
∞
main results of [7, 8], we obtain:
Proposition 3.1. Let C = V(F) ⊆ P2 be any irreducible algebraic curve of
degree d ≥ 2 and of class d∨. For a generic S ∈ P2, we have
deg(Σ (C)) = 3d+f −t −t and class(Σ (C)) = 2d∨+d−g−µ (C)−µ (C),
S 0 I J S I J
where g is the contact number of C with ℓ , i.e. g := Ω (C,ℓ ),
∞ Pm1∈C∩ℓ∞ m1 ∞
where f is the number of “inflectional branches” of C not tangent to the line
0
at infinity, i.e.
f := (i (B,T B)−2e ).
0 X P P B
(P,B)∈EC:iP(B,TPB)>2eB,TPB=6 ℓ∞
and where t is the number of branches of C tangent at P to the line at
P
infinity: t = e .
P PB∈BranchP(C):TPB=ℓ∞ B
Proof. According to Theorem 2.1, for a generic S in P2, the degree and class
(with multiplicity) of Σ (C) are equal to its degree and class. For the degree
S
formula, we use Theorem 20 of [7]. For the class formula, we use Theorem 2
of [8]. For a generic S ∈ P2 (S ∈ P2\(C∪ℓ ) not contained in an isotropic
∞
tangent to C), we have f = µ (C)+µ (C), f′ = 0, g′ = 0 and q′ = 0 (with
I J
the notations of Theorem 2 in [8]). (cid:3)
Acknowledgements
We thank Fabrizio Catanese for discussions having motivated the redac-
tion of this note on source generic results for caustics by reflection.
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Université de Brest, LMBA, UMR CNRS 6205, 29238 Brest cedex, France
E-mail address: [email protected]
Université de Brest, LMBA, UMR CNRS 6205, 29238 Brest cedex, France
E-mail address: [email protected]