Conchoidal transform of two plane curves
aa r X i v : . [ m a t h . AG ] A p r Conchoidal transform of two plane curves
Alberto Albano – Margherita RoggeroDipartimento di Matematica dell’Universit`a di TorinoVia Carlo Alberto 1010123 Torino, Italy [email protected]@unito.it
Abstract
The conchoid of a plane curve C is constructed using a fixed circle B in the affineplane. We generalize the classical definition so that we obtain a conchoid from anypair of curves B and C in the projective plane. We present two definitions, one purelyalgebraic through resultants and a more geometric one using an incidence correspondencein P × P . We prove, among other things, that the conchoid of a generic curve of fixeddegree is irreducible, we determine its singularities and give a formula for its degree andgenus. In the final section we return to the classical case: for any given curve C we givea criterion for its conchoid to be irreducible and we give a procedure to determine whena curve is the conchoid of another. The conchoid of a plane curve is a classical construction: given a curve C in the real affineplane, fix a point A and a positive real number r . The conchoid of C is the locus of points Q that are at distance r from a point P ∈ C on the line AP . Examples of this construction arethe conchoid of Nichomede and the lima¸con of Pascal (see for example [5], [7]). In [7] one canfind an analysis of the basic algebraic properties of the conchoid of an algebraic plane curveover an algebraically closed field of characteristic zero.When the curve C is algebraic it is easy to obtain the equation of the conchoid fromthe equation of C . One way to do this is by elimination of variables, using Gr¨obner bases.However, the conchoid of a curve may have multiple components and this procedure does notalways give the correct multiplicities. For example, choosing A = (0 ,
0) and r = 1, for theline x − y + x + x y − x − xy + 3 x = 0 whilefor the line x = 0 one finds x ( x + y −
1) = 0. In fact in this last case the component x = 0should be counted twice.In this paper we give two different ways to define correctly the conchoid. The first isalgebraic, and uses resultants instead of Gr¨obner bases to find the equation of the conchoid.The second is more geometric and uses techniques in algebraic geometry like correspondencesand multiple covers of P . Written with the support of the University Ministry funds. Mathematics Subject Classification 2000: 14H45, 14H50, 14Q05Keywords: conchoid, resultant, double planes R . Moreover, it is more convenientto work in a projective ambient, so for us curve will mean a divisor in P . However theconchoid is essentially an affine concept, and so we fix in P a line L ∞ as line at infinity anda point A in its complement. If B and C are two curves we define the conchoidal transformof C with respect to B as the locus of points Q intersection of the line AP , with P a pointin C , and the translate of B of a vector ~AP . The translation is well defined in the fixedaffine part. When B is a circle with center A and radius r , this definition is the same as theclassical one. In this description the two curves play different roles, but we will see that theconchoidal tranform is in fact symmetrical in B and C .Both our definitions are universal on the coefficients of the equations of the curves B and C . This will allow us to reduce many proofs to the case when one of them is a generic lineor a union of generic lines, and use deformations.After some preliminaries, in section 3 we give the definition of conchoid using resultantsand we prove some properties, in particular we determine the degree, the singularities and thespecial components of the conchoid. Then in sections 4 and 5 we give a geometric definition.The construction we make works only under suitable hypotheses, which are made explicit inAssumption 4.1 of Section 4. We then prove that under these hypotheses the two definitionscoincide.We use the word generic in the sense of algebraic geometry. For a family of objectsparametrized by an algebraic variety, for example the family of all plane curves of givendegree, generic means “in the complement of a proper closed algebraic subset”, i.e., outsidethe locus given by finitely many polynomial equations in the coordinates of the parameterspace. Sometimes it is possible to give these equations explicitely, as we do in Assumption 4.1where every geometric condition can be translated in the vanishing of some polynomials. Inother situations it’s enough to know that these equations exist, for example in the proof ofTheorem 5.3 where we use the classical Bertini’s theorem.We show that the conchoid of a generic curve is irreducible and give a formula for itsgenus that depends on the genera and the degrees of the curves B and C . The case B circleand C rational is studied extensively in [8], where an algorithm is given to determine whenthe conchoid of C is rational or splits in two rational components and to compute a rationalparametrization of each rational component. We also define the concept of proper conchoid in analogy of that of proper transfom.In the last section we go back to the classical case: in this situation the multiple coverof P is a double cover and we use the theory of double planes to give a criterion for theirreducibility of the proper conchoid of any curve. This part requires a bit more algebraicgeometry than the rest of the paper, in particular in the proofs we use freely the propertiesof branched coverings and of normalization. We introduce also the concept of n -iteratedconchoid and show that all the iterated conchoids of a fixed curve belong to a 1-dimensionalflat family. We end with a procedure to determine when an irreducible curve is either thecomplete or proper conchoid of another. We work over a fixed base field k . For a geometrical interpretation it is better to have k algebraically closed, but most definitions make sense on the field of definition of the startingcurves. We assume the characteristic of k to be 0 or a prime number p greater than thedegrees of the curves we consider, so we can use derivatives to study singularities.We will denote P the projective plane over k . As the concept of conchoid is an affineone, we fix a line L ∞ and we denote with A its complement. It is a fixed affine plane, andinside it we fix a point A . We choose homogeneous coordinates [ x : y : z ] in P so that L ∞ has equation z = 0, and A = [0 : 0 : 1] is the origin of A . If D ⊂ P is the curve given bythe homogeneous equation G ( x, y, z ) = 0, we denote by D ( a ) the affine part of D , i.e., D ( a ) is the curve in A given by the equation G ( x, y,
1) = 0.We fix two projective curves, denoted by B and C , with equations F ( x, y, z ) = 0 and G ( x, y, z ) = 0 of degrees d and δ and geometric genus g and γ respectively. To avoid trivialcases, we assume that B is the projective closure of B ( a ) , i.e., L ∞ is not a component of B .The following lemma will allow us to give different but equivalent definitions for theconcept of conchoidal transform of the curves B and C . Having fixed a point A , the affineplane has a natural vector space structure in which A is the zero element. In the statementof the lemma we are using this structure when we add points or multiply them by a scalar. Lemma 2.1.
Let B be a projective curve in P and B ( a ) its affine part. For every P , Q in A , such that P, Q = A , the following are equivalent:1. Q is on the line AP and on the translate of B ( a ) by the vector ~AP ;2. Q is on the line AP and the point Q − P (i.e., the translate of Q by the vector ~P A )belongs to B ( a ) ;3. Q = P + S with S ∈ B ( a ) and A , P and S are collinear;4. ∃ λ ∈ k such that P = λQ and (1 − λ ) Q ∈ B ( a ) . We do not give the proof, which is elementary; we only note that the main reason for theequivalence is the fact that the line AP is invariant under translation by the vector ~AP .Let C ( a ) be an affine curve. In the classical construction of a conchoid, to each point P ∈ C ( a ) one associates the two points on the line AP at distance 1 from P . These are thepoints Q satisfying condition of the previous Lemma, when B ( a ) is the circle of center A and radius 1. In this way one obtains an affine curve. More generally, one may think of theconchoidal transform of the curve C with respect to B as the projective closure of the set ofpoints satisfying one of the equivalent conditions of Lemma 2.1, as P varies on C ( a ) . Usingcondition , we see that the roles of C and B are in fact completely symmetrical. We will usecondition to give a general definition of the conchoidal transform of two projective curves:the definition will involve a resultant, and it will be useful both for theoretical purposes and asa computational device, instead of elimination of variables and Groebner basis computations. Let B and C be projective plane curves, with equations F ( x, y, z ) = 0 and G ( x, y, z ) = 0respectively. Writing down explicitely condition 4 of Lemma 2.1 we see that a point Q = [ a : b : 1] in A different from A is in the conchoid of C with respect to B if the system of twoequations in the single unknown λ (cid:26) F ((1 − λ ) a, (1 − λ ) b,
1) = 0 G ( λa, λb,
1) = 0has a solution. Using projective coordinates, we then define:
Definition 3.1.
The conchoidal transform C ( B, C ) of B and C (which we will often callsimply the conchoid ) is the divisor in P given by the resultant R ( F, G ) of the two polynomialsin the homogeneous variables λ and µF (( µ − λ ) x, ( µ − λ ) y, µz ) and G ( λx, λy, µz ) (1)Write F ( x, y, z ) = F d ( x, y ) + zF d − + . . . and G ( x, y, z ) = G δ ( x, y ) + zG δ − + . . . as poly-nomials in z so that F h e G h are homogeneous polynomials of degree h in the indeterminates x, y . We have: F (( µ − λ ) x, ( µ − λ ) y, µz ) = d X i =0 λ i µ d − i Φ i ( x, y, z )where Φ i ( x, y, z ) = ( − i P dj = i (cid:18) ji (cid:19) F j ( x, y ) z d − j , and G ( λx, λy, µz ) = δ X i =0 λ i µ δ − i G i ( x, y ) z δ − i hence: R ( F, G ) = Φ d Φ d − . . . . . . . . . Φ . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . d Φ d − . . . . . . . . . Φ G δ zG δ − . . . . . . . . . z δ G . . . . . . . . . . . .. . . . . . . . . . . . G δ zG δ − . . . . . . . . . . . . z δ G . Example 3.2.
Conchoidal transform of two lines. Let F = ax + by + cz and G = a ′ x + b ′ y + c ′ z .The conchoidal trasform is given by: (cid:12)(cid:12)(cid:12)(cid:12) − ( ax + by ) ax + by + cza ′ x + b ′ y c ′ z (cid:12)(cid:12)(cid:12)(cid:12) = − (cid:2) ( ax + by + cz )( a ′ x + b ′ y + c ′ z ) − cc ′ z (cid:3) . This polynomial does not define a curve only if B and C are both the line L ∞ given by z = 0 .In all the other cases it is the hyperbola passing through the origin A and with asymptotes thelines B and C . Example 3.3.
Conchoid with respect to a line B . Let F = ax + by + cz as before and G anyhomogeneous polynomial of degree δ ≥ . The conchoidal trasform is given by: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ( ax + by ) ax + by + cz . . . . . . . . . . . . . . . . . . − ( ax + by ) ax + by + czG δ zG δ − . . . z δ − G z δ G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . This polynomial is G ( x ( ax + by + cz ) , y ( ax + by + cz ) , ( ax + by ) z ) as we show by induction on the degree δ . For δ = 1 it is true by the computation in theprevious example; assume now the statement for deg G = δ − , and expand the determinantalong the last row: we obtain R ( F, G ) = ( ax + by + cz ) δ G δ ( x, y ) + z ( ax + by ) R ( F, G ) where G = G δ + zG , which is the thesis.We have again an effective divisor, unless B and C have both L ∞ as a component. We can obtain the following properties of the conchoidal transform from well knownproperties of the resultant. In particular, 4 . and 5 . say that if either B or C is in some specialposition with respect to A or L ∞ , then the conchoidal transform will have certain specialcomponents. Theorem 3.4.
Let B and C be as before. Then:1. deg C ( B, C ) = 2 δd ;2. C ( B, C ) = C ( C, B ) ;3. if C = C + C then C ( B, C ) = C ( B, C ) + C ( B, C ) ;4. if P ∈ L ∞ ∩ B ∩ C and the multiplicities in P of B and C are respectively η and ǫ , then P ∈ C ( B, C ) with multiplicity ≥ ǫδ + ηd and the line AP is a component of C ( B, C ) with multiplicity ≥ ǫ ( η − ǫ ) + ǫ ( ǫ +1)2 if ǫ ≤ η ;5. if A ∈ C with multiplicity ν , then the divisor νB is contained in C ( B, C ) .Proof. 1. and follow respectively from the definition and a property of the resultant ([2]Exercise 3 page 79).To prove we observe that the existence of a non trivial solution ( λ, µ ) is the same asthe existence of a non trivial solution ( λ ′ = µ − λ, µ ′ = µ ) and with respect to these newvariables λ ′ , µ ′ the roles of C and B are interchanged.To prove we may assume that P has coordinates [1 : 0 : 0] and so the line AP is given by y = 0. Since the η and ǫ are the multiplicities in P of B and C respectively, every monomialin F belongs to ( y, z ) η and every monomial in G belongs to ( y, z ) ǫ . Hence every entry in thefirst δ rows of the matrix whose determinant is R ( F, G ) belongs to ( y, z ) η and every entry inthe other d rows belongs to ( y, z ) ǫ . This facts clearly imply R ( F, G ) ∈ ( y, z ) ǫδ + ηd .For the same reason as above, y η − i divides Φ d − i for every i < η and y ǫ − i divides G δ − i forevery i < ǫ . Thanks to we may assume ǫ ≤ η . Moreover, after a change of coordinates x ′ = x , y ′ = y and z ′ = z + uy for a general constant u , we may assume that y ǫ is the maximalpower of y dividing G δ and y η is the maximal power of y dividing Φ d = F d (note that sucha change of coordinates does not modify A nor P ). Now the thesis is a consequence of thefollowing Lemma 3.5, where S = k [ x, y, z ] ( y ) and M is the matrix of the first η columns ofthe matrix whose determinant is R ( F, G ) (without the null rows).Finally, for we observe that, if G = · · · = G ν − = 0, then the resultant is multiple ofΦ ( x, y, z ) ν where Φ ( x, y, z ) = P dj =0 F j ( x, y ) z δ − j = F ( x, y, z ). Lemma 3.5.
Let S be a discrete valuation ring with valuation v , let a, b positive integers, a ≤ b , and let M be a b × b matrix with entries in S of the following type: M = g g . . . . . . g b g g . . . g b − . . . . . . . . . g f f . . . . . . f b f f . . . f b − . . . . . . . . . f such that v ( f ) = b , v ( g ) = a and v ( f i ) ≥ b + 1 − i and v ( g i ) ≥ a + 1 − i for i = 1 , .., b . Then v ( m ) ≥ a ( b − a ) + a ( a +1)2 for every minor m of maximal order of M .Proof. As g divides f , . . . , f b − a +1 , we can performe a row reduction on M obtaining amatrix of the following type (where we have deleted a final block of a rows containing onlyzero entries): M ′ = g g . . . . . . . . . g b . . . . . . . . . g . . . g a +1 . . . g . . . g a . . . . . . . . . . . . g . . . h b − a +1 . . . . . . h b . . . . . . . . . . . . . . . . . . h b − a +1 Note that the statement holds for M if and only if it holds for M ′ . Every b × b minor of M ′ is the product of the only non-zero ( b − a ) × ( b − a ) minor extracted from the first b − a columns, which is g b − a (whose valuation is a ( b − a )) and an a × a minor extracted from thelower right block M ′′ . All the entries of the i -colum of M ′′ have valuation ≥ a + 1 − i andthen every a × a minor of M ′′ has valuation ≥ a ( a +1)2 . Remark 3.6.
We computed several examples and we have always found that the multiplicityof the line AP in C ( B, C ) satisfies a stronger inequality than the one given in 4. above,namely it is always at least the product ǫη of the multiplicities at P of the curves B and C .We also have some theoretical justification for this fact, but not a proof. W B = C ( B, − ) The definition just given using resultants is applicable to any pair of curves, gives explicitelythe equation of the conchoidal transform and allows to prove some interesting consequences.However, it is hard in general to obtain geometrical properties from the equation alone.Hence we now present a different characterization of the conchoid of two curves, using a moregeometrical approach. In this construction the curves B and C will play different roles andthe conchoidal transform will appear as obtained from a fixed curve B acting over a generalcurve C .The definition will use a surface W B obtained from the curve B . In this section wedefine W B and study its properties. In the next section we will use it to define the conchoidof C . The geometrical construction makes sense only if B is generic enough, so we start byfixing the hypotheses on B . Assumption 4.1. B will always be a smooth curve in P of degree d and genus g (so that g = 1 / d − d − ), defined by the equation F ( x, y, z ) = 0 .We also assume that B intersects the fixed line L ∞ in d distinct points P i , it does notcontain the fixed point A and intersects every line through A in at least ( d − distinct points(i.e., no line through A is a multitangent to B or a flex tangent). We will denote by L i the d lines AP i and by D j the d ( d −
1) lines through A that aretangent to B : we do not exclude that L i = D j for some i and j may hold.Finally we will denote by B − the curve given by F ( − x, − y, z ) = 0, that is the curve whoseaffine part is symmetric to B ( a ) with respect to A .Let us consider the subset of P × P containing all the pairs of points ( P, Q ) that satisfythe equivalent conditions given in Lemma 2.1 and denote by W B its closure (with respect tothe Zariski topology). We can write the equations for the affine part of W B using condition 2as follows.Let P = [ x : y : z ] and Q = [ X : Y : Z ] two points not lying on L ∞ . Then ( P, Q ) ∈ W B ifand only if xY − yX = 0 and F ( zX − xZ, zY − yZ, zZ ) = 0. The first equation correspondsto “ A , P , Q collinear” and the second one to “ Q − P ∈ B ( a ) ”. In fact, in the affine open set z = 0, Z = 0 the point Q − P is given by ( XZ − xz , YZ − yz ) and so in P the correspondingpoint is [ XZ − xz : YZ − yz : 1] that is [ zX − xZ : zY − yZ : zZ ].This computation justifies the following definition in projective coordinates: Definition 4.2.
In the product of projective planes P × P with bihomogeneous coordinates [ x : y : z ; X : Y : Z ] , the incidence surface with respect to B is the subvariety W B defined bythe bihomogeneous ideal I = ( F ( zX − xZ, zY − yZ, zZ ) , xY − yX ) . (2) We will denote by π : W B → P e π : W B → P the projections on the first and on thesecond factor.In a similar way as before, we will use W ( a ) B for the affine part of W B , i.e., W ( a ) B is givenby the equations F ( X − x, Y − y,
1) = xY − yX = 0 in the affine space A given by z = 0 , Z = 0 . We note that the subscheme defined by the ideal I could be reducible or non reduced.However we will prove in the next proposition that under Assumption 4.1 W B is indeedirreducible and reduced, so it is a variety. Proposition 4.3.
In the above notation:1. π e π are surjective;2. the involution σ of P × P given by ( P, Q ) ( Q, P ) restricts to an isomorphism W B ∼ = W B − . Moreover π ◦ σ = π , π ◦ σ = π ;
3. the affine part W ( a ) B of W B is a product (though in a non-standard way). More precisely: W ( a ) B ∼ = B ( a ) × A (but W ( a ) B = π ( W ( a ) B ) × π ( W ( a ) B ) );4. W B is an irreducible and reduced suface and its affine part W ( a ) B is smooth.Proof. 1. By Assumption 4.1 a general line through A in A meets the affine curve B ( a ) in d points. So it is an easy consequence of condition 2. of Lemma 2.1 that for a general point Q on such a line there is a point P such that the condition holds for ( P, Q ) (and viceversa, fora general P there is at least a Q ). Then the image of π (or π ) is a dense subset of P . Asit is also closed, it must be the whole P . The isomorphism between W B and W B − given by the involution σ directly followsfrom condition 2. of Lemma 2.1, because Q − P ∈ B ( a ) if and only if P − Q ∈ B ( a ) − . In thesame way we can see that σ exchanges π and π on the affine subsets. Finally the relationsobtained on the affine subset can be extended to the projective closure, because σ is also aninvolution of ( P × P ) \ A . In the open subset Z = z = 1, the affine coordinates are ( x, y, X, Y ). The equationsdefining W ( a ) B are F ( X − x, Y − y,
1) = xY − yX = 0. With the change of coordinates x ′ = X − x , y ′ = Y − y these equations become F ( x ′ , y ′ ,
1) = x ′ y − y ′ x = 0. Thanks to thehypothesis A / ∈ B , all the solutions can be written as ( a, b, λa, λb ) where [ a : b : 1] ∈ B ( a ) (and so ( a, b ) = (0 , W ( a ) ∼ = B ( a ) × A .Finally, is a straightforward consequence of the previous item, because W B is theclosure of W ( a ) B in P × P .We now investigate the singular locus of W B , that must be contained in the part atinfinity W B \ W ( a ) B because of the previous result. Here we will use the hypothesis that eitherchar( k ) = 0 or char( k ) = p greater than the degree d of B . Proposition 4.4. If d = 1 , i.e., B is a line, then W B is smooth. If d ≥ , the singular locusof W B is the subvariety W B ∩ ( L ∞ × L ∞ ) cut by z = Z = 0 . More precisely, every point in W B ∩ ( L ∞ × L ∞ ) has multiplicity at least d .Proof. If (
P, Q ) belongs to the locally closed subset of W B where Z = 0 and z = 0, then ithas coordinates of type [ λa : λb : c ; a : b : 0] for some λ, a, b such that λ = 0 and either a = 0or b = 0. If for instance a = 0 we can choose a = 1 and consider ( P, Q ) as a point in theaffine 4-space given by z = X = 1 and with coordinates ( x, y, Y, Z ). In these coordinates, theequations of W B are F (1 − xZ, Y − yZ, Z ) = xY − y = 0 . We use F x , F y and F z to denote the derivatives of F ( x, y, z ) with respect to its originalvariables. Computing the derivatives with respect to the variables ( x, y, Y, Z ) using the chainrule and evaluating in the point ( P, Q ) = [ λ : λb : 1; 1 : b : 0], the Jacobian matrix of theequations of W B is: (cid:20) F y (1 , b, F z (1 , b, b − λ (cid:21) and has rank 2 because [1 , b, ∈ B and B is smooth. We observe that the last item inthe first row should be − λ [ F x (1 , b,
0) + bF y (1 , b, F z (1 , b, F (1 , b,
0) and [1 : b : 0] ∈ B . So wecan conclude that ( P, Q ) is a smooth point. In the same way we can prove the smoothnessof every point in the subset of W B given by Z = 0, z = 0 and Y = 0.The same holds if z = 0 and Z = 0, thanks to the symmetry between W B and W B − .If Z = z = 0 then ( P, Q ) = [ a : b : 0; a : b : 0] and either a or b does not vanish. If forinstance x = X = 1, the entries of the first row of the Jacobian matrix (with coordinates( y, z, Y, Z )) are homogeneous polynomials of degree d − zX − Zx, zY − Zy, zZ . If we evaluate the Jacobian matrix in (
P, Q ), that is if we set y = Y = b and z = Z = 0, then its rank is not maximal if and only if d ≥
2. Moreover, the rank is notmaximal also if we consider the higher derivatives up to the ( d − P, Q ) hasmultiplicity at least d .We study now the properties of the fibers of the projection π . We refer to the beginningof this section for the meaning of D j , P i and L i .The fibers of π will have the same properties. In fact π can be seen as the first projectionfrom the incidence surface W B − . Note that B and its symmetric curve B − share the sametangent lines through A and the same intersection points with the line at infinity L ∞ . Proposition 4.5.
Let P be any point in P .1. If P is general (more precisely if it is not one of the points considered in the followingitems), then π − ( P ) is a set of d = deg( B ) distinct points;2. if P ∈ D j \ L ∞ , then π − ( P ) is given by d − distinct points (exactly one of which withmultiplicity 2);3. π − ( A ) is the curve Γ in P × P of the points ( A, Q ) such that F ( Q ) = 0 , so that ina natural way Γ ∼ = π (Γ) = B ;4. if P ∈ L ∞ \ B , then π − ( P ) is a single point with multiplicity d ;5. if P = P i [ a i : b i : 0] ∈ L ∞ ∩ B , then π − ( P i ) is the rational curve Λ i of the points [ a i : b i : 0; λa i : λb i : µ ] , so that Λ i ∼ = π (Λ i ) = L i .Proof. If P is a point in A , then π − ( P ) can be obtained by first intersecting the line AP with B and then translating by the vector ~AP : this proves and Statement is thecase when P = A and easily follows from the equations (2) of W B .So it remains to prove the last two items. If P = [ a : b : 0], looking at the second equationin (2) we can see that every point Q in π − ( P ) is of the type Q = [ λa : λb : Z ]. If we evaluatethe first equation in ( P, Q ), we obtain F ( − aZ, − bZ,
0) = 0 that is ( − Z ) d F ( a, b,
0) = 0. Thereare two possibilities. If
P / ∈ B , then Z = 0 and π − ( P ) = { [ a : b : 0; a : b : 0] } contains asingle point Q = [ a : b : 0] with multiplicity d . If, on the other hand, P ∈ B , then all values for Z are possible and π − ( P ) is a rational curve with parametric equations [ a : b : 0; λa : λb : µ ]in the homogeneous parameters [ λ : µ ].We collect in the following corollary the main results obtained until now.0 Corollary 4.6. W B is a surface in P × P , that is a reduced and irreducible subvarietyof dimension . If the degree d = deg( B ) ≥ , its singular locus is the curve given by Z = z = xY − yX = 0 and every singular point is d -uple.The projection π : W B → P is a generically finite map of degree d , branched over the d ( d − lines D j , i.e., the lines containing A and tangent to B . The exceptional fibers arethe one over A , which is the curve Γ , isomorphic to B through π , and those over the d points P i ∈ B ∩ L ∞ , which are the rational curves Λ i , isomorphic (through π ) to the lines L i = AP i . C obtained from W B If C is a reduced curve and does not contain any special points (namely A and P i ∈ B ∩ L ∞ ),then the curve π ( π − ( C )) is well defined. Thanks to the equivalent conditions of Lemma 2.1,we can easily see that the curve π ( π − ( C )) is precisely the conchoidal transform C ( B, C )defined in Section 3. However, if either C is non reduced or it contains some of the specialpoints or some of the special divisors, the curve C ( B, C ) can have some non reduced compo-nents and also some components that are in some sense special components . This is a verycommon difficulty in algebraic geometry, when exceptional fibers of morphisms are involved.Similar to the definition of proper transform for a blowing-up morphism, we would like todefine a proper conchoid , not containing exceptional fibers of the transformation. To this end,we give a new definition of conchoid in a geometric way. We will prove that this definitionis equivalent to the previous one, but in it the two starting curves B and C play differentroles. More explicitly, for every B and C we will obtain not only a curve C B ( C ), but alsoa set of exceptional divisors: the curve C B ( C ) is the same as C ( B, C ), but the exceptionaldivisors will depend only on B , so that they are in general a different set from that of C C ( B ).Removing the exceptional divisors, we will finally obtain the definition of the proper conchoid(Definition 5.5).We recall the definitions from algebraic geometry of pull-back and push-forward of cycles.To keep things simple we state them only for the special case we need. For more informationon the general notions, see [3], Chapter 1 or [4], Appendix A .Let W be a projective variety and φ : W → P a surjective morphism. A k -cycle is aformal linear combination with integer coefficients of reduced and irreducible subvarieties ofdimension k . For C a reduced and irreducible subvariety in W , we define the push-forward φ ∗ ( C ) to be 0 if the dimension of the image φ ( C ) is strictly less than dim C , otherwise we set φ ∗ ( C ) = m · φ ( C )where m is the degree of the map φ | C , i.e., the number of points in the generic fiber. Weextend φ ∗ to all cycles by linearity. We note that the push-forward of a k -cycle is again a k -cycle.We define the pull-back only for divisors, i.e., cycles of codimension 1. If D is a reducedand irreducible curve in P , it is given by a single equation G = 0. The pull-back φ ∗ ( D ) isgiven by “pulling back” this equation to W . To make sense of this we cover P with openaffine sets { U i } , for example the standard ones obtained by dehomogenizing one variable atthe time, and we set g i the corresponding inhomogeneous equation of C on U i . Then the W i = φ − ( U i ) are an affine open cover of W and we let φ i : W i → U i be the restriction of φ W i . Then φ ∗ ( D ) is the divisor with local equation g i ◦ φ i = 0 in the open set W i . We notethat even if D is reduced, φ ∗ ( D ) may have multiple components since the differential of φ may not have maximal rank everywhere. Again we extend φ ∗ to all divisors by linearity.We can now give the: Definition 5.1. If C is a curve of P , that is a -cycle in P , we will call conchoid of C (with respect to B ) the cycle C B ( C ) = π ∗ ( π ∗ ( C )) . If C is reduced and does not contain any special point or divisor for π and π , then C B ( C ) is precisely π ( π − ( C )). We can obtain an equation for its affine part ( C B ( C )) ( a ) afterelimination of the variables x, y from the ideal: I = ( F ( X − x, Y − y, , xY − yX, G ( x, y, . (3)In all the other cases, we can consider a flat family of curves C t depending on one or moreparameters t such that C t = C and, for a general t , C t is of the previous type. The conchoidof C is the limit of C B ( C t ) for t = t .We can for instance consider the family C t of all degree δ curves whose equation is adegree δ polynomial with indeterminate coefficients. Then we can formally performe theelimination of the variables x, y and, at the end, specialize t .It can also be useful to think of the general degree δ polynomial as an element of a vectorspace generated by all the products of δ linear forms, corresponding to curves split in lines. Example 5.2.
Let us consider the classical case, when B is the circle x + y − z = 0 and A = [0 : 0 : 1] . The conchoid of a general line ax + by + cz = 0 , obtained as just indicated,is given by the equation ( aX + bY + cZ ) ( X + Y ) − ( aX + bY ) Z = 0 . If we specializethe coefficients a, b, c in order to obtain the conchoid of the line L of equation x = 0 (whichcontains A ), we get X ( X + Y − Z ) = 0 , i.e., the divisor L + B that has degree . Ifinstead we eliminate, for example using a Gr¨obner basis computation, the variables ( x, y ) from the ideal I in ( ) above, the resulting curve has (affine) equation X ( X + Y −
1) = 0 ,i.e., it is L + B . So we see that specialization does not commute with elimination.We can also obtain the conchoid of the infinity line L ∞ : its equation is Z ( X + Y ) = 0 ,i.e., the conchoid is L ∞ + L + L . We now state and prove the main result for B and C generic. Theorem 5.3.
Let B be a curve as in Assumption 4.1, and let C be a generic curve ofdegree δ and geometric genus γ . Then:1. C B ( C ) = C ( B, C ) .2. C B ( C ) is irreducible and reduced;3. C B ( C ) is birational to π − ( C ) (via π );4. C B ( C ) has genus ˜ g = dγ + δg + ( d − δ − C B ( C ) goes through the origin A with multiplicity δd ; the tangent cone in the origin isthe union of the lines joining A to the δd points of B ∩ C − ; C B ( C ) meets the line L ∞ in the points at infinity of B with multiplicity δ and in thepoints at infinity of C with multiplicity d .Proof. Let f δ : P → P N be the δ -uple embedding of P and f = f δ ◦ π : W B → P N .Since W B is an irreducible surface and π is surjective, we can apply Bertini’s Theorem (see[6, Theorem 3.3.1, pg. 207]) to obtain that f − ( H ) is reduced and irreducible for a generalhyperplane H ⊆ P N . By definition of f δ , a generic plane curve C of degree δ is the inverseimage of a generic hyperplane of P N and hence π − ( C ) = f − ( H ) is reduced and irreduciblefor C generic.As the image of an irreducible subvariety is irreducible we obtain that C B ( C ) = π ∗ ( π ∗ ( C )) = mD where D is a reduced and irreducible plane curve, and m is the degree of π | π − ( C ) . We notethat m ≤ d = degree of π = degree of B .We now show that the cycle C B ( C ) has degree 2 dδ . The degree is the homology class ofthe cycle C B ( C ) = π ∗ ( π ∗ ( C )) in H ( P , Z ) ∼ = Z , where π and π are the restrictions to W B of the projections p and p defined on P × P . Let H be the class of a line in P and let p be the class of a point. The homology module of P × P is free with generators: A = H × P A = P × Ha = p × P b = H × H c = P × pα = p × H β = H × pγ = p × p As homology classes we have: π ∗ ( π ∗ ( C )) = p ∗ (( p ∗ ( C ) · W ) W is a surface, complete intersection of two hypersurfaces of bidegree (1 ,
1) and ( d, d ) andhence its homology class is:[ W ] = ( A + A ) · ( dA + dA ) = d ( A + A ) = d ( a + 2 b + c ) C is a plane curve of degree δ and hence [ C ] = δH . Then p ∗ ( C ) = δH × P = δA . Intersectingwith W we get p ∗ ( C ) · W = dδA · ( a + 2 b + c ) = dδ (2 α + β )We have p ∗ ( β ) = 0 since the image of β is a point, while p ∗ ( α ) = H . We conclude[ π ∗ ( π ∗ ( C ))] = p ∗ (( p ∗ ( C ) · W ) = 2 dδH ∈ H ( P , Z ) . We now let B = L a line not through the point A and let C be reduced and irreducible,not through A and such that B ∩ C ∩ L ∞ = ∅ . Since m ≤ d = 1, we have that C L ( C ) = D is reduced and irreducible and has the same degree as C ( L, C ). As the affine parts of thesetwo curves have the same support, we conclude that they are equal as cycles. In particular, C ( L, C ) is reduced and irreducible for C generic.3Since the definition via resultants is symmetric, taking B as in Assumption 4.1 and L ageneric line, we have that C ( B, L ) is reduced and irreducible, and by Theorem 3.4 C ( B, C ) isreduced when C is the union of δ distinct generic lines. The property of being reduced is anopen property, since one can detect multiple roots with the vanishing of system of resultants(the discriminants), and hence we have that C ( B, C ) is reduced for C generic.Consider again C B ( C ) and C ( B, C ) for C generic: their affine parts have the same support, C ( B, C ) is reduced, C B ( C ) = mD has only one irreducible component, and they have the samedegree as cycles. We conclude that they are equal as cycles, and hence they are both reducedand irreducible. Moreover, m = 1 and so π | π − ( C ) is a birational map from π − ( C ) to C B ( C ).This proves , , and By what we have just proved, it is enough to compute the genus of π − ( C ) = ˜ C .The map π : ˜ C → C is a covering of degree d , ramified over the points where C meets theramification of π , i.e., the d ( d −
1) lines through A tangent to B . By our assumption on B the ramification index is 1 for all these points, and so the Riemann-Hurwitz formula gives:2˜ g − d (2 γ −
2) + δd ( d − B is smooth of degree d , its genus g equals ( d − d − C and B . and now follow from what we have proved, and the fact that they are true when C is a generic line (see Example 3.3). In fact, if C is the line of equation ax + by + cz = 0, C B ( C ) has equation F ( x ( ax + by + cz ) , y ( ax + by + cz ) , ( ax + by ) z ) = 0Setting z = 1, the homogeneous part of minimum degree is F ( cx, cy, ax + by ) and setting thisto zero gives the tangent cone in the origin. Hence we see that the tangent cone is given bythe lines through the origin and the points of intersection of B and the line ax + by − cz = 0,which is C − . Corollary 5.4.
Let B be a curve as in Assumption 4.1. Then for every curve C we have C ( B, C ) = C B ( C ) . Moreover 5. and 6. of Theorem 5.3 still hold, with the multiplicitiesgreater than or equal to the ones given (instead of just equal).Proof. Denote with G t ( x, y, z ) the generic form of degree δ in the indeterminates x, y, z ,denote with t its coefficients which we take as indeterminates, and let C t be the correspondingcurve. Using the definition via resultants, we can determine C ( B, C t ): as a function of thevariables t it is given by a polynomial R t .Also C B ( C t ) is given by a polynomial function of the variables t , and the two polynomialsmust coincide up to a constant factor since the curves obtained by generic specialization of t coincide thanks to Theorem 5.3. Hence the two curves C ( B, C ) and C B ( C ) coincide for allchoices of C .Since also and are given by properties of the polynomials defining C ( B, C t ) and C B ( C t ), the same reasoning shows that they hold in general, as inequalities, by semicontinuityin t .4We note that if the surface W B is reducible our proof does not work since we cannot useBertini’s theorem. Our Assumption 4.1 is used not only to ensure the irreducibility of W B ,but also to obtain the genus formula given in the previuos theorem.We want to emphasize that specialization of parameters does not commute with elimi-nation of variables, as we have seen in Example 5.2, while it commutes with the resultants.Our geometric definition coincides with the one given by resultants, and so it commutes withspecialization.Recall the definition and the properties of the divisors Γ and Λ i in W B that we willconsider as special . Γ is the divisor π − ( A ) and Λ i = π − ( P i ) where P i ∈ B ∩ L ∞ . We have π ( π − ( A )) = π (Γ) = B and π (Λ i ) = L i . Definition 5.5.
Let C be a curve. The proper conchoid of C with respect to B is thecurve ˜ C B ( C ) that does not have B and the L i ’s as components and such that C B ( C ) = aB + P i b i L i + ˜ C B ( C ) . By what has been proved, the integers a, b i are greater than or equal to the multiplicitiesof C in the points A, P i ; they are strictly greater if the tangent cone to C in one of thesepoints contains one of the lines L i . Remark 5.6.
This definition has one drawback: it always eliminates the curve B from theconchoid of another curve, even when B should be considered as a non-exceptional component.For example, if B is the circle with center A and radius and C is the circle with the samecenter A and radius , B should be considered a non-exceptional component of the conchoidof C with respect to B , since C does not go through A . W B and double planes We want now to apply our results in the classical case, i.e., when B is a circle with center A .In this case we show that W B is the blow-up in three points of a ramified double cover of P . The geometry of these surfaces, classically known as double planes , is well-known andthis will allow us to determine sufficient conditions on the curve C so that its conchoid isirreducible.The same approach could be followed for curves B of any degree d . W B is again a blow-upof a ramified cover of P of degree d , but in this case the geometry of multiple covers is muchless known, and little can be said in general.For a clear exposition in modern language of the classical theory of double planes see forinstance the paper by Sernesi [9]. In particular, in that paper one can find necessary andsometimes sufficient conditions on a curve in P so that its pullback to the double cover isreducible. Stated loosely, the condition is that the curve must be everywhere tangent to thebranch locus. We do not use directly this, since it requires that the branch locus is smoothand the curve generic and in our case the branch locus is a pair of lines. However, thestatement turns out to be true for the particular double plane we are interested in and validfor all irreducible curves as we will prove.Let B be a circle with center A or, more generally, a conic with center in A . There aretwo points P and P in B ∩ L ∞ and hence two lines L and L . These lines are also thetangents to B passing through A , previously denoted D i , since the center of the conic is thepole of the line at infinity.5Let D be the cycle L + L + 2 L ∞ in P , i.e., the curve (reducible and not reduced) withequation ℓ ℓ z = 0, E the double plane branched over D and p : E → P the correspondingfinite morphism of degree 2.A point on the surface E is singular if and only if its image under p is a singular pointof D , and in this case it is a double point on E . Since D has a multiple component, E is notnormal and has a curve of double points that projects onto L ∞ . Moreover, ˜ A = p − ( A ) is anordinary double point of E .Let n : F → E be the normalization morphism (see, e.g., [4, I, Exercise 3.17]): thecomposition q = p ◦ n : F → P is a double plane branched over the divisor L + L , andhence F is a quadric cone. For a proof of all of the preceding assertions, see [9], especiallyTeorema page 19 and Esempio page 8 and 9.It follows that E is obtained from a quadric cone by identifying two rational curves (thatare not lines on the cone, since they project onto the line z = 0). In particular, we obtain anisomorphism between the open sets π − ( A \ { } ) of W B and q − ( A \ { } ) of F .We summarize the construction in the following diagram: F n / / q ' ' OOOOOOOOOOOOOOO E p @@@@@@@@ W Bf o o π } } {{{{{{{{ π ! ! CCCCCCCC P P (4)where f is the blow-up of E in ˜ A and the two points over P and P , the points at infinity of L and L , as can be seen from the description of the geometry of W B given in Proposition 4.5.So the proper conchoid ˜ C B ( C ) of Definition 5.5 is birational to the corresponding propertransfom in F . We already proved that a generic irreducible curve has irreducible conchoid(and hence irreducible proper conchoid). Using this description via double planes we can nowcharacterize completely the curves whose proper conchoid is irreducible. Theorem 6.1.
Let C be a reduced and irreducible curve in P of degree δ with equation G ( x, y, z ) = 0 . Then ˜ C B ( C ) is reducible if and only if there exist homogeneous polynomials H and H such that:1. G = H − ℓ ℓ H if δ is evenor2. G = ℓ H − ℓ H if δ is oddwhere ℓ and ℓ are the equations of the lines L and L respectively.Proof. By Diagram (4) ˜ C B ( C ) is irreducible if and only if q − ( C ) is. So it is enough to provethe claim for q − ( C ), and even restrict ourselves to the affine case. Identifying F with thequadric cone of equation ℓ ℓ − t in A , the map q is the projection onto the plane A givenby t = 0. The quadric cone is normal and the degree map is an isomorphism from its divisorclass group to Z / Z . In particular a divisor is a complete intersection if and only if it haseven degree (see [4], Ch. I, Exercise 3.17 and Ch. II, Example 6.5.2).If C has equation G ( x, y ) = 0, then q − ( C ) is given by the intersection of the cone withthe hypersurface in A of equation G ( x, y ) = 0. If this divisor on the cone is reducible thenit has exactly two components, since q is a finite map of degree two. If Y is one of the6components, the other component Y is obtained using the involution t
7→ − t on the cone.Hence both components have degree δ .If δ is even, then Y is a complete intersection of the cone with a hypersurface H ( x, y, t ) ofdegree δ/
2, and since we are taking intersection with the cone t = ℓ ℓ , we can assume that H = H + tH , where H , H ∈ k [ x, y ]. In this case, H − tH gives the other component Y and H − t H cuts on the cone the sum Y + Y = q − ( C ). Using again the equation t = ℓ ℓ of the cone, we see that H − ℓ ℓ H cuts the same divisor Y + Y . Finally, H − ℓ ℓ H = 0as a curve in A contains the curve C and has the same degree, and hence coincide with C .So the equation G ( x, y ) of C is as claimed.The proof is similar in the case δ odd. The divisor Y is not principal, and we considerthe principal divisor Y + L , where L is the line ℓ = t = 0. As before, Y + L is cut on thecone by a hypersurface that in this case will be of the form ℓ H + tH with H , H ∈ k [ x, y ]because it must vanish on L . Then Y + L is cut by ℓ H − tH , Y + Y + 2 L is cut by ℓ H − ℓ ℓ H and finally Y + Y by ℓ H − ℓ H as ℓ cuts 2 L on the cone. Example 6.2.
Let B be the circle with center A and any radius. By the previous theorem, aconic C has reducible conchoid if and only if C has equation ℓ − ℓ ℓ = 0 , where ℓ has degree .Notice that ℓ is the equation of the polar line of A with respect to C . Since B is a circle, thepoints P and P are the cyclic points. Then ℓ and ℓ are two tangents to the conic C fromthe cyclic points and so their intersection A is a focus of C (by definition of focus, see, e.g., [1,pag. 171]). Note that a complex conic which is not a parabola has foci, and if the conic is realonly two of them are real. These are the usual foci of ellipses and hyperbolas. Since a parabolais tangent to the line at infinity, it has always only focus. We conclude that a conic hasreducible conchoid if and only if A is one of the foci of C . For instance if B is the circle withcenter A and radius and C is the parabola ( y + z ) − ( x + y ) , then ˜ C B ( C ) is the union of thetwo quartics x +( y − yz ) x − y z + y z = 0 and x +( y − yz − z ) x − y z − y z = 0 .For a different proof and many explicit computations, see [8, Theorem 6]. We assume now that C is reducible. We can again ask if its proper conchoid is reducibleor not. To answer this question we introduce the notion of iterated conchoid . We begin withan example. Example 6.3.
Let C be a generic line. Let C = C B ( C ) be its conchoid, which is againirreducible, and let us consider C B ( C ) . This is a divisor of degree , whose components arethe circle B with multiplicity , the two lines L and L each with multiplicity , the line C with multiplicity and an irreducible curve C of degree (to check this computation, take C the line of equation x − hz = 0 and use resultants). The curve C is in fact the conchoidof C with respect to the circle B with center A and radius twice that of B . This behaviour is not special to the lines and we prove:
Proposition 6.4.
Let C be a generic curve of degree δ , and let C = C B ( C ) . Then theconchoid C = C B ( C ) is a divisor of degree δ , whose components are:1. the circle B , with multiplicity δ ;2. the two lines L and L each with multiplicity δ ;3. the curve C with multiplicity ;
4. a curve C of degree δ , which is the conchoid of C with respect to the circle B withcenter A and radius twice that of B .Proof. As we did earlier, we can consider C as specialization of the curve C t of degree δ with generic coefficients. The linear system C t is generated by curves that are product ofgeneric linear forms. Hence every curve C B ( C B ( C t )), and so especially C := C B ( C B ( C )),must contain B , L and L with at least multiplicity as stated.Now, let us consider the affine part C a ) of C . It contains all the points Q of the form Q = P ′ + ( P + S ), where S ∈ C ( a ) and P ′ , P ∈ B ( a ) collinear with A (see Lemma 2.1). Theintersection B ∩ AS consists of two points P + and P − and hence there are two possibilitiesfor P and two for P ′ . If either P = P ′ = P + or P = P ′ = P − , the corresponding point Q belongs to B and hence Q belongs to C B ( C ). Since the total degree is 16 δ the componentsappear with the stated multiplicity, and not higher, and their sum is the whole divisor C B ( C ). Definition 6.5.
The curve C defined in the previous Proposition is called proper secondconchoid of C . In this case we discard from C B ( C ) not only the exceptional components, but also thecurve C . Remark 6.6.
We can define inductively the proper n -th conchoid C n of C and see in thesame way that it turns out to be the conchoid of C with respect to the circle B n with center A and radius n times that of B . The infinitely many curves C n belong to a -dimensional flatfamily. In fact C n = C B n ( C ) = C ( B n , C ) can be obtained using the resultant R ( F t , G ) , where F t = x + y − t z , and specializing the parameter t to n . Proposition 6.7.
The proper conchoid of a reducible curve is reducible.Proof.
Let C be a reducible curve and ∆ := π − ( C ) \ { exceptional components } . C isreducible and so ∆ has at least two components, since π (∆) = C has a number of componentsless than or equal to that of ∆. Assume that the proper conchoid ˜ C ( C ) is irreducible. As themap π is generically 2 : 1, π − ( ˜ C ( C )) has at most two components and since it contains ∆it must be π − ( ˜ C ( C )) = ∆ and hence π ( π − ( ˜ C ( C ))) = π (∆) = C . Since B = B − , thecurve π ( π − ( ˜ C ( C ))) contains the proper second conchoid of C and hence it has at least 3irreducible non exceptional components, those of C and C B ( C ). This contradiction proves ourclaim, since a component of C cannot be equal to C B ( C ): in fact any curve different from L ∞ or an exceptional curve has only finitely many points in common with its conchoid.We conclude giving a computational procedure to establish when an irreducible curve D is either the conchoid or the proper conchoid of another curve C with respect to some point A (not necessarily the origin) and radius r , i.e., with respect to the circle B with equation ( x − a ) + ( y − b ) − r z = 0.In order to decide if D is a complete conchoid, we start by checking some obvious necessaryconditions: first of all the degree must be a multiple of 4. If we set deg( D ) = 4 δ , then D must meet the line at infinity z = 0 in the two cyclic points ([1 : i : 0] and [1 : − i : 0])with multiplicity at least δ and all the other points at infinity of D must be at least doublepoints. Hence, if H ( x, y, z ) = 0 is an equation defining D , then H ( x, y,
0) must split as8( x + y ) δ H δ ( x, y ) . Moreover, there must be a point on D (namely the point A ) in the affineopen set A with multiplicity at least 2 δ .When all these conditions are fulfilled, the distance r must be twice the distance betweena pair of points on D and collinear with A .Hence the only possibilities for A and r are finite, and we can check all cases to see if theconchoid of D with respect to the circle with center A and radius r contains a non-exceptionalcomponent with multiplicity 2: for what we proved above this component, if it exists, is acurve whose conchoid is D .In order to check if D is the proper conchoid of a curve C we can use Theorem 6.1 andProposition 6.4. Excluding the trivial case deg( D ) = 1, a first necessary condition is theexistence of the pair of lines ℓ , ℓ , each containing a cyclic point, which are everywheretangent to D . If they exist and they meet in the affine subset A , their common point is A and, as above, the distance r must be twice the distance between a pair of points on D and collinear with A . Hence there are finitely many possibilities for r and we can check allcases to see if the conchoid of D with respect to the circle with center A and radius r splitsas described in Proposition 6.4: if D = C B ( C ), the curve C is a non-exceptional componentwith multiplicity 2 of C B ( D ). References [1] Coolidge, J. L.: A Treatise on Algebraic Plane Curves, Dover Publ. Inc., New York,1959[2] Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry, Graduate Texts in Mathe-matics, Volume 185, Springer, 2005[3] Fulton, W.: Intersection Theory, Springer-Verlag, New York (1984)[4] Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics, Volume 52,Springer-Verlag, New York (1977)[5] Lawrence, J. D.: A catalog of special plane curve, Dover Publ. Inc., New York, 1972[6] Lazarsfeld, R.: Positivity in Algebraic Geometry I, Springer-Verlag, (2004)[7] Sendra, J. R., Sendra, J.:
An algebraic analysis of conchoids to algebraic curves , Appl.Algebra Eng., Commun. Comput., , 5, 2008[8] Sendra, J. R., Sendra, J.: Rational Parametrization of Conchoids to Algebraic Curves ,Appl. Algebra Eng., Commun. Comput., this issue[9] Sernesi, E.: