Capítulo 4 Análisis de resultados
4.2 Diagnóstico operativo
4.2.1 Hallazgos por subcategorías de acuerdo con el proceso
Since this material is somewhat outside of the topic of the book, we state some of the facts without proof, referring to classical sources for the invariant theory (e.g. [114], t. 2, [493]).
We know that the ring of invariants of ternary cubic forms is generated by the Aronhold invariantsSandT. Let us look for covariants and contravariants. As we know from Subsection1.5.1, any invariant of binary form of degree 3 is a power of the discriminant invariant of order 4, and the algebra of covariants is generated over the ring of invariants by the identical covariantU: f 7→f, the Hessian covariantHof order 2 with symbolic expression(ab)axbx, and the covariantJ=Jac(f,H)of degree 3 and order 6 with symbolic expression (ab)2(ca)b
xc2x. Clebsch’s transfer of the discriminant is a contravariantFof degree4and class6. Its symbolic expression is(abu)2(cdu)2(acu)(bdu). Its value on a general ternary cubic form is the form defining the dual cubic curve. Clebsch’s transfer ofHis a mixed concomitantΘof degree2, order 2 and class
2. Its symbolic expression is(abu)2axbx. Explicitly, up to a constant factor, Θ = det f00 f01 f02 u0 f10 f11 f12 u1 f20 f21 f22 u2 u0 u1 u2 0 , (3.33) wherefij= ∂ 2f ∂ti∂tj.
The equation Θ(f, x, u) = 0, for fixed x, is the equation of the dual of the polar conic Px(V(f)). The equationΘ(f, x, u) = 0, for fixedu, is the equation of the locus of pointsxsuch that the first polarPx(V(f))is tangent to the lineV(u). It is called thepoloconicof the lineV(u). Other description of the poloconic can be found in Exercise 3.3.
The Clebsch transfer ofJis a mixed concomitantQof degree3, order 3 and class 3. Its symbolic expression is(abu)2(cau)c2xbx. The equationQ(f, x, u) = 0, for fixedu, is the equation of the cubic curve such that second polarsPx2(V(f))
of its points intersectV(u)at a point conjugate toxwith respect to the polo- conic ofV(u). A similar contravariant is defined by the condition that it van- ishes on the set of pairs(x, u)such that the lineV(u)belongs to the Salmon envelope conic of the polars of xwith respect to the curve and its Hessian curve.
An obvious covariant of degree 3 and order 3 is the Hessian determinant H= detHe(f). Its symbolic expression is(abc)2a
xbxcx. Another covariantG is defined by the condition that it vanishes on the locus of pointsxsuch that the Salmon conic of the polar ofxwith respect to the curve and its Hessian curve passes throughx. It is of degree8and order6. Its equation is the following bordered determinant f00 f01 f02 h0 f10 f11 f12 h1 f20 f21 f22 h2 h0 h1 h2 0 , wherefij = ∂ 2f ∂ti∂tj, hi = ∂H(f)
∂ti (see [77],[114], t. 2, p. 313). The algebra of
covariants is generated byU,H,Gand theBrioschi covariant[56]J(f,H,G) whose value on the cubic (3.7) is equal to
(1 + 8α3)(t31−t32)(t23−t30)(t30−t31).
Comparing this formula with (3.16), we find that it vanishes on the union of 9 harmonic polars of the curve. The square of the Hermite covariant is a polyno- mial inU,H,G.
3. Its symbolic expression is(abc)(abu)(acu)(bcu).Its value on the curve in the Hesse form is given in (3.27). There is also a contravariantQ of degree 5 and class 3. In analogy with the form of the word Hessian, A. Cayley gave them the names thePippianand theQuippian[78]. IfC =V(f)is given in the Hesse form (3.7), then
Q(f) =V((1−10α3)(u20+u31+u32)−6α2(5 + 4α3)u0u1u2). The full formula can be found in Cayley’s paper [77]). He also gives the for- mula
H(6aP+bQ) = (−2T a3+ 48S2a2b+ 18T Sab2+ (T3+ 16S2)b3P +(8Sa3+ 3Ta2b−24S2ab2−TS2b3)Q,
where the product of a covariant and a contravariant is considered as the com- position of the corresponding equivariant maps.
According to A. Clebsch,Q(f)vanishes on the locus of lines whose polo- conics with respect to the Cayleyan ofC are apolar to their poloconics with respect toC. Also, according to W. Milne and D. Taylor,Q(f)is the locus of lines which intersectC at three points such that the polar line of the Hessian curveH(f)with respect two of the points is tangent toH(f)at the third point (see [384]). This is similar to the property of the Pippian which is the set of lines which intersectCat three points such that the polar line with respect to two of the points is tangent toCat the third point. The algebra of contravari- ants is generated byF,P,Qand theHermite contravariant[288]. Its value on the cubic in the Hesse form is equal to
(1 + 8α3)(u31−u32)(u23−u30)(u30−u31).
It vanishes on the union of nine lines corresponding to the inflection points of the curve. The square of the Hermite contravariant is a polynomial inF,P,Q.
Exercises
3.1Find the Hessian form of a nonsingular cubic given by the Weierstrass equation.
3.2LetH =He(C)be the Hessian cubic of a nonsingular plane cubic curveCthat is not an equianharmonic cubic. Letτ :H →H be the Steinerian automorphism ofH
that assigns toa∈Hthe unique singular point ofPa(C).
(i) LetH˜ ={(a, `) ∈ H×(P2)∨ :`⊂ Pa(C)}. Show that the projectionp1 :
˜
H→His an unramified double cover.
3.3LetC=V(f)⊂P2be a nonsingular cubic.
(i) Show that the setK(`)of second polars ofCwith respect to points on a fixed line`is a dual conic of the poloconic ofCwith respect to`.
(ii) Show that K(`) is equal to the set of poles of`with respect to polar conics
Px(C), wherex∈`.
(iii) What happens to the conicK(`)when the line`is tangent toC?
(iv) Show that the set of lines`such thatK(`)is tangent to`is the dual curve ofC. (v) Let`=V(a0t0+a1t1+a2t2). Show thatK(`)can be given by the equation
g(a, t) = det 0 a0 a1 a2 a0 ∂ 2f ∂t2 0 ∂2f ∂t0∂t1 ∂2f ∂t0∂t2 a1 ∂ 2f ∂t1∂t0 ∂2f ∂t2 1 ∂2f ∂t1∂t2 a2 ∂ 2f ∂t2∂t0 ∂2f ∂t2∂t1 ∂2f ∂t2 2 = 0.
(vi) Show that the dual curveC∨ ofCcan be given by the equation (theSchl¨afli equation) det 0 ξ0 ξ1 ξ2 ξ0 ∂ 2g(ξ,t) ∂t2 0 (ξ) ∂∂t2g(ξ,t) 0∂t1 (ξ) ∂2g(ξ,t) ∂t0∂t2 (ξ) ξ1 ∂ 2g(ξ,t) ∂t1∂t0 (ξ) ∂2g(ξ,t) ∂t2 1 (ξ) ∂∂t2g(ξ,t) 1∂t2 (ξ) ξ2 ∂ 2g(ξ,t) ∂t2∂t0 (ξ) ∂2g(ξ,t) ∂t2∂t1 (ξ) ∂2g(ξ,t) ∂t2 2 (ξ) .
3.4LetC ⊂ Pd−1 be an elliptic curve embedded by the linear system OC(dp0)
,
wherep0is a point inC. Assumed=pis prime.
(i) Show that the image of anyp-torsion point is an osculating point ofC, i.e., a point such that there exists a hyperplane (anosculating hyperplane) which intersects the curve only at this point.
(ii) Show that there is a bijective correspondence between the sets of cosets of(Z/pZ)2 with respect to subgroups of orderpand hyperplanes inPp−1which cut out inC the set ofposculating points.
(iii) Show that the set ofp-torsion points and the set of osculating hyperplanes define a(p2p+1, p(p+ 1)p)-configuration ofp2points andp(p+ 1)hyperplanes (i.e. each point is contained inp+ 1hyperplanes and each hyperplane containsppoints). (iv) Find a projective representation of the group(Z/pZ)2 inPp−1such that each
osculating hyperplane is invariant with respect to some cyclic subgroup of orderp
of(Z/pZ)2 .
3.5A point on a nonsingular cubic is called asextactic pointif there exists an irreducible conic intersecting the cubic at this point with multiplicity 6. Show that there are 27 sextactic points.
3.6The pencil of lines through a point on a nonsingular cubic curveCcontains four tangent lines. Show that the twelve contact points of three pencils with collinear base points onClie on 16 lines forming a configuration(124,163)(theHesse-Salmon con-
figuration).
3.7Show that the cross ratio of the four tangent lines of a nonsingular plane cubic curve which pass through a point on the curve does not depend on the point.
the Hessian He(C)is equal to the tangent lineTb(He(C)), wherebis the singular point of the polar conicPa(C).
3.9Leta, bbe two points on the Hessian curve He(C)forming an orbit with the respect to the Steinerian involution. Show that the lineabis tangent to the dual of the Caylean curve Cay(C)at some pointd. Letcbe the third intersection point of He(C)with the lineab. Show that the pairs(a, b)and(c, d)are harmonically conjugate.
3.10Show that from each pointaon the He(C)one can pass three tangent lines to the dual curve of Cay(C). Letbbe the singular point ofPa(C). Show that the set of the three tangent lines consists of the lineaband the components of the reducible polar conicPb(C).
3.11LetC=V(P
0≤i≤j≤k≤2aijktitjtk). Show that the Cayleyan curve Cay(C)can be given by the equation
det a000 a001 a002 ξ0 0 0 a110 a111 a112 0 ξ1 0 a220 a221 a222 0 0 ξ2 2a120 2a121 2a122 0 ξ2 ξ1 2a200 2a201 2a202 ξ2 0 ξ0 2a010 2a011 2a012 ξ1 ξ0 0 = 0 [114], p. 245.
3.12Show that any general net of conics is equal to the net of polars of some cu- bic curve. Show that the curve parameterizing the irreducible components of singular members of the net coincides with the Cayleyan curve of the cubic (it is called the
Hermite curveof the net).
3.13Show that the group of projective transformations leaving a nonsingular plane cubic invariant is a finite group of order 18, 36 or 54. Determine these groups.
3.14Find all ternary cubicsCsuch that VSP(C,4)o=∅ .
3.15Show that a plane cubic curve belongs to the closure of the Fermat locus if and only if it admits a first polar equal to a double line or the whole space.
3.16Show that any plane cubic curve can be projectively generated by three pencils of lines.
3.17Given a nonsingular conicKand a nonsingular cubicC, show that the set of points
xsuch thatPx(C)is inscribed in a self-polar triangle ofKis a conic.
3.18A complete quadrilateral is inscribed in a nonsingular plane cubic. Show that the tangent lines at the two opposite vertices intersect at a point on the curve. Also, show that the three points obtained in this way from the three pairs of opposite vertices are collinear.
3.19Letobe a point in the plane outside of a nonsingular plane cubicC. Consider the six tangents toCfrom the pointo. Show that there exists a conic passing through the six points onCwhich lie on the tangents but not equal to the tangency points. It is called thesatellite conicofC[142]. Show that this conic is tangent to the polar conic
Po(C)at the points where it intersects the polar linePo2(C).
3.20Show that two general plane cubic curvesC1andC2admit a common polar pen- tagon if and only if the planes of apolar conics|AP2(C1)|and|AP2(C2)|intersect.
3.21LetCbe a nonsingular cubic andKbe its apolar cubic in the dual plane. Prove that, for any point onC, there exists a conic passing through this point such that the remaining five intersection points withCform a polar pentagon ofK[500].
3.22Letp, qbe two distinct points on a nonsingular plane cubic curve. Starting from an arbitrary pointp1find the third intersection pointq1of the linepp1withC, then define p2 as the third intersection point of the lineqq1 withC, and continue in this way to define a sequence of pointsp1, q1, p2, q2, . . . , qk, pk+1onC. Show thatpk+1=p1if and only ifp−qis ak-torsion point in the group law onCdefined by a choice of some inflection point as the zero point. The obtained polygon(p1, q1, . . . , qk, p1)is called theSteiner polygoninscribed inC.
3.23Show that the polar conicPx(C)of a pointxon a nonsingular plane cubic curve
Ccuts out onCthe divisor2x+a+b+c+dsuch that the intersection pointsab∩cd,
ac∩bdandad∩bclie onC.
3.24Show that any intersection point of a nonsingular cubicCand its Hessian curve is a sextactic point on the latter.
3.25Fix three pairs(pi, qi)of points in the plane in general position. Show that the closure of the locus of pointsxsuch that the three pairs of linesxpi, xqiare members of ag12in the pencil of lines throughxis a plane cubic.
3.26Fix three pointsp1, p2, p3in the plane and three lines`1, `2, `3in general position. Show that the set of pointsxsuch that the intersection points ofxpiwith`iare collinear is a plane cubic curve [262].
Historical Notes
The theory of plane cubic curves originates from the works of I. Newton [414] and his student C. MacLaurin [376]. Newton was the first to classify real cubic curves, and he also introduced the Weierstrass equation. Much later, K. Weier- strass showed that the equation can be parameterized by elliptic functions, the Weierstrass functions℘(z)and℘(z)0. The parameterization of a cubic curve by elliptic functions was widely used for defining a group law on the cubic. We refer to [496] for the history of the group law on a cubic curve. Many geo- metric results on cubic curves follow simply from the group law and were first discovered without using it. For example, the fact that the line joining two in- flection points contains the third inflection point was discovered by MacLaurin much earlier before the group law was discovered. The book of Clebsch and Lindemann [114] contains many applications of the group law to the geometry of cubic curves.
The Hesse pencil was introduced and studied by O. Hesse [289],[290]. The pencil was also known as thesyzygetic pencil(see [114]). It was widely used as a canonical form for a nonsingular cubic curve. More facts about the Hesse pencil and its connection to other constructions in modern algebraic geometry can be found in [14].
The Cayleyan curve first appeared in Cayley’s paper [72]. The Schl¨afli equa- tion of the dual curve from the Exercises was given by L. Schl¨afli in [497]. Its modern proof can be found in [240].
The polar polygons of plane cubics were first studied by F. London [367]. London proves that the set of polar 4-gons of a general cubic curve are base points of apolar pencils of conics in the dual plane. A modern treatment of some of these results is given in [178] (see also [461] for related results). A beautiful paper by G. Halphen [276] discusses the geometry of torsion points on plane cubic curves.
Poloconics of a cubic curve are studied extensively in Dur`ege’s book [195]. The term belongs to L. Cremona [142] (conic polar in Salmon’s terminology). O. Schlessinger proved in [500] that any polar pentagon of a nonsingular cubic curve can be inscribed in an apolar cubic curve.
The projective generation of a cubic curve by a pencil and a pencil of conics was first given by M. Chasles. Other geometric ways to generate a plane cubic are discussed in Dur`ege’s book [195]. Steiner polygons inscribed in a plane cubic were introduced by J. Steiner in [542]. His claim that their existence is an example of a porism was given without proof. The proof was later supplied by A. Clebsch [107].
The invariantsS andT of a cubic ternary form were first introduced by Aronhold [11]. G. Salmon gave the explicit formulas for them in [493]. The basic covariants and contravariants of plane cubics were given by A. Cayley [77]. He also introduced 34 basic concomitants [90]. They were later studied in detail by A. Clebsch and P. Gordan [111]. The fact that they generate the algebra of concomitants was first proved by P. Gordan and M. Noether [254], and by S. Gundelfinger [273]. A simple proof for the completeness of the set of basic covariants was given by L. Dickson [168]. One can find an exposition on the theory of invariants of ternary cubics in classical books on the invariant theory [259], [213].
Cremona’s paper [142] is a fundamental source of the rich geometry of plane curves, and in particular, cubic curves. Other good sources for the classical geometry of cubic curves are books by Clebsch and Lindemann [114], t. 2, by H. Dur`ege [195], by G. Salmon [493], by H. White [602] and by H. Schroter [504].