Journal of Topology 1 (2008) 362–390
c 2008 London Mathematical Society doi:10.1112/jtopol/jtm015
Projection genericity of space curves C. T. C. Wall Abstract If Γ is a smooth space curve, we consider the family of projections of Γ from a variable point not on Γ to a fixed plane. For a residual set of curves Γ, this family versally unfolds those singularities that occur in it. To obtain a family of curves which is open as well as dense in the space of smooth maps, we must compactify the parameter space, so we study curves in real projective space, and include projections from points of the curve itself. If Γ is smoothly embedded, the projection CP of Γ from P ∈ Γ is a well-defined smooth curve, and for generic Γ the family CP has generic singularities. However, when the point of projection moves off Γ, the projection varies discontinuously. We define a family of plane curves, parametrised by the blow-up X of P 3 along Γ, such that for a point in the exceptional locus lying over P ∈ Γ, we have the union of the projection CP of Γ from P and a straight line L through the image of the tangent at P . A key result asserts that this is a flat family. We give an explicit list of restrictions on the family CP ∪ L (the key condition is that the total contact order of CP with L never exceeds 2), and show that these hold for a dense open set of curves Γ, and that if they do hold, there is a neighbourhood U of Γ, such that the family of projections from points of U \ Γ is generic. Combining this list of conditions with those obtained previously gives a natural definition of a dense set of space curves Γ, for which the complete family of projections has generic singularities, and we show that this set is also open.
Introduction ∞
If Γ is a smooth (that is, C ) space curve we can define a three-parameter family of maps by considering the family of projections of Γ from a variable point of space to a fixed plane. It was shown by Soares David in [7] (see also [8]) that; for a residual set of curves Γ, this family is generic — that is, transverse to the natural stratification of the family of singular smooth curves in the plane by equisingularity (and hence, stable under perturbations). Thus only a short list (see equation (1.1) below) of singularities may occur. We recall these results in § 1. While attractive, this result leaves some business unfinished. A full statement ought, in particular, to produce a family of curves which is open as well as dense in the space of smooth maps φ : S 1 → R3 (with the C ∞ -topology). To obtain an openness result, one should compactify the parameter space. Thus, first, from now on we regard R3 as embedded in real projective space, which we denote P 3 ; and second, we have to consider projections from points of the curve itself. The projective setting offers further advantages, as the geometry is then simplified. However, while provided that Γ is smoothly embedded, its projection from a point of itself is a well-defined smooth curve, it is not now the case that varying the point gives a smooth projection, or even a continuous family of projections, and the study of this situation plays an essential rˆole in the proof of openness.
Received 6 September 2007; published online 5 February 2008. 2000 Mathematics Subject Classification 57R45 (primary), 51N15, 53A04 (secondary).
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We will see that the correct procedure gives a family of plane curves parametrised by the blow up of P 3 along Γ, in which a point lying over x ∈ Γ corresponds to the projection of Γ from x, together with a straight line. Theorem 3.1, one of our key results, states that this gives a flat family of curves. We describe, in terms of the geometry, the restrictions to impose on this family, and show (in § 4) that these hold for a dense open set of curves Γ, and then that if these conditions hold, there is a neighbourhood U of Γ, such that the family of projections from points of U is generic. This leads to our main result, that we have a dense open set of curves Γ, defined by an explicit list of conditions, for which the family of projections is stable in a certain sense. The details of the proof require many calculations, which we defer to the final section. 1. Recall of earlier results We recall the main results of [7], with details slightly altered to fit the context of this paper. Write f : S 1 → P 3 for a C ∞ -map, with image a smoothly embedded curve Γ. We will adhere to this notation throughout and, for example, have a family {fu } with image Γu . Choose a neighbourhood U of Γ, and fix a plane P 2 ⊂ P 3 and a compact subset K of P 3 disjoint from U ∪ P 2 . Then for any map f : S 1 → U , there is an induced map Hf : S 1 × K → P 2 , where Hf (t, x) is the point where the straight line from f (t) to x meets P 2 : the projection of f (t) from x. It is easily seen that the singularities of this map do not depend on the choice of the plane P 2 , which we thus suppress from the notation. We regard Hf as a K-parameter family of maps S 1 → P 2 , so that the jet extensions are taken as maps j Hf : K × S 1 → J (S 1 , P 2 ), and similarly for multi-jets. Theorem 1.1 [7, Theorem 1.1]. For any sub-manifold W of r J (S 1 , P 2 ), the set {f ∈ C (S 1 , U ) | r j Hf W } is residual in C ∞ (S 1 , U ). ∞
By applying this result to particular sub-manifolds W , Soares David obtains the complete list of singularities which appear in such families of projections. We shall use Arnold’s notations An , Dn , X9 , which usually refer to the equations defining these singularities, for the singularities of the image curve Hf (S 1 ). Since a generic map from S 1 to P 2 has A1 singularities, we call these non-degenerate and will usually ignore them. We list types of (degenerate) singularities as follows: codimension 1 A2 A3 D4 codimension 2 A4 A5 D5 D6 X9 . (1.1) codimension 3 A6 A7 D8 . Rough pictures of these singularities are given in [7]; equations in normal form and versal unfoldings are given in Subsection 6.1. Theorem 1.2. (see[7, Theorem 5.1]; see also [6]). There is a residual set U of maps f : S 1 → U , such that for f ∈ U, the family {Cx = Hf (S 1 × x) | x ∈ K} of image curves has the following properties. There is a regular smooth stratification of K, such that if x belongs to a (3 − c)-dimensional stratum, Cx has degenerate singularities, all in equation (1.1), only at a finite set with codimensions that add up to c. We next describe these conditions geometrically. We also wish to dispense with K and U . Since P 3 is the set of lines through the origin in R4 , we have a tautological bundle ξ over P 3 which is a sub-bundle of the trivial bundle of rank 4. Write η for the quotient R3 bundle, and E = P (η) for the associated plane bundle. For any x ∈ P 3 , there is a well-defined projection R4 → ηx and hence, πx : P 3 \ {x} → Ex . We can thus define our family of projections to be the
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family of restrictions {πx |Γ : x ∈ P 3 \ Γ}. However, as our considerations are (multi-)local, we only deal at any time with points x in a neighbourhood of one point, and a subset of Γ in a neighbourhood of a fixed finite subset. Moreover, all conditions that we impose are projectively invariant. We will thus continue to speak of a family of projections to P 2 , and for calculations can take local coordinates with a fixed target plane R2 . We can now impose the conditions of Theorem 1.2, taking K to be the complement of a (small) neighbourhood of Γ. Suppose that Γ is an embedded curve with nowhere vanishing curvature (this is itself an open dense condition). Thus, at each point P ∈ Γ, there are a well-defined tangent line TP Γ and osculating plane OP Γ. We say P is a stall of Γ, if the torsion vanishes at P , or equivalently, if the local intersection number Γ.OP Γ > 3, and a transverse stall, if Γ.OP Γ = 4. An r-secant is a line, not a tangent, meeting Γ in at least r points: if r = 2 we omit the r. We will call it a T-r-secant, if the tangents at two of its points P, Q on Γ lie in a plane (or, equivalently, since we are in projective space, intersect); we call this plane the T-plane. The projection of Γ from a point x ∈ Γ is an immersion at P , unless x lies on TP Γ. In this case, the image has an A2 singularity at the image of P , unless either TP meets Γ again (when if f ∈ U, we have a D5 singularity); or P is a stall (if f ∈ U, all stalls are transverse). If P is a transverse stall, there is a unique point x0 — which we call the P-centre of the stall — on TP Γ, such that the projection from a point x = x0 on TP Γ has an A4 -type singularity, but the projection from x0 has a higher singularity (if f ∈ U, an A6 ). It was pointed out by Soares David [7] that these points were omitted in [8]. If f ∈ U, Γ has no bitangent, tangent which is a 3-secant, or tangent at a stall meeting Γ again. It also follows that there is no 5-secant, T-4-secant, or T-3-secant with all three tangents coplanar. Each T-secant contains a unique point y0 — its T-centre — such that under the projection from a point x not equal to y0 (and not equal to P, Q) on the T-secant, the intersection number of the images of the branches at P and Q is 2 (an A3 singularity), but under projection from y0 , is greater than 2. In general, this is 3 (type A5 ); otherwise, it is 4 (type A7 ), and we speak of a special T-secant (if f ∈ U, we do not have a higher singularity). Projecting along a general 3-secant gives a D4 singularity. A T-3-secant has a T-centre; the projection from another point has a D6 singularity, but from the T-centre, a D8 (for f ∈ U, we do not have any special T-3-secant). Finally, projecting along a 4-secant gives an X9 singularity. While there are ∞2 secants, there are only ∞1 tangents, T-secants or 3-secants, and only finitely many tangents meeting Γ again, stalls, special T-secants, T-3-secants or 4-secants. Consideration of these lines explains why equation (1.1) does not include all singularities of Ae -codimension 3. For as only finitely many tangents meet Γ again, in general, none of them occurs at a stall, meets Γ a third time or has the tangent at the second point contained in the osculating plane at the first; similarly, of the finitely many 4-secants, in general, none are T-4-secants or 5-secants. We may give this argument an alternative shape by observing that each of the remaining singularity types of codimension 3, if it occurred in our situation, would have to occur along a line, and so would occur in codimension 2.
2. Openness of transversality conditions Given smooth manifolds A and B, where A (at least) is compact, we will use the C ∞ -topology on the space C ∞ (A, B): since A is compact, there is no distinction between the fine and the ordinary topology. Our crucial tool in proving openness will be the following. Proposition 2.1 [2, Corollary 3.4.12]. Let N and P be C ∞ -manifolds, with N compact. Then a subset U is open in C ∞ (N, P ), if and only if for all one-parameter families
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{fu : N → P | u ∈ U } (for U , a neighbourhood of 0 in R), defining a smooth map F | N × U → P × U , the set {u ∈ U | fu ∈ U} is open in U . We begin with two easy applications. Lemma 2.2. The set O0 of maps f : S 1 → P 3 , such that (Tim) f is an immersion, (Tinj) f is injective, and (Tcur) Γ := f (S 1 ) has nowhere zero curvature is dense and open in C ∞ (S 1 , P 3 ). Proof. Density follows from standard transversality theorems, since each condition involves avoidance of a subset of (multi-)jet space of codimension greater than the source dimension. Openness for the first two conditions follows, since embeddings are open in the space of maps. As to (Tcur), let {fu } be a smooth family with f0 ∈ O0 , and suppose there is a sequence ui → 0, such that for some xi ∈ Γ, fu i (S 1 ) has zero curvature at xi . Passing to a subsequence, we may suppose (since S 1 is compact) that xi converges to a point x0 . But then it follows that f0 (S 1 ) has zero curvature at f0 (x0 ), contrary to hypothesis. Lemma 2.3. The set of maps f : S 1 → P 2 , such that f (S 1 ) has only non-degenerate singularities, is dense and open in C ∞ (S 1 , P 2 ). Proof. Since the set Σ2 of singular 1-jets has codimension 2, Thom’s transversality theorem implies that the set of immersions is dense; since Σ is closed, it is open. The subset Σ4 of 0 1 2 3 j (S , P ) corresponding to jets with a common target has codimension 4; the subset Σ3 of 1 1 2 2 j (S , P ) of jets with common target and the two images tangent has codimension 3. Thus, for a dense set of maps, we avoid both these, so have no degenerate singularities. Now suppose fu a smooth family, such that f0 (S 1 ) has only non-degenerate singularities; given a sequence ui → 0, we will write fi for fu i and Ci for fi (S 1 ). Since immersions are open, we may suppose each fu an immersion. If, for some sequence ui → 0, Ci has a triple point fi (Pi ) = fi (Qi ) = fi (Ri ), we may suppose, passing to a subsequence, that Pi converges to a limit P0 ; similarly for Qi and Ri . If for example, P0 = Q0 , the curvatures of Ci would be unbounded in the small intervals between Pi and Qi , whereas these must all converge to the curvature of C0 at P0 , a contradiction. But if P0 , Q0 and R0 are all distinct, C0 has a triple point, which is also a contradiction. Similarly, if Ci has a self-tangency at fi (Pi ) = fi (Qi ), we may suppose that Pi → P0 , Qi → Q0 and P0 = Q0 . But then C0 has a self-tangency, a contradiction. From now on, we will only consider maps f ∈ O0 . In this case, the curve Γ determines f up to composition with a diffeomorphism of S 1 , and at each point P ∈ Γ, Γ has a well-defined tangent line TP Γ and osculating plane OP Γ. For Γ ∈ O0 , the projection πP of Γ from the point P of itself is well defined and smooth, since if f (t) is a smooth function of one variable which vanishes when t = 0, the quotient f (t)/t is again smooth. We write CP for the image πP (Γ). Similarly, this depends smoothly on P , and defines a smooth, continuous map pr1 : O0 −→ C ∞ (S 1 , C ∞ (S 1 , P 2 )). Strictly speaking, we should be discussing sections of a bundle, but again this does not affect the local theory.
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We thus need to consider one-parameter families F : U × S 1 → P 2 , where U also is a copy of S 1 . We define F ∈ C1 : if fu is an immersion with only non-degenerate singularities for all, but finitely many u ∈ U , and for each u, fu has at most one degenerate singularity, which has codimension 1 (that is, has type A2 , A3 or D4 ), and is versally unfolded by the family {fu | u ∈ U }. We define O1 := {f ∈ O0 | pr1 (f ) ∈ C1 }. Lemma 2.4. (i) The set C1 is dense and open in C ∞ (U × S 1 , P 2 ). (ii) The set O1 is dense and open in O0 . Proof. We establish density by an explicit listing of jet transversality conditions. It is convenient to take the two cases together. We begin with a list of strata we can avoid for dimensional reasons. (TE1) CP has no quadruple point: Γ has no 5-secant. (TE2) CP has no triple point with two branches tangent: Γ has no T-4-secant. (TE3) CP has no double point with one branch singular: Γ has no tangent 3-secant. (TE4) CP has no double point where the branches have the same tangent and curvature: Γ has no T-3-secant with T-centre on Γ. (TE5) CP has no singular point of type higher than A2 : the tangent at a stall of Γ does not meet Γ again. With these excluded, the only possible types of degenerate singularities are A2 , A3 and D4 . For part (i), we need each of these to be versally unfolded in the family F . This requires a slight variation of Mather’s multi-transversality theorem [4]. For a family F : N × U → P , we define partial jet extensions r jUk F : N (r ) × U → r J k (N, P ) by k r jU F (x1 , . . . , xr , u)
:= (j k fu (x1 ), . . . , j k fu (xr )).
A singularity type corresponding to a sub-manifold Σ of r J k (N, P ) is versally unfolded by the family F , if and only if r jUk F is transverse to Σ. The set of F such that this holds is residual. It now suffices to apply this to the sub-manifolds Σ2 , Σ3 and Σ4 of multi-jet space defined above. For part (ii), CP has a point of type A2 , if P lies on the tangent at some Q ∈ Γ, of type A3 , if P is on a T-3-secant P QR and of type D4 , if P lies on a 4-secant P QRS. We will show in Lemma 6.8 that the versality conditions in the definition of O1 in cases A2 , A3 and D4 are equivalent respectively to (TD) If P ∈ TQ Γ, then TP Γ ⊂ OQ Γ. (TC) No plane is tangent to Γ at three collinear points. (TX) The cross-ratio of the planes through a 4-secant L containing the four tangent lines is not equal to the cross-ratio of the four points on L. Each of these corresponds a subset of some jet space of such codimension that, in general, it is avoided. We turn to openness. It suffices to consider C1 , since openness of O1 then follows from the fact that pr1 is continuous. By Proposition 2.1, it suffices to consider a one-parameter family {Ft } with F0 ∈ C1 and a sequence ti converging to 0, such that Ft i ∈ C1 , so there exist ui ∈ S 1 and a ‘bad’ singular point Pi of Ft i ,u i (S 1 ). Passing to a sub-sequence if necessary, we may assume that ui converges to u0 ∈ S 1 and Pi to P0 ∈ F0,u 0 (S 1 ). Arguing as for Lemma 2.3, we see that Pi cannot be of one of the types excluded by (TE1)–(TE5), so has type A2 , A3 or D4 . By hypothesis, such a singularity is versally unfolded by the family F0 . But then, by openness of versality [9, 3.7.1], the same is true for all small enough ti : a contradiction.
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A simple consequence of the above conditions is as follows. Lemma 2.5. If Γ ∈ O1 , and the projection CP from P has a singular point ZP , then (i) ZP is distinct from the image YP under projection of the tangent at P ; (ii) the line YP ZP is transverse to CP at ZP . Proof. In the A2 case, if ZP = YP , the line P Q is tangent to Γ at both P and Q, contradicting (TD). If the line is tangent to CP at ZP , we have TP Γ ⊂ OQ Γ, again contradicting (TD). In the A3 case, if ZP = YP , TP Γ would be a tangent 3-secant, contradicting condition (TE3). If tangency occurs, the T-plane would contain also the tangent at P , contradicting (TC). For D4 , if ZP = YP , TP Γ would be a tangent 3-secant, again contradicting condition (TE3). If tangency occurs, we have a T-4-secant, contradicting (TE2). We obtain further open conditions by using the arguments of [10]. Consider curves Γ in P 3 , given as the image of maps f : S 1 → P 3 . If the local intersection number of Γ with a plane Π at a point P is k, define the contact number of Γ with Γ at P to be k − 1, and define the total contact number κ(Γ, Π) of Γ with Π to be the sum of the contact numbers at all their common points, or, equivalently, the total intersection number Γ.Π diminished by the number #(Γ ∩ Π) of points of intersection. Here we use ‘intersection number’ in the real sense: if Π is given locally by x = 0 and f by f (t) = (a(t), a (t), a (t)), then the intersection number at t = 0 is the multiplicity of 0 as root of a(t), that is, the order of a(t) at t = 0. If a(t) has order k at 0, and we have a one-parameter at values of t family Γu of curves, leading to au (t), then the sum of orders of vanishing of au k near 0 is at most k, as we can reduce to the case of a polynomial au (t) = tk + 1 ai (u)tk −i . We will use this semi-continuity property below. Define a condition (TKO) For each plane Π, κ(Γ, Π) ≤ 3. Proposition 2.6. The family Ok of smooth curves Γ satisfying (TK0) is a dense open set. Proof. Density follows by counting constants: the conditions on a multi-jet of a plane corresponding to having κ(Γ, Π) ≥ 4 define a subset of multi-jet space, such that multitransversality to it implies avoidance. Let F = {ft : S 1 → P 3 | t ∈ U } be a smooth one-parameter family, with f0 ∈ Ok . Suppose that there exist arbitrarily small values of t, for which there is a plane Πt for which the total contact number with Γt is at least 4. The set of all planes forms the dual projective space P 3∨ and is, in particular, compact. Thus, we may pick a sequence of values of t converging to 0, for which Πt converges to a line Π0 . For each point P ∈ Γ0 ∩ Π0 , with intersection number a + 1, say, choose a neighbourhood U of P not containing any other point of Γ0 ∩ Π0 . Then the sum of the intersection numbers of Πt with Γt at points of U is upper semi-continous. For small t, either the number of points of Γt ∩ U ∩ Πt is 0, so the intersection number and contact number are also 0, or it is at least 1, so as the total intersection number is at most a + 1, the contact number is at most a. So the total contact number also is upper semi-continuous. Hence, the total contact number of Γ0 with Π0 is at least 4, a contradiction. We show in Proposition 6.12 that if W111 ⊂ 3 J 1 (S 1 , P 3 ), W21 ⊂ 2 J 2 (S 1 , P 3 ) and W3 ⊂ J 3 (S 1 , P 3 ) are the sub-manifolds corresponding to planes having κ(Π, Γ) ≥ 3 with contact numbers 1+1+1, 2+1 and 3 respectively, the multi-jet of f fails to be transverse if and only if either κ(Π, Γ) > 3 or (in the 1+1+1 case) the three contact points are collinear.
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The condition (TK0) can be expressed as the union of conditions corresponding to partitions of 4: (1111) no plane is tangent at more than three points, (211) no plane is osculating at one point and tangent at 2 more, (31) the osculating plane at a torsion zero point does not touch the curve again, (22) no plane is osculating at more than one point, (4) the zeros of the torsion are non-degenerate. Define a curve Γ to be section-generic, and write Γ ∈ Os , if (TK0), (TC) and (TD) hold. Proposition 2.7. Os is dense and open in C ∞ (S 1 , P 3 ). This result is proved in [1] by another method: the authors construct an auxiliary map φf and show that φf depends continuously on f , and that φf is (C ∞ -)stable, if and only if Γ is section-generic. The result then follows from the openness of the set of stable maps. Lemma 2.8. Suppose that p is not a double point of CP . Then for each line L through p, with pre-image the plane Π through TP Γ, we have Π.Γ = L.CP + 1, so that if #(Π ∩ Γ) = #(L ∩ CP ), we have κ(Π, Γ) = κ(L, CP ) + 1. Proof. At a point Q ∈ Π \ TP Γ, the local equation of Π is the composite of projection with the local equation of L, so the respective intersection numbers coincide. Under the hypothesis, the only point of TP Γ ∩ Γ is P itself. At P , we need a local calculation. Take local coordinates with Γ given by (t, φ(t), ψ(t)), where φ, ψ each have order at least, 2 at t = 0, so TP is the x-axis, and we may take Π as the plane y = 0. The projection CP from P is then parametrised by (φ(t)/t, ψ(t)/t). We thus have L.CP = ordt (φ(t)/t) = ordt (φ(t)) − 1 = Π.Γ − 1. 3. Flatness of the family of enhanced projections The crucial stage in our analysis is the passage from the family of projections of Γ from points of itself to the projections from nearby points. First suppose, in suitable local coordinates, that we have f (t) = (t, tφ(t), tψ(t)), where φ and ψ vanish at t = 0, so that the tangent there is the x-axis. We consider projection from the point (0, Y, Z) on the normal plane (any point close to Γ lies on a unique normal plane locally) to the fixed plane x = 1. This gives the map t → (φ(t)+Y (1−t−1 ), ψ(t)+Z(1−t−1 )). We are interested in what happens for Y and Z small. For t very small, the result is close to the line t → (Y (1 − t−1 ), Z(1 − t−1 )), or Zy = Y z, and for t large, the result is close to t → (φ(t) + Y, ψ(t) + Z), which is a translation of the projection from the origin. Thus, in some sense, the image of projection from (0, Y, Z) converges to the union of the line Zy = Y z with the image of the projection from (0, 0, 0). In other terms, the projection of Γ from a point near P ∈ Γ is close to the union of the projection CP of Γ from P and a line L through the point YP corresponding to P itself (the image of the tangent line TP Γ). The following sketch illustrates the case (φ(t), ψ(t)) = (t2 , t), (Y, Z) = (1, 0). '
& To make this precise, we need to deal with values of t which are neither very small nor very large. However, since Y and Z themselves are small, the image points in question are close to the origin, so that the crucial question is local.
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Write B : X → P 3 for the blow-up of P 3 along Γ, and E for the P 2 bundle over X given by pulling back the universal bundle over P 3 . Define a family of curves in the fibres of E as follows. If z = B(x) ∈ Γ, then Γx = πz (Γ). If z ∈ Γ, then x defines a plane Πx through the tangent Tz Γ, and we define Γx := πz (Γ \ {z}) ∪ πz (Πx ). Theorem 3.1. Suppose that no plane has infinite order of contact with Γ. Then near the exceptional hypersurface π −1 Γ, the above is a flat family. This means that near any point, there is a smooth function with zero set that intersects each fibre in the given curve. Proof. The assertion is local, so we will work in local coordinates X, Y, U . We may suppose Γ given by a smooth parametrisation (x, y, z) = (t, φ(t), ψ(t)),
(3.1)
where each of φ, ψ has order at least 2 at t = 0. We define local coordinates on the blow-up by setting Z − ψ(X) = U (Y − φ(X)).
(3.2)
The behaviour of the projection is unaltered if we regard it as projection onto the plane x = 1: denote coordinates in this image plane by (v, w) (rather than (y, z)) to avoid confusion. Thus the image of the projection of the point (x, y, z) from the point (X, Y, Z) is given by (1 − x)Z + (X − 1)z (1 − x)Y + (X − 1)y , w= . X −x X −x Substituting for Z from equation (3.2) and for x, y, z from equation (3.1), we find v=
v=
(1 − t)Y + (X − 1)φ(t) , X −t
(1 − t)(ψ(X) − U φ(X)) + (X − 1)(ψ(t) − U φ(t)) . X −t Clearing the denominator in equation (3.3) and rearranging gives w − Uv =
(vX − Y ) + t(Y − v) = (X − 1)φ(t).
(3.3)
(3.4)
(3.5)
On the other hand, since φ(t) and ψ(t) are C ∞ near t = 0, the right-hand side of equation (3.4) extends to a C ∞ function Ψ(X, U, t) near the origin. Letting t tend to X, we find that Ψ(X, U, X) = (1 − X)(ψ (X) − U φ (X)) + (ψ(X) − U φ(X)). Moreover, Ψ(0, 0, t) reduces to ψ(t)/t. We now distinguish cases according to the order of ψ(t) at t = 0. First let the order be 2. Then Ψ(X, U, t) has non-zero coefficient of t. By the implicit function theorem, there is a C ∞ function τ (W, X, U ) (defined near the origin), such that t = τ (W, X, U ), if and only if W = Ψ(X, U, t). We can thus substitute t = τ (w − U v, X, U ) in equation (3.5), and obtain v(X − τ (w − U v, X, U )) = Y (1 − τ (w − U v, X, U )) + (X − 1)φ(τ (w − U v, X, U )).
(3.6)
Conversely, suppose that equation (3.6) holds. Define t := τ (w − U v, X, U ). Then by the definition of τ , equation (3.4) holds, and substituting in equation (3.6) gives equation (3.5). If t = X, we can divide by X − t to obtain equation (3.3); hence, (v, w) is indeed the projection of (t, φ(t), ψ(t)) from (X, Y, Z) with Z given by equation (3.2). If, however, the solution is t = X, then equation (3.5) reduces (since X is small, so X = 1) to Y = φ(X). Thus, Z − ψ(X) = U (Y − φ(X)) = 0, and the point of projection lies on the curve Γ. Our other equation is w − U v = Ψ(X, U, X), which we can rewrite as w − ψ(X) − (1 − X)ψ (X) = U (v − φ(X) − (1 − X)φ (X)).
(3.7)
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But the plane through the tangent line at t = X to Γ corresponding to the auxiliary parameter U is given by z − ψ(X) − (x − X)ψ (X) = U (y − φ(X) − (x − X)φ (X)), and its projection is given by setting x = 1, y = v, z = w, and thus coincides with equation (3.7). We follow a similar plan in general. Let the order of ψ(t) at t = 0 be n + 1. Since this is the order of contact of the curve with the plane Z = U Y , it is finite by hypothesis. Then by Malgrange’s preparation theorem [3] (a generalisation of the implicit function theorem), for any C ∞ function h(t, X, U )of t, X and U , we can find C ∞ functions αi (W, X, U ) (for 1 ≤ i ≤ n), n such that h(t, X, U ) = 1 αi (W, X, U )tn −i when W = Ψ(X, U, t). Since the difference vanishes when W = Ψ(X, U, t), it is divisible by it, so there exists a further C ∞ function C(W, X, U, t), such that n h(t, X, U ) − αi (W, X, U )tn −i = C(W, X, U, t)(W − Ψ(X, U, t)). (3.8) 1
We apply this twice, first taking h to be (X − 1)φ(t) and second, taking h to be tn : denote the functions αi and C in the second case by adding a prime. Replacing (X − 1)φ(t) using equation (3.5), we have two polynomial equations for t which hold when W = Ψ(X, U, t) (vX − Y ) + t(Y − v) =
n
αi (W, X, U )tn −i ,
(3.9)
1
tn =
n
αi (W, X, U )tn −i .
(3.10)
1
Form the resultant of these two equations (for example, write down the Sylvester determinant), and denote it by R(W, X, Y, U, v). Then we claim that the equation R(w − U v, X, Y, U, v) = 0 defines the required locus. Suppose given a point (X, Y, U, v, w) on which R vanishes. Since R was defined as a resultant, there exists at least one value of t for which equations (3.9) and (3.10) both hold, with W = w − U v. Hence, C (w − U v, X, U, t)(w − U v − Ψ(X, U, t)) = 0. We will see below that C does not vanish at, and hence near, the origin. It then follows that equation (3.4) holds. We also have the identity (X − 1)φ(t) −
n
αi (W, X, U )tn −i = C(W, X, U, t)(W − Ψ(X, U, t)).
1
Substituting W = w − U v we see that the right-hand side vanishes, hence, so does the left: combining this with equation (3.9), we deduce that equation (3.5) also holds. It follows, as before, from equations (3.4) and (3.5), that if t = X, the point (v, w) lies on the projection of Γ from (X, Y, Z); while if t = X, then (X, Y, Z) ∈ Γ and the point lies on the line (equation (3.7)) which is the projection of the plane (parametrised by U ) through the tangent to Γ. It remains to show C does not vanish. Setting W = X = U = 0 in the defining identity n that n of C gives (t − 1 αi (0, 0, 0)tn −i ) = C (0, 0, 0, t)(−ψ(t)/t). Here the left-hand side has order at most n; the right-hand side has order exceeding n unless C (0, 0, 0, 0) = 0. The parameter n in this proof is the local intersection number of Γ with the plane Π : z = 0, which projects to the line L. Since Π passes through the tangent line, n ≥ 2. We have n = 2, if Π is not the osculating plane; if it is, n = 3 if the point is not a stall, and n = 4 if it is a transverse stall.
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The elimination can be written down more easily in the case n = 2, but this does not essentially simplify the argument. The equations obtained are too complicated to be of direct use (even for the case when Γ is given by (t, t2 , t3 ), the resultant has 85 terms). When n = 2, we begin with two transverse non-singular branches, forming an A1 , so no degenerate singularity occurs. When n = 3, we begin with an A3 singularity: we can illustrate what happens by taking Γ to be a twisted cubic. Then the projection from X ∈ Γ is a rational cubic, nodal if X does not lie on the surface of tangents, cuspidal if it does. The projection from X ∈ Γ is a conic; L touches this, if and only if Π osculates Γ. When n = 5, we have a deformation of an A5 singularity, which cannot be versal for dimensional reasons (indeed, the above example illustrates that versality fails also for the A3 case). Note that if the tangent at P passes through another point Q of Γ, we have constructed a flat family by considering a neighbourhood of P . The projection of the part of Γ near Q gives a branch BQ (which is non-singular as we exclude bitangents), so the total projection from points near P is the union of the relevant member of the flat family and a smooth deformation of BQ . 4. Enhanced projections from points of Γ We saw in Theorem 3.1 that, as X converges to a point P ∈ Γ, the image of the projection CX of Γ from X converges, not to the projection CP of Γ from P , but to the union of CP with a line L through the image point YP of the tangent TP Γ. We thus have a one-parameter family of based curves {CP , YP } and the two-parameter family obtained by adjoining a variable line through YP . It turns out that the families obtained as above by projection have properties not shared by general one-parameter families of based curves, so we confine ourselves to families {CP , YP } obtained by projection from a space curve Γ ∈ O1 . Thus, CP is non-degenerate for all but finitely many P , in which cases it has a single singularity of one of the types A2 , A3 or D4 . Denote these cases α, β and γ, respectively. By Lemma 2.5, in each of these cases the degenerate singular point ZP is distinct from YP , and the line YP ZP is transverse to CP at ZP . Thus, if YP is a singular point of CP , it is an ordinary double point. We will denote by δ the case when it is. For a single based curve (C, Y ), we consider the family of lines L through Y and their contact with C. The condition κ(L, C) = 0 is open: not only for nearby lines L with fixed C, but also for nearby pairs (L, C). Next suppose that C is non-degenerate and κ(L, C) = 1. The contact occurs at a unique point u ∈ C, and either a: is a simple tangency at a non-singular point of C, or b: is a transverse intersection at a double point (which must be ordinary). We distinguish the cases when u = Y and denote them by a∗ and b∗ ; thus, the latter only occurs in case δ. The condition κ(L, C) = 1 also has a certain stability property. In case a, u = Y is a nonsingular point of C, and for any (C , Y ) near (C, Y ), there is a unique line L through Y and near L which touches C , and other nearby lines are transverse to C. In case a∗ , L is the tangent to C at Y and so deforms smoothly, as we deform the pair (C, Y ). In case b, as we deform C, the point of transverse self-intersection will persist for nearby curves, and again there is a unique way to deform L, while for other nearby lines, the total contact number is 0. We will give more detail in the proof of Proposition 6.17. We now impose two further conditions (TK1) A line L through YP and a degenerate singular point of CP passes through no other singular point, and is transverse to CP . Transversality at the singular point itself was analysed in Lemma 2.5.
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(TK2) For any P , and any L through YP , κ(L, CP ) ≤ 2. If CP has no double point on L, this condition is equivalent to (TK0), as follows from Lemma 2.8, but the new condition excludes many further possibilities. Now suppose that κ(L, CP ) = 2, and that L does not pass through a degenerate singular point of CP . If the contact occurs at two points, the cases may be enumerated as aa, aa∗ , ab, ab∗ , ba∗ , bb, bb∗ . If L has contact number 2 at one point u, we may have a2 : L is an inflexional tangent at u = YP , a∗2 : L is an inflexional tangent at YP , b2 : L is tangent to one branch at a double point u = YP , or b∗2 : L is tangent to one branch at the double point YP ; when necessary, we distinguish the case b∗2+ , when L corresponds to the osculating plane at P from b∗2− , when L is tangent to the other branch. Thus, in case δ, there are several lines with κ(L, CP ) = 2: the two tangents at YP (each in case b∗2 ), tangents from YP to CP (each of type ab∗ ) and lines from YP to other double points (each of type bb∗ ). The singularities of CP ∪ LP are thus as follows, where for the cases α, β and γ, we take LP as the line joining YP to the degenerate singular point; in the rest, LP is the line through YP with κ(LP , CP ) = 2: Type α Sings D5
β D6
γ X9
aa, aa∗ 2A3
ab, ab∗ , ba∗ A3 + D4
bb, bb∗ 2D4
a2 , a∗2 A5
b2 , b∗2 D6 .
(4.1)
We next interpret the cases in terms of the geometry of the space curve. In each of the following list of twelve cases, we first describe the configuration of the projected curve CP , and give the singularities of CP ∪ LP . Here, we write p, q, etc. for the images of P, Q, etc. α: The curve CP has a cusp at q. The point P lies on the tangent to Γ at another point Q. β: The curve CP has a self-tangency at q = r. The point P lies on a T-trisecant P QR with the tangents at Q, R coplanar. γ: CP has a triple point q = r = s. The point P lies on a 4-secant P QRS. δ: p = q is a double point of CP . The tangent to Γ at P meets Γ again in Q. aa: LP touches CP at q and r. The tangents to Γ at P, Q and R are coplanar. aa∗ : LP touches CP at p and q. The tangent to Γ at Q lies in the osculating plane OP . ab: LP touches CP at q and passes through a double point r = s. The plane πP contains the tangents to Γ at P, Q and the trisecant P RS. ba∗ : LP touches CP at p and passes through a double point r = s. The osculating plane OP contains the trisecant P RS. bb: LP passes through double points q = r and s = t. The plane πP contains the tangent to Γ at P and the trisecants P QR and P ST . a2 : LP is an inflexional tangent at q. The tangent to Γ at P lies in the osculating plane OQ . a∗2 : LP is an inflexional tangent at p. P is a stall on Γ. b2 : LP is tangent at q to a double point q = r of CP . We have a T-trisecant P QR: the tangents at P, Q lie in a plane πP . Observe that the same geometrical situation may give rise to two cases by permuting the roles of the points P, Q, etc.: these pairs are α and δ (a tangent to Γ meets the curve again),
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β and b2 (there is a T-trisecant), and a2 and aa∗ (the osculating plane at one point contains the tangent at another). We will find that the cases with no b in the notation (no collinearity condition) are easier to treat. In these cases, the plane πP satisfies κ(πP , Γ) = 3, where 3 is partitioned as 3 (case a∗2 ), 21 (cases a2 , aa∗ ), and 111 (case aa). These cases occurred in the discussion of section genericity, as did α and δ. The cases ab, bb, a∗ b and b2 will require deeper calculations. We require each case to occur transversely. A convenient precise formulation is (TMS) the multi-jet extensions of f are transverse to the sub-manifolds of jet space defining each of the twelve cases. However, in all cases except a∗ b, ab, and bb, this already follows from conditions previously considered. Proposition 4.1. In the cases below, (TMS) is equivalent to the named condition. aa (T K0)
aa∗ (T K0)
a2 (T K0)
a∗2 (T K0)
α (T D)
β (T E4)
γ (T X)
δ (T D)
b2 (T E4)
These follow from the more detailed statements of Propositions 6.12 and 6.13. Thus, for these nine cases, the conditions hold for all Γ ∈ O1 ∩ Os . Each of the cases a∗ , a and b occurs for (P, Π) on a curve in E: as P moves along Γ we can deform Π (or equivalently, the line L = πP Π), so that (P, Π) remains on the curve. A two-point singularity aa, ab, etc., occurs at an intersection of two branches of such curves. (Trot): At a two-point singularity, the two curves in E are transverse. Equivalently, if we deform P to Pt and θt is the angle between the two deformed positions of L, then ∂θt /∂t = 0 at t = t0 . We show in Proposition 6.17 that in cases a∗ b, ab and bb, (TMS) is equivalent to (Trot). In cases aa and aa∗ , (Trot) always holds. We define Γ ∈ O2 if Γ ∈ O1 , the family (ΓP , YP ) of its self-projections satisfies (TK1) and (TK2), and (TMS) (or, equivalently, (Trot)) holds. Proposition 4.2. O2 is dense and open in the set of families of space curves. Proof. As to density, we can suppose that Γ ∈ O1 . Then the values of t corresponding to types α, β and γ are isolated, and they are stable under deformation of the family. For a single based curve of one of these types, the condition that YP ZP is transverse to CP holds for a dense set of such based curves. It follows that (TK1) holds on a dense set. The condition that for some P , there exists a line L through YP with κ(L, CP ) ≥ 3 defines subsets of the relevant multi-jet spaces of codimension exceeding that of the source. Hence, (TK2) holds on a dense set. Since (TMS) is a finite collection of multi-transversality conditions, it too holds densely. We turn to openness. By Lemma 2.4, O1 is open. Now again, cases α, β, γ occur discretely in the family, and for a single curve of one of these types, the condition that YP ZP is transverse to CP defines open sets of such based curves. It follows that (TK1) gives an open set. For (TK2), we use the criterion of Proposition 2.1. Thus suppose given a one-parameter family Γu of space curves, projecting to a one-parameter family {(Ct,u , Yt,u )} of based families, with Γ0 satisfying (TK2), and suppose, if possible, that there is a sequence of values u = un tending to 0 for which this condition fails. Thus, there is a line Ln through Yn := Y(t n ,u n ) having total contact number at least 3 with Cn := C(t n ,u n ) . Passing to a subsequence, we may assume (since the tn belongs to the compact set S 1 ) that tn tends to a limit t0 , and similarly, that Ln converges to a line L0 through Y0 . By semi-continuity, L0 has total contact number at least 3 with C0 . But this contradicts our hypothesis.
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Now suppose Γu a one-parameter family, such that Γ0 satisfies (TMS), and that there is a sequence of values u = un tending to 0 for which this condition fails. Passing to a subsequence, we may suppose that the clause of (TMS) which fails is the same on each occasion, that the corresponding values tn converge to a limit, and also that the corresponding lines Ln converge to a line L0 , so that the contact points of Ln with Cn converge to those of L0 with C0 . By hypothesis, the transversality condition holds for (C0 , L0 ). Hence, it also holds for (Cn , Ln ) for n large enough. We thus have a contradiction. For a family in O2 , the exceptional cases α, β, γ, δ, aa, aa∗ , ab, a∗ b, bb, a2 , a∗2 and b2 each occur for isolated points P on Γ (and the subcases involving b∗ only occur for P in case δ). We can thus define a further condition (Tdis) For each P not of type δ, there is at most one line LP through YP , which either passes through a degenerate singular point of CP or satisfies κ(LP , CP ) = 2. Addendum 4.3.
The families in O2 satisfying (Tdis) form an open dense set.
Density follows since for a based curve (CP , YP ) to admit two such lines imposes a codimension 2 condition. Openness is immediate since under the hypothesis O2 the values of P giving exceptional behaviour form a finite set and each deforms continuously. It is optional whether or not we require (Tdis). It is not essential for our main arguments, but clarifies the geometry: our list of cases for pairs (P, L) can now be regarded as a list of cases for P . 5. Conclusion We define O3 to be the class of curves Γ ∈ O2 , such that the family of projections from points not on Γ satisfies the conditions of Theorem 1.2. We aim to show that O3 is dense and open in C ∞ (S 1 , P 3 ). First, we need to strengthen the conclusions of § 3. Proposition 5.1. For any curve in Γ ∈ O2 , and any smooth family {fu } with f0 (S 1 ) = Γ0 = Γ, there is a neighbourhood U of Γ × 0 in P 3 × R, such that the family of projections of Γu from points (P, u) ∈ U (with P ∈ Γu ) satisfies the conditions of Theorem 1.2. Proof. It follows from the O2 condition that the singularities occurring in the family CP ∪L belong to the list enumerated above; in particular, they occur in equation (1.1) and have codimension at most 2. By Theorem 3.1, the family of deformations of Γu from nearby points of P 3 is a flat deformation of this family. But any such flat deformation can only produce singularities which again occur in equation (1.1) and have codimension at most 2. We have seen in Lemmas 6.1, 6.2, 6.4 that those of codimension 1 (A2 , A3 , D4 ) are automatically versally unfolded by the family of projections from all points of 3-space. For one-point singularities of codimension 2, the conditions for versality are as follows. Case
Condition
Reference
A4 A5 D5 D6 X9
(TK0) (4 case) (TK0) (22 case) No condition (TE4) (TX)
Lemma Lemma Lemma Lemma Lemma
6.1 6.2 6.3 6.4 6.5
It follows from the definition that these conditions always hold for curves Γ ∈ O2 . For two-point singularities, it follows from Lemma 6.7 that the condition for versality is that the tangent planes to the two codimension 1 strata meet transversely.
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Under (TK0), versality always holds in the cases 2A2 , A2 + A3 and 2A3 . For the tangent plane to the A2 stratum is the osculating plane; that to the A3 stratum is the T-plane. Thus, a plane Π satisfying two such conditions has κ(Π, Γ) ≥ 4, contradicting (TK0). In the case A2 + D4 , X lies on a tangent TP and a 3-secant QRS; versality fails only if the osculating plane at P coincides with the tangent plane to the D4 stratum, and so, in particular, passes through Q, R and S. As X approaches Γ, P and S (say) both converge to the same point P0 on the space curve. By Lemma 6.11(iii), a point which is not a stall has a neighbourhood U , such that the osculating plane at a point of U does not meet Γ again in U . Thus, P0 is a stall, and the osculating plane at P0 contains a trisecant P0 Q0 R0 . But then the projection from P0 has a line L with κ(L, CP 0 ) ≥ 3: namely, 2 at YP and 1 at the image of Q0 , thus contradicting (TK2). Next, consider the case A3 + D4 . We see from the list (equation (4.1)) that for this to arise for X arbitrarily close to P0 ∈ Γ, the limiting plane Π must have contact of type ab, ab∗ or a∗ b. However, case a∗ b does not arise, since in a neighbourhood of a projection of type a∗ , projections from points off Γ cannot produce an A3 . Suppose that the A3 comes from a T-secant XP Q and the D4 from a 3-secant XRST , and that as X approaches P0 , P and R approach P0 . Then P Q converges to a T-secant P0 Q0 and RST to a 3-secant P0 S0 T0 ; Π is the T-plane of P0 Q0 , and the limit of the tangent plane to the D4 stratum at X, which contains the tangent TP 0 Γ. Since by (TE3), Γ has no tangent 3-secant, case ab∗ does not arise, so we have ab. However, it is shown in Proposition 6.17 that in case ab, (TMS) is equivalent to condition (Trot) for the family of curves ∆t ∪ L, and in Lemma 6.10, that (Trot) is equivalent to such compound singularities being versally unfolded in that family. It follows by openness of versality that they are then also versally unfolded in the family of nearby projections which deforms this family: note that since the projection is not singular at YP 0 , this is indeed a parametrised family of curves. The argument for 2D4 is very similar. The limiting plane Π must correspond to case bb or bb∗ ; again bb∗ is excluded by (TE3), and the argument for bb is the same as for ab. We are now ready to prove our main result. Theorem 5.2. The class O3 is dense and open in C ∞ (S 1 , P 3 ). Proof. We have already seen that O2 is dense, and that for any curve in Γ ∈ O2 and any smooth family {fu } with f0 (S 1 ) = Γ0 = Γ, there is a neighbourhood U of Γ × 0 in P 3 × R, such that the family of projections of Γu from points (P, u) ∈ U (with P ∈ Γu ) satisfies the conditions of Theorem 1.2. By that theorem, we may approximate Γ by a curve which in addition satisfies the conditions for projections from other points; since O2 is open, this is still in O2 , and hence, in O3 . To prove openness, consider a smooth family {fu : S 1 → P 3 } with f0 ∈ O3 . By Proposition 4.2, for all small enough u, we have fu ∈ O2 . Were there arbitrarily small u for which the projection from some point Xu ∈ Γu had singularities otherwise than allowed in Theorem 1.2, we could pick a subsequence with Xu converging to a point X0 (here, of course, we use compactness of P 3 ). If X0 ∈ Γ0 , this contradicts the hypothesis that f0 ∈ O3 . Otherwise, for small enough u, Xu will be in the neighbourhood of Γu provided by Proposition 5.1, again giving a contradiction. Now suppose that for arbitrarily small u, there is some point Xu ∈ Γu , for which the projection from Xu has singularities not versally unfolded. Again, we may suppose Xu convergent to X0 ; we may also suppose that each Xu corresponds to the same list of singularities. But the projection of Γ0 from X0 is versally unfolded by hypothesis. Openness of versality thus gives a contradiction.
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The condition O3 is equivalent to O2 together with the requirement that, away from Γ, the various strata meet transversely. For we have verified that under O2 , each one-point singularity that occurs belongs to equation (1.1) and is versally unfolded in the family of projections. We have also shown this for the two-point singularities 2A2 , A2 A3 and 2A3 . There are, however, numerous other cases. Our arguments allow us to infer properties of the stratification induced on P 3 by the equisingularity type of the corresponding projection of a given curve Γ ∈ O3 . Away from Γ itself, the stratification near any point is locally equivalent to that of the versal unfolding of the family of degenerate singularities on the projected curve. Thus, for example, the A3 surface has a cuspidal edge along the A5 curve. This extends to a stratification of the blow-up X of P 3 along Γ. We expect that there is a unique local model for each of our named cases (in some cases, depending on parameters). 6. Calculations For our calculations, which are local, we work throughout in R3 . We denote a typical point by X = (x, x , x ); points of Γ are denoted P = (p, p , p ), Q, R, etc. We regard the coordinates subscript p, p , p as functions of a local parameter tp on Γ, which vanishes at P(we omit the ∞ ∞ p if there is no ambiguity). Their Taylor expansions are denoted p = 0 pr trp , p = 0 pr trp , etc. Successive derivatives of the vector P with respect to tp are denoted by suffices: P1 , P2 , . . .. Thus, at tp = 0, we have Pr = r!(pr , pr , pr ). We regard P , P1 , etc. as vectors in R3 , and use the ordinary notations of vector calculus. Projections are made onto the plane x = 0 and, where possible, along a line close to the x -axis. Consider a neighbourhood of A, where a0 = a0 = 0. We project from a point X, considered as a deformation of (0, 0, x0 ) (where x0 = 0), to the plane x = 0. Then the image point has coordinates xa (t) − x a(t) x a (t) − x a (t) a (t)(x, x , x ) − x (a(t), a (t), a (t)) = , ,0 . (6.1) a (t) − x a (t) − x a (t) − x 6.1. Versality criteria In this section, we give direct calculations of the conditions for transversality of the family of projections of Γ to the given one-point singularities (at the end, we deal with compound singularities). We begin by recalling the situation when the singularities are in normal form. We use the notations Ex for the ring of germs of C ∞ -functions of x and x , θ to denote tangent spaces, tf and ωf for the maps induced by f : see, for example, [9] for a full introduction. In particular, we write T Ae (f ) = tf (θt ⊕ θu ) + ωf (θx,x ) for the extended A-tangent space (in the case when we have a bi-germ, with source variables t and u), and use the symbol ≡ to denote congruence of elements of θ(f ) modulo it. First consider the case A2k , with normal form t → (t2 , t2k +1 ). The ring f ∗ Ex of function germs induced on R by f has codimension k in Et : it contains all monomials in t except the t2r −1 (1 ≤ r ≤ k). This describes ωf : θx → θ(f ); we obtain tf : θt → θ(f ) by differentiating f with respect to t, and the quotient θ(f )/{tf (θt ) + ωf (θx )} also has dimension k. We regard θ(f ) as the free element ∂/∂x where, Et –module with basis ∂/∂x, ∂/∂x , and write i as V := A∂/∂x+A a typical i r Ai t . All monomials t ∂/∂x, tr ∂/∂x as usual, we have Taylor expansions A = Ai t , A = 2r −1 lie in T Ae (f ) with the exception of the t ∂/∂x (1 ≤ r ≤ k), which thus give a basis for the quotient. The projection onto the quotient is thus given by the maps πr (V ) = A2r −1 for 1 ≤ r ≤ k. Secondly, consider A2k −1 , with normal form given by the bi-germ t → (t, tk ), u → (u, 0). Here, f ∗ Ex has codimension k in Et ⊕ Eu ; the image, generated by (t, u) and (tk , 0), contains all monomials except (tr , 0) and (0, ur ) with 0 ≤ r ≤ k − 1, and also contains (tr , ur ) for these
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values of r. Here, we denote a typical element of θ(f ) by V = (A, B)∂/∂x + (A , B )∂/∂x . All monomials in θ(f ) belong to T Ae (f ), with the exception of tr ∂/∂x ≡ ur ∂/∂x for 0 ≤ r ≤ k − 2, and the quotient has dimension k − 1. Projections to the quotient are given by πr (V ) = Ar −1 − Br −1 for 1 ≤ r ≤ k − 1. Thirdly, we have D2k +3 , with normal form t → (t2 , t2k +1 ), v → (0, v). The monomials not in ∗ f Ex are (1, 0) ≡ −(0, 1), (t2r −1 , 0) (1 ≤ r ≤ k) and (t2k +1 , 0) ≡ −(0, v). All monomials in θ(f ) belong to T Ae (f ) with the exception of (1, −1)∂/∂x and t2r −1 ∂/∂x (1 ≤ r ≤ k), so we have codimension k + 1. Projection to the quotient is given, for V = (A, C)∂/∂x + (A , C )∂/∂x , by π1 (V ) = A0 − C0 and πr +1 (V ) = A2r −1 for 1 ≤ r ≤ k. Fourthly, we have D2k +2 , with normal form given by the tri-germ t → (t, tk ), u → (u, 0), v → (0, v). Here, f ∗ Ex omits the monomials (tr , 0, 0), (0, ur , 0) (0 ≤ r ≤ k) and (0, 0, 1), (0, 0, v), but contains (1, 1, 1), (tr , ur , 0) (1 ≤ r ≤ k) and (tk , 0, v), so has codimension k + 2. The monomials in θ(f ) not in T Ae (f ) are (tr , 0, 0)∂/∂x ≡ −(0, ur , 0)∂/∂x (0 ≤ r ≤ k − 1) and (0, 0, 1)∂/∂x ≡ −(1, 0, 0)∂/∂x ≡ k(tk −1 , 0, 0)∂/∂x , so the codimension here is k. We take πr +1 (V ) = Ar − Br for 0 ≤ r < k − 1 and πk (V ) = Ak −1 − Bk −1 − k(A0 − C0 ). For our calculations, we need to consider maps not in normal form; fortunately, only the cases with codimension at most 3 are required; moreover, the above give a useful guide, for example, giving the degrees of determinacy of these germs. In each case, we give explicit maps πr : θ(f ) → R for 1 ≤ r ≤ k, such that (π1 , . . . , πk ) induces an isomorphism θ(f )/T Ae (f ) → Rk . Since our maps are clearly independent, to check the calculation it suffices to verify that each map vanishes on the images ωf (θx ) and, for each monogerm, tf (θt ). ∞ For A2k , we write the germ as f (t) = (α(t), α (t)), where, as usual, α(t) = 1 αr tr . In this case, we may suppose that α1 = α1 = α2 = 0 and α2 = 0. We have k = 1 ⇔ α3 = 0. If α3 = 0, we subtract a suitable multiple of α2 from α to eliminate the coefficient of t4 : then the condition that we have an A4 is that the coefficient of t5 does not vanish, that is, that 2α3 α4 = α2 α5 . Similarly, we may obtain the condition for k to equal 3, which involves α7 : we do not need it explicitly. To find the corresponding projections πr we proceed as follows. If we have A2 , then any element of Et of order at least 2 is in f ∗ Et . The term tf (θt ) gives multiples of (∂α/∂t, ∂α /∂t), which has a non-zero coefficient of (t, 0), so, as before, we may take π1 = A1 . Now suppose that α3 = 0 so any element of Et of order at least 2 is in f ∗ Et , and we must consider coefficients of (t, 0), (t3 , 0), (0, t) and (0, t3 ). For V ∈ θf , we may suppose that A1 = 0. Subtract (A1 /2α2 ) ∂f /∂t to remove the coefficient of (t, 0); α2−1 (A2 − (3α3 /2α2 )A1 )(α, 0) to remove the coefficient of (t2 , 0); and A3 −
α3 2α4 3α2 A2 − A1 + 32 A1 α2 α2 2α2
t2 ∂f 2α2 ∂t
to remove the coefficient of (t3 , 0); and (A2 /α2 )(0, α) to remove the coefficient of (0, t2 ). This leaves the coefficient of (0, t3 ) as A3 − (α3 /α2 )A2 − 2(α4 /α2 )A1 , which we can take as π2 . We can proceed similarly in the A6 case, or suppose that A0 = 0 and subtract {A(t)∂f /∂t} /(∂α/∂t) from V to eliminate the first component. Then subtract appropriate multiples of α, α2 from the second to eliminate the coefficients of t2 and t4 . The coefficients of t, t3 and t5 in the result give possible choices for π1 , π2 and π3 (the latter may be changed by adding a multiple of π2 ). The details can be done by hand or with a computer algebra programme. In summary, π1 = A1 , π2 = α2 A3 − α3 A2 − 2α4 A1 π3 =
α2 A5
−
2α3 A4
+
2α4 A3
−
α5 A2
−
2α4 A3
+
α5 A2
+
−3α6
+
α α 5 α3 2 5
(6.2)
A1 .
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For A2k +1 , we write the germs as f (t) = (α(t), α (t)), f (u) = (β(u), β (u)). We may suppose that k ≥ 1, and then that α1 = β1 = 0, while α1 = 0, β1 = 0. Then k = 1 ⇔ α1−2 α2 = β1−2 β2 . If equality does hold, then k = 2 ⇔ α1−4 (α1 α3 − 2α2 α2 ) = β1−4 (β1 β3 − 2β2 β2 ). We do not require the precise condition for k to equal 3. The corresponding projections πr may be taken as given by
π3 =
π1 = A0 − B0 , −1 π2 = −α A + β1−1 B1 + 2α1−2 α2 A0 − 2β1−2 β2 B0 , −2 1 −31 α1 A2 − α1 (α2 A1 + 2α2 A1 ) − 3α1−4 (α1 α3 − 2α2 α2 )A0 − β1−2 B2 − β1−3 (β2 B1 + 2β2 B1 ) − 3β1−4 (β1 β3 − 2β2 β2 )B0 .
(6.3)
For D2k +1 , we take the first germ as for A2k −2 and the second as (γ(v), γ (v)), with γ1 = 0 = γ1 , so the condition for k = 2 is α3 = 0. In fact, we require only the case k = 2, and here we have (as before) π1 = A0 − C0 , π2 = A1 ,
(6.4)
though it is easy to calculate further that when α3 = 0, we can take π3 = α2 A3 − 2α4 A1 . Finally, for D2k +2 , we have a third germ (γ(v), γ (v)) in addition to two as for A2k −1 . The condition for k = 1 is that no two of the vectors (α1 , α1 ), (β1 , β1 ), (γ1 , γ1 ) are proportional. If the perturbation parameter is denoted s, then, to first order, the perturbed lines are given by (α1 t + A0 s, α1 t + A0 s), etc., that is, by α1 x − α1 y = (α1 A0 − α1 A0 )s, etc., and we can define π1 as the invariant α1 α α A0 − α1 A 1 1 0 β1 β1 β1 B0 − β1 B0 , γ1 γ1 γ1 C0 − γ1 C0 which is proportional to the area of the triangle cut by the three lines. If we require α1 = γ1 = 0, then α1 and γ1 are non-zero; the condition for D4 is that neither β1 nor β1 vanishes; and we normalise α1 = γ1 = 1. Here, π1 reduces to β1 A0 +β1 B0 −β1 B0 −β1 C0 . If k ≥ 2, we further normalise β1 = 0; then β1 = 0 and the formula for π1 (divided by β1 ) simplifies to the same formula A0 −B0 as for A3 . The condition defining k = 2 (D6 ) is, as in the case of A3 , that α2 /(α1 )2 = β2 /(β1 )2 . As in the normal form case, we see that the contribution of the third component involves only the constant term C0 , and we find that the formulae for π2 and π3 are the same as in the cases of A5 and A7 , with A0 and B0 replaced by (A0 − C0 ) and (B0 − C0 ), respectively, in π2 for A5 and in π3 for A7 : π2 = −α1−1 A1 + β1−1 B1 + 2α1−2 α2 (A0 − C0 ) − 2β1−2 β2 (B0 − C0 ), π3 = (α1−2 A2 − α1−3 (α2 A1 + 2α2 A1 ) − 3α1−4 (α1 α3 − 2α2 α2 )(A0 − C0 )) −(β1−2 B2
−
β1−3 (β2 B1
+
2β2 B1 )
−
3β1−4 (β1 β3
−
2β2 β2 )(B0
(6.5)
− C0 )).
We observe that the condition defining the case D6 is just that the coefficient of C0 in the expression for π2 is non-zero; similarly, the condition defining D8 is that the coefficient of C0 in the expression for π3 does not vanish. 6.2. Projections from points off Γ This completes our discussion of local conditions: we turn to the formula defining central projection. We project Γ from a point X (which in this section does not lie on Γ) to the plane x = 0. The image πX (P ) of P is given by equation (6.1). We will take this as the curve (α(t), α (t)) of the preceding section; to check the transversality conditions of equation (6.2), etc., we will require the values obtained on substituting (0, 0, x0 )
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for (x, x , x ) not only in this expression, but also in its partial derivatives with respect to x, x and x . Dropping the superfluous third coordinate, these are a (t) ∂πX (P ) ∂πX (P ) a (t) = ,0 , = 0, , (6.6) ∂x a (t) − x0 ∂x a (t) − x0 −a (t) ∂πX (P ) = (a(t), a (t)). ∂x (a (t) − x0 )2 To perform calculations, we may substitute α(t) = x0 a(t)/x0 −a (t), α (t) = x0 a (t)/x0 − a (t), take in turn, ∂πX (P )/∂x, ∂πX (P )/∂x , ∂πX (P )/∂x (which we abbreviate to ∂x , ∂x , ∂x ) as (A, A ), and evaluate the πr on such instances of them as we need; and correspondingly if we have more than one branch. Thus, for example, if a0 = a1 = 0 we have
α = x0 ρa a2 t2 + (a3 + a2 a1 ρa )t3 + a4 + a3 a1 ρa + a2 a2 ρa + a1 2 ρ2a t4 + . . . , where ρa denotes 1/(x0 − a0 ) (we will define ρb , etc., similarly). And for Vx , we have A = 0 and A0 = a0 /(a0 − x0 ) = −a0 ρa . First, consider projection along a tangent: this will give rise to singularities of type A2k . We thus have a1 = a1 = 0, so by (Tim), a1 = 0. Setting a2 = 0 (so that the osculating plane is x = 0), it follows from (Tcur) that a2 = 0. We have an A2 , if and only if α3 = 0, that is, a3 = 0, which is equivalent to the point not being a stall. If, however, a3 = 0, we have an A4 unless α2 α5 = 2α3 α4 , which reduces to a2 a5 = a4 (a2 a1 ρa + 2a3 ), which is equivalent to x0 = a0 + a2 a4 a1 /(a2 a5 − 2a3 a4 ), giving the coordinate of the T-centre of the stall (if the denominator vanishes, the T-centre is at infinity in the chosen coordinates). In this case, we have an A6 unless a rather complicated expression vanishes, which can be regarded as giving an expression for a7 in terms of lower coefficients. We consider only the cases k ≤ 3; we have defined πi for i ≤ k. In the following (and later) lemmas, the values noted ? are not needed to determine whether or not the matrices are non-singular, so we ignore them. Lemma 6.1. The values of πi on the basis elements for A2k are given by i 1 2 3
πi (∂x ) −a1 x0 ρ2a ? ?
πi (∂x ) 0 2a4 a1 x0 2 ρ3a ?
πi (∂x ) 0 . 0 a2 a4 a1 x0 2 ρ4a
An A2 is always versally unfolded; an A4 is, if and only if a4 = 0 (that is, the stall is transverse) and when this holds, an A6 also is versally unfolded. Proof. The values of πi given are obtained by substituting as above and simplifying. We have already seen that, in the relevant cases, a1 and a2 do not vanish; neither do x0 and ρa . Thus, transversality always holds in the A2 case, and holds in both the others, if and only if a4 = 0, which is equivalent to the stall being transverse. Next, consider projection along a T-secant: this will give singularities of type A2k +1 with k ≥ 1. We may assume that the T-secant not a tangent, and that the T-plane is x = 0, so that the germs (a(ta ), a (ta ), a (ta )), (b(tb ), b (tb ), b (tb )) satisfy a0 = a0 = b0 = b0 , a1 = 0, b1 = 0 and a1 = b1 = 0 as well as having x0 , a0 and b0 all distinct. Thus, we have
α = x0 ρa a1 ta + (a2 − a1 a1 ρa )t2a + . . . ,
α = x0 ρa a2 t2a + (a3 − a2 a1 ρa )t3a + . . . ,
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and the leading terms of Vx are (a0 ρa , 0), those of Vy are (0, a0 ρa ) and those of Vz are (−a1 a0 ρ2a ta , −a2 a0 ρ2a t2a ). The condition for an A5 is α2 /α12 = β2 /β12 , which reduces to a2 (x0 − a0 )/a21 = b2 (x0 − b0 )/b21 , giving the value 2 b1 a2 − a21 b2 x0 = b21 a2 a0 − a21 b2 b0 for the T-centre. The singularity is higher than A5 , if and only if (α1 α3 − 2α2 α2 )/(α1 )4 = (β1 β3 − 2β2 β2 )/(β1 )4 , giving the condition for a special T-secant. Lemma 6.2. For A2k +1 , the values of πi (when 1 ≤ i ≤ k ≤ 3) on the basis elements are i 1 2 3
πi (∂x ) −x0 (a0 − b0 )ρa ρb ? ?
πi (∂x ) 0 2a2 (a0 − b0 )ρb /a21 ?
πi (∂x ) 0 . 0 2 a2 (a0 − b0 )ρb /a1
An A3 is always versally unfolded; an A5 is, if and only if a2 = 0, that is, the T-plane does not osculate at one (hence, both) points of contact; and then an A7 also is versally unfolded. Proof. Clearly, π1 takes the values −a0 ρa + b0 ρb , 0 and 0, so the first assertion follows. Next, π2 (∂x ) = 0, while π2 (∂x ) = 2α1−2 α2 a0 ρa − 2β1−2 β2 b0 ρb . Since α2 /α12 = β2 /β12 , this is equal to 2α2 (α1 )−2 = 2a2 /(a21 x0 ρa ) multiplied by a0 ρa − b0 ρb , giving the stated result. For π3 (∂x ), since A0 = 0, A1 = −a1 a0 ρ2a , A1 = 0 and A2 = −a2 a0 ρ2a (and similarly for b), we obtain α1−2 A2 − 2α2 α1−3 A1 − β1−2 B2 + 2β2 β1−3 B1 , which reduces to (2 − 1)(α2 /α12 )(a0 ρa /x0 ) − (2 − 1)(β2 /β12 )(b0 ρb /x0 ), and hence to (α2 /α12 )(a0 ρa − b0 ρb )/x0 . For projection along a tangent meeting the curve, again we normalise the first (tangent) component as before, and take the second as (c, c , c ). Then we have the following lemma. Lemma 6.3. The values of π1 and π2 for D5 are i 1 2
πi (∂x ) − a0 )ρa ρc ?
x0 (c0
πi (∂x ) 0 −a1 a0 ρ2a
πi (∂x ) 0 . 0
Thus, a D5 is always versally unfolded. Next, we project along a trisecant. Lemma 6.4. If a1 = c1 = 0, we have π1 (∂x ) = 0 and
π1 (∂x ) = a1 b1 c1 (b0 − c0 )x0 4 ρa ρ2b ρ2c ,
π1 (∂x ) = −a1 b1 c1 (a0 − b0 )x0 4 ρ2a ρ2b ρc .
If b1 = 0, we have πi (∂x ) i −2 2 −2 b21 a2 (a0 − c0 ) − a21 b2 (b0 − c0 ) a−2 1 b1 ρc ? 3
πi (∂x ) . 0 a2 (a0 − b0 )ρb /a21
A singularity of type D4 is always versally unfolded; a D6 is, provided that the T-centre is not the third point on Γ; and then a D8 is versally unfolded unless the T-plane osculates the two branches.
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Proof. The calculation of π1 is obtained by substituting in the determinant, which has several zero entries. Now a1 and c1 must both be non-zero, so π1 (∂x ) = π1 (∂x ) = 0, if and only if b1 = b1 = 0; that is, the trisecant is tangent at the point Pu . Now suppose that b1 = 0, b1 = 0; then π1 (∂x ) = 0 and π1 (∂x ) = 0. Then π2 (∂x ) = 0, since all the relevant coefficients of ∂x vanish. For π2 (∂x ), since A1 = B1 = 0, we have 2
β2 α2 (−a ρ + c ρ ) − 2 (−b0 ρb + c0 ρc ), a c 0 0 α12 β12
and as α2 /α12 = a2 /(a21 x0 ρa ), this reduces to the expression shown. We recognise the vanishing of this as the condition that (0, 0, c0 ) is the T-centre of the bi-germ given by the two other components. Finally, suppose that α2 /(α1 )2 = β2 /(β1 )2 ; then we need to evaluate π3 (∂x ). However, since here the term C0 = 0, this is exactly the same as the calculation of π3 (∂x ) for the A7 case, and we have seen above that this vanishes only if the T-plane osculates both the first two branches.
We will also require the calculation for a 4-secant. The conclusion is as follows. Lemma 6.5. The family of projections fails to versally unfold the X9 singularity stratum corresponding to a given 4-secant, if and only if the cross-ratio of the planes through the 4-secant containing the four tangent lines is equal to the cross-ratio of the four points on the line. Proof. For the normal form, we choose coordinates so that the 1-jets satisfy γ1 = δ1 = 0 (so that γ1 = 0, δ1 = 0). Then f ∗ Ex,x is generated in degrees at most 1 by (1, 1, 1, 1), (α1 t, β1 u, γ1 v, 0) and (α1 t, β1 u, 0, δ1 w). To calculate Te A, it suffices to study the constant terms. The quotient θf /Te A is 3-dimensional, and a projection (π1 , π2 ) to it is given by two terms of degree 0 and one of degree 1. In degree 0, we set π1 = (A0 − D0 )α1 + (C0 − A0 )α1 ,
π2 = (B0 − D0 )β1 + (C0 − B0 )β1 .
In degree 1, we find a unique projection π3 given by α1 β1 β1 α1 −1 −1 −1 −1 −1 −1 A1 α1 − B1 β1 + D1 δ1 + − . − − A1 α1 + B1 β1 + C1 γ1 α1 β1 α1 β1 However, this is not relevant for us, since the tangent space to the stratum E˜7 as opposed to the orbit contains one extra vector, which is not killed by π3 . Now (π1 , π2 ) vanishes on ∂x and has image on (∂x , ∂x ) given by the matrix
a1 (d0 − a0 )x0 ρa ρd b1 (d0 − b0 )x0 ρb ρd , a1 (a0 − c0 )x0 ρa ρc b1 (b0 − c0 )x0 ρb ρc and the desired transversality is equivalent to the non-vanishing of the determinant of the matrix. But the determinant vanishes, if and only if the cross-ratio of the slopes (a1 /a1 , b1 /b1 , 0, ∞) is equal to the cross-ratio of (a0 , b0 , c0 , d0 ). From the above calculations we deduce the following proposition. Proposition 6.6. (i) The tangent plane to the A2 stratum at a point on the tangent to Γ at P is the osculating plane at P . (ii) The tangent plane to the A3 stratum at a point on the T-secant P Q is the T-plane containing the tangents to Γ at P and Q.
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(iii) The tangent plane to the D4 stratum at a point X on the trisecant P QR is projectively related to the point X, in such a way that the point P corresponds to the plane through P QR containing the tangent to Γ at P , and similarly for Q, R. (iv) The tangent line to the A5 stratum at the T-centre of the T-secant P Q is the secant itself. Proof. The tangent plane is essentially the kernel of the map π1 above, so the conclusion is immediate in cases (i), (ii) and (iv). For case (iii), we calculate as follows. We see, from Lemma 6.4, that the kernel of π1 is generated by β1 ∂x (c0 − x0 )(a0 − b0 ) − β1 ∂x (a0 − x0 )(c0 − b0 ). This is linear in x0 , so the plane — which, of course, passes through the 3-secant — is projectively related to the point x0 on it. Moreover, at x0 = c0 , this expression reduces to (a multiple of) ∂x , corresponding to the plane x = 0 through the tangent at (0, 0, c0 ); at x0 = a0 it reduces to ∂x , giving the plane x = 0 through the tangent at (0, 0, a0 ); and at x0 = b0 to β1 ∂x − β1 ∂x , corresponding to the plane β1 x = β1 x through the tangent at (0, 0, b0 ). The tangent lines to the strata A4 , D5 , D6 and E˜7 are of course, the lines themselves, since each of these strata is a finite union of lines (tangent at a stall, tangent meeting Γ again, T-trisecant or 4-secant, respectively). From this, we can deduce the conditions for transversality to multistrata. Lemma 6.7. If the projection of Γ from X has 2 (or, respectively, 3) singularities, each of one of the types A2 , A3 or D4 , then the corresponding family of singularities is versally unfolded at X, if and only if the tangent planes to the (2 or 3) strata at X intersect transversely. Proof. This is an immediate consequence of the fact that each of the singularities by itself is versally unfolded. 6.3. Projections from points on Γ Next, we apply the criteria of § 6.1 to versality for the one-parameter family of based curves Ct obtained by projecting Γ from a variable point of itself. Thus the point X = (x, x , x ) above must be taken as a variable point of Γ and, therefore, depends only on one parameter. We need only consider possible singularities of types A2 , A3 and D4 . Lemma 6.8. An A2 in this family is versally unfolded, if and only if TX Γ ⊂ OP Γ. An A3 in this family is versally unfolded, if and only if TX Γ is not contained in the T-plane. A D4 in this family is versally unfolded, if and only if (TX) holds for this 4-secant. Proof. By Lemma 6.1, the condition for versal unfolding of A2 is (in the notation of that lemma) that we have a non-zero coefficient of ∂x ; that is, that TX Γ has a non-zero component in the x direction. Since we have normalised coordinates so that OP Γ is the plane x = 0, this gives the condition stated. An essentially identical argument holds in the A3 case; here, however, it is the T-plane which is given by x = 0. Applying Lemma 6.4, and substituting β1 = b1
−x0 , b0 − x0
β1 = b1
−x0 b0 − x0
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shows that versality fails in the D4 case, if and only if b0 b0 c0 a0 − + b − b1 x x1 = 0, 1 1 b0 − x0 c0 − x0 b0 − x0 a0 − x0 which reduces to (a0 − x0 )(c0 − b0 ) = −b1 x1 /b1 x1 . (c0 − x0 )(a0 − b0 ) Since the planes through the respective points A, B, C, X containing the tangent lines at those points are x = 0, b1 x = b1 x, x = 0 and x1 x = x1 x, the corresponding cross-ratio is −b1 x1 /b1 x1 , and the result follows. Next, we consider versality for the two-parameter family consisting of a one-parameter family of based curves (Ct , Yt ) with a line L through Yt . We may choose coordinates so that for each t, Yt is at the origin, and the line L is the x-axis. Write u for the coordinate on L. Thus, for deformations at the origin, we have A0 = A0 = A1 = A1 = 0; at a point other than the origin on the x-axis, there is no restriction on A and A . Rotating L gives (u cos θ, u sin θ). Differentiating and setting θ = 0 gives B = 0 and B = u. First, we consider the cases κ(L, C0 ) = 1. Lemma 6.9. For cases a and b, the singularity (A3 or D4 , respectively) is versally unfolded by rotating the line L. For cases a∗ and b∗ , it is not versally unfolded. Proof. For case a, we have π1 = A0 − B0 ; since we have a point u = 0, we have B0 = 0, so rotating the line is sufficient to yield a versal unfolding. For a∗ however, we have A0 = B0 = 0, so rotating the line together with deforming Ct does not suffice. For case b, we take L to be the component-labelled α to conform with the notation of § 6.1, and have π1 = β1 A0 + β1 B0 − β1 B0 − β1 C0 , where β1 and β1 are both non-zero. Again, at a point other than Yt , we have A0 = u = 0, so rotating the line gives a versal unfolding. However, for case b∗ , we have (as before) B0 = B0 = C0 = A0 = 0. We next consider the cases κ(L, C0 ) = 2 of type aa, ab or bb. Lemma 6.10. In the cases aa, ab, bb, the two-point singularity is versally unfolded if and only if (Trot) holds. Proof. We have two singularities, each of type A3 or D4 , and we know by Lemma 6.9 that each is versally unfolded by rotating the line L. Condition (Trot) states that the rotations corresponding to the two separate points are essentially independent, which is exactly what we require for versality. 6.4. Multi-jet transversality conditions In this subsection, we give explicit forms to the transversality conditions in condition (TMS), and obtain geometric equivalents for most of them. First, however, we obtain geometric characterisations of stalls. Lemma 6.11. (i) A point on Γ which is not a stall has a neighbourhood U , such that no two tangents at points of U lie in a plane. (ii) Near a stall there is an involution, given in local coordinates in the form τ (t) = −t+O(t2 ), such that the tangents at t and t are coplanar, if and only if t = τ (t).
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(iii) A point on Γ which is not a stall has a neighbourhood U , such that an osculating plane at a point of U does not meet Γ again in U . (iv) In local coordinates at a stall, there is a function φ with φ(t) = −3t + O(t2 ), such that the osculating plane at t meets Γ ∩ U again at t only if t = φ(t). Proof. (i) Take local coordinates with respect to which the curve is given by A(t) = (a(t), a (t), a (t)), with our usual notation. We may suppose that: a0 = a0 = a0 = 0; since the curve is embedded, that a1 = 0 = a1 = a1 ; and since the curvature is non-zero, that a2 = 0 = a2 . The condition for the tangents at t and t to lie in a plane is that the vectors A(t ) − A(t), A1 (t ) and A1 (t) are dependent. The least-order terms in the expansions give the determinant a1 (t − t) a2 (t2 − t2 ) a3 (t3 − t3 ) 2a2 t 3a3 t2 , a1 a1 2a2 t 3a3 t2 which reduces to 6a1 a2 a3 (t − t)4 . Observe that A(t) − A(t ) is divisible by t − t, with quotient Q, say, and 2Q − A1 (t) − A1 (t ) by (t − t)2 , so the triple product is divisible by (t − t)4 . The quotient is a unit and thus has no other zeros nearby. (ii) If, however, a3 = 0 but a4 = 0, a corresponding calculation yields the leading term 12a1 a2 a4 (t − t)4 (t + t). We have just seen that the factor (t − t)4 is to be removed, so we have uniquely t = −t added to higher order terms. (iii) For t to lie on the osculating plane at t, we need the vectors A(t ) − A(t), A1 (t) and A2 (t) to be dependent. Here, the determinant of leading terms gives 2a2 a3 (t − t)3 , but we see, as above, that the factor (t − t)3 must be removed, leaving a unit. (iv) However, in the case of a stall, the corresponding determinant of leading terms reduces to 2a2 a4 (t − t)3 (t + 3t), and our assertion follows as before. Recall that the cases corresponding to a plane Π with κ(Π, Γ) = 3 are aa (1 + 1 + 1), a2 , aa∗ (2 + 1) and a∗2 (3). These correspond to the following sub-manifolds of multi-jet spaces. Write W111 for the set of multi-jets in 3 J 1 (S 1 , P 3 ) such that the three image points are distinct and not collinear, and the three image tangent vectors lie in the plane Π of the points. Let W21 be the set of multi-jets in 2 J 2 (S 1 , P 3 ), such that the image points P and Q are distinct, Q ∈ TP Γ, and both TQ Γ and OP Γ lie in the plane Π defined by TP Γ and Q. Let W3 be the set of jets in 1 3 3 J(S , P ), such that the curvature at P is non-zero, and the third derivative lies in the plane Π = OP Γ. Proposition 6.12. The multi-jet of f fails to be transverse to W111 , if and only if Π osculates Γ at one of P, Q, R; to W21 , if and only if either P is a stall or Π osculates Γ at Q; to W3 , if and only if κP (Γ, Π) ≥ 4. Proof. (111) Take affine coordinates, so that none of the points in question is at infinity. The vector X := P ∧ Q + Q ∧ R + R ∧ P is non-zero (since the points are not collinear) and orthogonal to Π. The sub-manifold W111 is defined by the vanishing of the three expressions X.P1 , X.Q1 and X.R1 . The condition for transversality is obtained by differentiating these expressions with respect to tp , tq and tr , and taking the determinant of the resulting matrix. Since TP Γ lies in Π, ∂X/∂tp = P1 ∧ (Q − R) is perpendicular to Π, and hence is orthogonal to Q1 and R1 ; similarly for ∂X/∂tq and ∂X/∂tr . It follows that the matrix is diagonal, and transversality fails if and only if one of the diagonal entries vanishes. But ∂(X.P1 )/∂tp = X.P2 vanishes only if Π osculates Γ at P .
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(21) The vector Y := P1 ∧ (Q − P ) orthogonal to Π, and W21 is defined by the vanishing of Y.P2 and Y.Q1 . We have ∂Y /∂tp = P2 ∧ (Q − P ) and ∂Y /∂tq = P1 ∧ Q1 . Thus, ∂(Y.P2 )/∂tp = ∂Y /∂tp .P2 + Y.P3 = Y.P3 , ∂(Y.P2 )/∂tq = ∂Y /∂tq .P2 = [P1 , Q1 , P2 ], ∂(Y.Q1 )/∂tp = ∂Y /∂tp .Q1 = [P2 , Q − P, Q1 ], ∂(Y.Q1 )/∂tq = ∂Y /∂tq .Q1 + Y.Q2 = Y.Q2 . The middle two expressions vanish since, in each case, all three of the vectors in the scalar triple product are tangent to Π; thus, for failure of transversality, one of the other two expressions must be zero; that is, either P3 or Q2 is tangent to Π. (3) Here, P1 and P2 are given to be non-parallel, and W3 is defined by the condition [P1 , P2 , P3 ] = 0. Transversality fails if the derivative of this with respect to tp vanishes. As this is [P1 , P2 , P4 ], the condition is that P4 also is tangent to Π. We next deal with the ‘Greek letter cases’ α, β, γ. The next result shows that in these cases, (TMS) is equivalent to the condition previously obtained in Lemma 6.8. Define Wα to be the set of multi-jets in 2 J 2 (S 1 , P 3 ), such that the image points P and Q are distinct and Q ∈ TP Γ (this corresponds to cases α, δ of § 4). Let Wγ ⊂ 0 J 4 (S 1 , P 3 ) be the set of jets, such that P, Q, R and S are collinear and no plane contains the tangents at three of them (this relates to case γ); let Wβ ⊂ 3 J 2 (S 1 , P 3 ) be the set of jets, such that P, Q, R are collinear and the tangents at P and Q (but not that at R) lie in a plane Π (this relates to cases β, b2 ). Proposition 6.13. Transversality to Wα fails, if and only if TQ Γ ⊂ OP Γ; to Wγ , if and only if the cross-ratio of the four collinear points P QRS coincides with that of the planes through the line containing the four respective tangents; to Wβ , if and only if R is not the T-centre of the T-secant P Q. Proof. (α) The condition Q ∈ TP Γ translates as P1 ∧ (Q − P ) = 0. Although this appears to give three conditions, if we take coordinates, so that P and Q lie on the x -axis, it suffices to take the first and second coordinates. Now ∂(P1 ∧ (Q − P ))/∂tp = P2 ∧ (Q − P ) and
∂(P1 ∧ (Q − P ))/∂tq = P1 ∧ Q1 .
These vectors are proportional, if and only if the plane OP Γ through P Q contains TQ Γ. (γ) Take the line P QRS to be the x -axis. Then Wγ is defined by the vanishing of (the first and second coordinates of) X := P ∧ Q + Q ∧ R + R ∧ P and Y := P ∧ Q + Q ∧ S + S ∧ P . We have ∂X/∂tp = P1 ∧ (Q − R) = (p1 , p1 , p1 ) ∧ (0, 0, q0 − r0 ) = (q0 − r0 )(p1 , −p1 , 0), and similarly for the others. The first two coordinates of the derivatives of X and Y with respect to the local coordinates tp , tq , tr , ts give the 4 × 4 matrix ⎛ ⎞ (q0 − r0 )p1 −(q0 − r0 )p1 (q0 − s0 )p1 −(q0 − s0 )p1 ⎜(r0 − p0 )q1 −(r0 − p0 )q1 (s0 − p0 )q1 −(s0 − p0 )q1 ⎟ ⎜ ⎟. ⎝(p0 − q0 )r1 −(p0 − q0 )r1 ⎠ 0 0 0 0 (p0 − q0 )s1 −(p0 − q0 )s1 To simplify the determinant, choose coordinates so that p1 , q1 , r1 and s1 are all non-zero; divide the rows successively by these quantities, and write θp := p1 /p1 , etc. Remove the common factor (p0 − q0 ) from each entry in the two lower rows. Evaluate explicitly and collect terms to obtain (θp θq + θr θs )(p0 − q0 )(s0 − r0 ) + (θp θr + θs θq )(p0 − r0 )(q0 − s0 ) + (θp θs + θq θr )(p0 − s0 )(r0 − q0 ).
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But this same expression is obtained by multiplying the difference of cross-ratios (p0 , q0 ; r0 , s0 )− (θp , θq ; θr , θs ) by the denominator. Thus, if transversality fails, either the two cross-ratios are equal or the second is indeterminate, which occurs only when three of the θ values coincide; that is, there is a plane containing the tangents at three of the collinear points P, Q, R, S. (β) We may take the line P QR as the x -axis, and the plane Π as x = 0. Then Wγ is defined by the vanishing of (the first and second coordinates of) X := P ∧ Q + Q ∧ R + R ∧ P , and of Z := [P1 , Q1 , Q − P ]. The first derivatives of X and Z are ∂X/∂tp = (0, p1 (q0 − r0 ), 0), ∂X/∂tr = (p0 − q0 )(r1 , −r1 , 0), ∂Z/∂tp = [P2 , Q1 , Q − P ] = 2p2 q1 (p0 − q0 ), ∂Z/∂tq = [P1 , Q2 , Q − P ] = −2q2 p1 (p0 − q0 ), We thus obtain the determinant 0 0 (p0 − q0 )r1
p1 (q0 − r0 ) q1 (r0 − p0 ) −(p0 − q0 )r1
∂X/∂tq = (0, q1 (r0 − p0 ), 0),
∂Z/∂tr = 0.
2p2 q1 (p0 − q0 ) −2q2 p1 (p0 − q0 ) . 0
In the determinant, we may divide the last row and column by (p0 − q0 ). The result reduces to 2r1 (p21 q2 (q0 − r0 ) − q12 p2 (r0 − p0 )). By hypothesis, r1 = 0. On the other hand, projecting the part of Γ near P from R to x = 0 gives r0 (p1 t + O(t2 ), p2 t2 + O(t3 )). r0 − p0 The radius of curvature of this is
p21 r0 . r0 − p0 2p2
A similar result holds for Q. The final term above thus vanishes, if and only if the radii of curvature of the projections at P and at Q from R coincide, that is, if and only if R is the T-centre of P Q. We have not succeeded in obtaining a geometric argument for Proposition 6.17, so will obtain it by direct calculation of the conditions. To prepare for the proof, we discuss the curves a∗ , a and b in E. We retain the notation and conventions at the beginning of this section. We denote by P0 the result of substituting tp = 0 in P . We may suppose, all the points P0 , Q0 , . . . below have distinct x -coordinates and (since we no longer need to calculate projections) that P0 is at the origin, with tangent along the x -axis, and Π is the plane given by x = 0. Thus, the line P0 Q0 is determined by the ratio λq := q0 /q0 . Since C is non-singular at P , p1 = 0. We may take x (scaled by p1 , which we retain to preserve homogeneity in our formulae) as local coordinate at P , so pr = 0 for r = 1. The blow-up E is defined by a new coordinate obtained by adapting x /x: more precisely, if C is locally parametrised as (φ(tp ), φ (tp ), p0 + p1 tp ), we define z := (x − φ (tp ))/(x − φ(tp )). Thus, a variable point (1 − λ)P + λQ on the line P Q has blow-up coordinate z :=
(1 − λ)p + λq − φ ({(1 − λ)p + λq }/p1 ) . (1 − λ)p + λq − φ({(1 − λ)p + λq }/p1 )
This meets the exceptional set E in the limit as λ → 0. As numerator and denominator both vanish, we evaluate the limit by l’Hˆ opital’s rule, obtaining q − p − ∂φ /∂tp (q − p )/p1 . (6.7) z= q − p − ∂φ/∂tp (q − p )/p1
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Provided that q0 = 0, the terms of order at most 1 in the numerator are q1 tq − (2p2 tp )(q0 )/p1 , while at tp = tq = 0, the denominator reduces to q0 . Thus, if q0 = 0 (that is, Q0 ∈ TP 0 C) the curve in E defined by the chord P Q has slope with respect to coordinates (tp , z) q1 tq p q − 2 2 0 . q0 t p p1 q 0
(6.8)
If (P0 , Π) lies on the a∗ curve, x = 0 is the osculating plane at P0 , so p2 = 0. Lemma 6.14. If (P0 , Π) is not of type (a∗2 ), the a∗ curve is non-singular at (P0 , Π), with slope 3p3 /p2 . Proof. Here we cannot use equation (6.7), but calculate directly. A point on the tangent at P can be taken as P + λP1 = p2 t2p + 2λtp + . . . , p3 t3p + 3λt2p + . . . , p0 + p1 (tp + λ) , giving z=
p3 (t3p + 3λt2p ) + . . . − p3 (tp + λ)3 − . . . . p2 (t2p + 2λtp ) + . . . − p2 (tp + λ)2 − . . .
Cancelling λ2 from numerator and denominator gives (−3p3 tp + . . .)/(−p2 + . . .). Thus, indeed, the slope is 3p3 /p2 . For the a curve at a point (P0 , Π) given by a T-secant P0 Q0 , since TQ 0 C lies in Π, we have q0 = q1 = 0. Lemma 6.15. If (P0 , Π) is not of type (a2 ) or (b∗2 ), the a curve is non-singular at (P0 , Π), with slope −2(p2 q0 /p1 q0 ) = −2p2 /p1 λq . Proof. If P Q is a T-secant, P − Q, P1 and Q1 are coplanar, so for P and Q near P0 and Q0 , we have 2 p2 tp + . . . − q0 − . . . p2 t2p + . . . − q2 t2q + . . . p1 tp − q0 − . . . = 0. 2p2 tp + . . . p1 2p2 tp + . . . q1 + . . . 2q2 tq + . . . q1 + . . . As will be the case in similar examples below, all terms in the middle column have order at least 1, so to find those of order 1 in the determinant, we only need the constant terms in the other columns. Thus, −q0 0 −q0 2p2 tp p1 = −2p2 (−q1 q0 + q0 q1 )tp + 2q0 q2 p1 tq . ∆1 := 0 q1 2q2 tq q1 The hypothesis gives q2 = 0 and q0 = 0, so we can express tq as a multiple of tp ; further terms in the expansion then allow us to solve for tq as a power series in tp . So we can take tp as local coordinate along the a curve, which is non-singular. Since q1 = 0, the result now follows from equation (6.8). For the b curve, we need the condition for collinearity of P, Q, R, namely, that the matrix ⎛ ⎞ 1 p p p ⎝1 q q q ⎠ (6.9) 1 r r r
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have rank 2. Since the first and last columns are independent for P0 , Q0 , R0 and hence nearby, it suffices to equate to zero the determinants formed by omitting the third and second columns, which we denote respectively by ∆ and ∆ . Since the points are collinear, λq = λr . We have a T-trisecant if (case (b2 )) TQ 0 C or TR 0 C meets TP 0 C, that is, if q1 = 0 or r1 = 0, or if (case β) TR 0 C meets TQ 0 C: this holds if 0 = (q1 − λq q1 )r1 − (r1 − λr r1 )q1 = ξq , say. Lemma 6.16. If (P0 , Π) is not of type b2 or β, the b curve in EC is non-singular at (P0 , Π), with slope p q r p (r − q ) − 1 1 1 0 0 − 2 2 . ξq q0 r0 λ q p1 Proof. The terms of order at most 1 in tp , tq , tr in equation (6.9) are ⎞ ⎛ 1 0 0 p1 tp ⎝1 q0 + q1 tq q1 tq q0 + q1 tq ⎠ . 1 r0 + r1 tr r1 tr r0 + r1 tr Since λq = λr , the linear terms in the determinants are ∆ = p1 (r0 − q0 )tp + (q1 − λq q1 )r0 tq + (λr r1 − r1 )q0 tr = 0, ∆ = q1 r0 tq − r1 q0 tr . Eliminate tr by forming ∆e := r1 ∆ + (λr r1 − r1 )∆ . Then ∆e = r1 p1 (r0 − q0 )tp + {(q1 − λq q1 )r1 − (r1 − λr r1 )q1 }r0 tq = r1 p1 (r0 − q0 )tp + r0 ξq tq . Since r1 , q0 and ξq are non-zero, we can solve for tq and tr in terms of tp and the curve is non-singular. The slope is again given by equation (6.8). We can now easily see that (Trot) holds for cases aa∗ and aa. In the first case, since p2 = 0, one slope vanishes and the other does not; in case aa, the two slopes differ, since λq = λu as the three points are not collinear. Proposition 6.17. In cases a∗ b, ab and bb, condition (TMS) is equivalent to (Trot). Proof. It follows from (Tdis) that at most, one condition can hold for a given P0 , so we may suppose throughout that none of a2 , a∗2 , b2 , b∗2 , α, β occur. (a∗ b) The sub-manifold of multi-jet space is defined by: (P, Q, R) are collinear and (C0) P Q ⊂ OP Γ; that is, (P − Q, P1 , P2 ) are coplanar. This holds when the parameters (tp , tq , etc.) all vanish; we will expand the conditions to first order in the parameters, and (TMS) states that the determinant of the coefficients in these expressions vanishes. Now (C0) is equivalent to the vanishing of the determinant of the matrix of (P − Q, P1 , P2 ); and to order 1, this matrix is ⎞ ⎛ −q0 − q1 tq −q1 tq p1 tp − q1 tq ⎠. ⎝ 2p2 tp 0 p1 2p2 + 6p3 tp 6p3 tp 0 Again, the middle column vanishes if tp = tq = 0, so we can ignore all terms except −q0 −q1 tq 0 0 p1 = 2p1 (3q0 p3 tp − p2 q1 tq ). ∆0 = 0 2p2 6p3 tp 0
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Thus, (TMS) is the condition that ∆, ∆ and ∆0 are independent, that is, that ∆e and ∆0 are, that is, that r1 p1 (r0 − q0 ) r0 ξq = 0. 3q0 p3 −p2 q1 This reduces to (Trot), which since p2 = 0 is 3p3 /p2 = −q1 r1 p1 (r0 − q0 )/ξq q0 r0 . (ab) The subset of multi-jet space is defined by: (P, Q, R) are collinear, and TU Γ ⊂ Π, so (C1) (P − U, P1 , U1 ) are coplanar and (C3) (P − Q, P1 , P − U ) are coplanar. We have already evaluated the collinearity condition and (in Lemma 6.15) the condition for (C1). To order 1, the matrix for (C3) is ⎛ ⎞ −q0 − q1 tq −q1 tq p1 tp − q0 − q1 tq ⎝ 2p2 tp 2p2 tp p1 + 2p2 tp ⎠ , −u0 − u1 tu −u1 tu p1 tp − u0 − u1 tu where again the second −q0 ∆3 = 0 −u0
column has only first-order terms, so −q1 tq −q0 2p2 tp p1 = 2p2 (q0 u0 − u0 q0 )tp + u0 q1 p1 tq − q0 u1 p1 tu . −u1 tu −u0
We will need this expression in full below; however, in the present case, u1 = 0, so ∆3 does not involve tu . As tu has non-zero coefficient in ∆1 , condition (TMS) reduces to independence of ∆e and ∆3 , that is, to r1 p1 (r0 − q0 ) r0 ξq 2p (q0 u − u0 q ) u0 q p = 0. 2
0
Multiplying this out, and dividing by q1 r1 (r0
−
0
q0 )p1
q0 r0 ξq
1 1
q0 u0 r0 p1 ξq , −
gives the expression
p 2 2 (1/λu p1
− 1/λq ),
whose non-vanishing is equivalent to (Trot). (bb) The multi-jet conditions reduce to (P, Q, R) are collinear, (P, U, V ) are collinear, and (C3) holds. As before, we eliminate tr from ∆q and ∆q and tv from ∆u and ∆u to obtain ∆q = p1 (r0 − q0 )tp + ξq
r0 tq , r1
∆u = p1 (v0 − u0 )tp + ξu
v0 tu . v1
Thus, (TMS) reduces to non-vanishing of p (r0 − q0 ) 1 p1 (v0 − u0 ) 2p (q u − u q ) 0 0 2 0 0 Evaluating the
ξq
r 0 r 1
0 u0 q1 p1
v ξu v0 . 1 −q u p 0
0 1 1
determinant, and dividing by q0 u0 p1 , gives p q r p −1 2 2 λ−1 − (r0 − q0 ) 1 1 1 + (v0 u − λq p1 ξq q0 r0
− u0 )
u1 v1 p1 , ξu u0 v0
and we see at once that (Trot) is equivalent to the non-vanishing of this expression. We make one final remark. Although the transversality arguments do not apply directly, the analysis of cases is essentially also valid in the complex case, so applies to algebraic curves in
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complex projective space. Our transversality conditions hold for curves which are general in a certain sense. For these, the number of 0-dimensional strata of each type is finite, and there are formulae (many of which are known) expressing these numbers in terms of the degree and genus of Γ. References 1. T. Banchof, T. Gaffney and C. McCrory, ‘Counting tritangent planes of space curves’, Topology 24 (1985) 15–24. 2. A. A. Du Plessis and C. T. C. Wall, The geometry of topological stability, London Mathematical Society monographs N.S. 9 (Oxford University Press, Oxford, 1995). 3. B. Malgrange, Ideals of differentiable functions (Oxford University Press, Oxford, 1966). 4. J. N. Mather, ‘Transversality’, Advances in Math. 4 (1970) 301–335. 5. D. M. Q. Mond, ‘Singularities of the tangent developable surface of a space curve’, Quart. Jour. Math. Oxford 40 (1989) 79–91. 6. D. M. Q. Mond, ‘Looking at bent wires’, Math. Proc. Camb. Phil. Soc. 117 (1995) 213–222. 7. J. M. Soares David, ‘Projection-generic curves’, J. London Math. Soc. (2) 27 (1983) 552–562. 8. C. T. C. Wall, ‘Geometric properties of generic differentiable manifolds’, Geometry and topology: III Latin American school of mathematics (ed. J. Palis and M. P. do Carmo), Springer Lecture Notes in Mathematics 597 (Springer, Berlin, 1977) 707–774. 9. C. T. C. Wall, ‘Finite determinacy of smooth map-germs’, Bull. London Math. Soc. 13 (1981) 481–539. 10. C. T. C. Wall, ‘Openness and multitransversality’, Real and complex singularities (VI S˜ ao Carlos workshop) (ed. D. Mond and M. J. Saia), Lecture Notes in Pure and Applied Mathematics 232 (Marcel Dekker, New York, NY, 2003) 121–136.
C. T. C. Wall Department of Mathematical Sciences University of Liverpool Liverpool L69 7ZL United Kingdom
[email protected]
