Two projections, two questions
The diagram of §22.6 is taken as given. Inside ℙT*ℙⁿ — pairs (point, hyperplane through it), of dimension 2n−1 — sits the conormal variety Con(X), Legendrian, of dimension n−1 whatever the dimension of X. It carries two projections, and the whole subject is the difference between them.
π₁ is easy. Over a smooth point of a hypersurface its fibre is a single point, because the embedded tangent hyperplane is determined; over a smooth point of a variety of codimension c the fibre is a ℙc−1 of hyperplanes containing the tangent space. Either way the fibre dimension is constant on the smooth locus, and π₁ loses nothing.
π₂ is where the content is. Its fibre over a hyperplane H is the set of points of X at which H is tangent. Generically that is one point and π₂ is birational. Over special hyperplanes it is not: H may be tangent at two points, or tangent to higher order at one. Those are exactly the hyperplanes at which X∨ fails to be smooth, and the book-keeping of that failure is the rest of this chapter.
Contact locus. For H ∈ X∨, the contact locus is π₁(π₂−1(H)) ⊆ X — the set of points where H is tangent. A general H has finite contact locus; a positive-dimensional contact locus for every H is the defective case of §23.4.
The dual variety is a discriminant
The word discriminant arrives here rather than being imported. Consider the family of all binary forms of degree d, that is, all homogeneous polynomials f(u,v) = c₀ud + c₁ud−1v + ⋯ + cdvd. The coefficients (c₀ : ⋯ : cd) are a point of ℙd, and the forms with a repeated root make a hypersurface in that ℙd: the discriminant.
Separately, the rational normal curve Cd ⊂ ℙd is the image of [u:v] ↦ [ud : ud−1v : ⋯ : vd]. A hyperplane of ℙd is a linear condition on the coefficients, that is, a binary form; it is tangent to Cd exactly when that form has a repeated root. So:
The discriminant is a dual variety. The dual of the rational normal curve Cd ⊂ ℙd is the discriminant hypersurface of binary forms of degree d, and its degree is 2(d−1).
Checked
Computing Res(f, f′) in the coefficients and dividing out the leading coefficient — the resultant carries one extra factor of c₀ — gives degrees 2, 4, 6, 8 for d = 2, 3, 4, 5, which is 2(d−1) in each case. Block [7] of ch23-verify.py. For d = 2 this agrees with the smooth-plane-curve formula of §23.3, since C₂ is a conic and 2(2−1) = 2·1 = 2; the two formulas compute the same number by different routes, which is the only reason to trust either.
This is what the projection costs, stated positively. Passing from a variety to its dual converts a geometric object into the vanishing locus of an algebraic condition on a family. Every discriminant, resultant and determinant appearing anywhere in this corpus is π₂ applied to something.
What the singularities cost
For a plane curve the accounting closes exactly. Let X ⊂ ℙ² be an irreducible curve of degree d whose only singularities are δ ordinary nodes and κ ordinary cusps. Then the dual curve — classically the class of X, the number of tangent lines through a general point — has degree
Read that as a budget. A smooth curve of degree d has class d(d−1); every node refunds 2, every cusp refunds 3. The refunds are not conventions — they count how many of the d(d−1) tangent lines are absorbed into the singular point and are no longer available to pass through a general point elsewhere.
| Curve | d | δ | κ | d∨ | Check |
|---|---|---|---|---|---|
| smooth conic | 2 | 0 | 0 | 2 | dual of a conic is a conic |
| smooth cubic | 3 | 0 | 0 | 6 | a sextic, with 9 cusps at the 9 flexes |
| nodal cubic | 3 | 1 | 0 | 4 | 6 − 2 |
| cuspidal cubic | 3 | 0 | 1 | 3 | 6 − 3 — computed independently below |
The last row is the one worth doing by hand, because it can be checked without the formula. Parametrise the cuspidal cubic as (x, y) = (t², t³). The tangent line at parameter t is a x + b y + c = 0 subject to the two conditions that it meets the point and matches the direction; eliminating t between them gives the dual curve in line coordinates,
Block [6]. Now read its singularities, because this is where the chart matters and where a careless reading goes wrong. The gradient (12a², 54bc, 27b²) vanishes on X∨ at the single point [0 : 0 : 1]. That point is not visible in the affine chart b = 1, where the equation reads 4a³ + 27c = 0 — a smooth graph c = −4a³/27, with no singular point at all. It lives in the chart c = 1, where the same curve reads
So the dual of a curve with a cusp is a curve with a cusp: the correspondence does not smooth anything out, it moves the singularity to the other side. But the cusp is only in view from the right chart, and a dual curve plotted in the wrong one looks smooth while being nothing of the kind. Block [11] locates the singular point projectively and then checks both charts, which is the only honest way to make the claim.
When the dual is not a hypersurface
For a plane curve the dual is again a curve, and for most X the dual is a hypersurface. Not always. If every tangent hyperplane is tangent along a positive-dimensional locus, π₂ contracts, and X∨ comes out too small.
Dual defect. def(X) = codim X∨ − 1. It is zero exactly when the dual is a hypersurface. Biduality still holds; it is only the expectation of codimension one that fails.
The standard example is the Segre embedding ℙ¹ × ℙ² ⊂ ℙ⁵, which is the rank-one locus in the space of 2 × 3 matrices. Its dual is the locus where a general 2 × 3 matrix drops rank, and for a non-square matrix that locus has codimension 2, not 1. The defect is 1. Block [8].
Squareness is the whole of it: for m × m matrices the same construction gives the determinant, a genuine hypersurface with defect 0. The determinant is the dual variety of the Segre ℙm−1 × ℙm−1 — the cleanest statement in this chapter, and the one to keep if only one survives.
Legendre, in an affine chart
Everything above is projective. Take an affine chart and the same correspondence turns into an object this book has used since Chapter 1 without naming it as duality.
On J¹(ℝ,ℝ) with coordinates (x, u, p) and contact form α = du − p dx, define
The Legendre transformation is a contactomorphism, and an involution. L*α = −α, so L preserves ker α; and L ∘ L = id.
Checked
Both by direct computation, blocks [9] and [10]. The second is worth pausing on. Chapter 22 proved biduality (X∨)∨ = X from the isomorphism ℙT*ℙⁿ ≅ ℙT*(ℙⁿ)∨ exchanging the two projections. In an affine chart that isomorphism is L, and biduality is the identity L ∘ L = id, which takes one line to verify. The classical theorem and the undergraduate exercise are the same statement in two charts.
And L is not the prolongation of anything happening on the base. It sends the base coordinate x to the slope coordinate and the slope p to a base coordinate; a transformation of (x, u) alone could never do that. Hold that fact for one section.
Bäcklund, and Chapter 5's Theorem 5.1
Chapter 5 identifies the Sator Square with J¹(ℝ, ℝ²) and states, as Theorem 5.1, that every contact symmetry of J¹(ℝ, ℝⁿ) is either a prolonged point transformation or a genuine contact transformation mixing x, yi and pi, the latter being the Galilean Contact Transformations of Chapter 6. As printed, the second alternative is empty on the space the chapter works in.
Bäcklund (1876). For n ≥ 2, every contact transformation of J¹(ℝ, ℝⁿ) — every diffeomorphism preserving the Cartan distribution — is the prolongation of a point transformation of the base. Genuine contact transformations exist only for n = 1.
The mechanism is dimensional, and visible before any theorem is invoked. A contact structure on a manifold of dimension 2m+1 is a maximally non-integrable field of hyperplanes — corank exactly one. On J¹(ℝ, ℝⁿ) the Cartan distribution is cut out by the n independent forms αi = dyi − pi dx, so its corank is n.
| n | dim J¹(ℝ,ℝⁿ) | Cartan rank | corank | contact? |
|---|---|---|---|---|
| 1 | 3 | 2 | 1 | yes |
| 2 | 5 | 3 | 2 | no — a rank-2 Pfaffian system |
| 3 | 7 | 4 | 3 | no |
So J¹(ℝ, ℝ²) is not a contact manifold at all, and "contact symmetry" there means symmetry of a Pfaffian system of rank 2. Bäcklund's theorem says such systems are rigid. The exceptional case n = 1 is exceptional precisely because corank 1 is the only case in which the distribution is a hyperplane field — and the Legendre transformation of §23.5 is the witness that the exception is not vacuous.
Theorem 5.1, as printed, is false for n ≥ 2. Its dichotomy offers a second alternative — genuine contact transformations of J¹(ℝ, ℝ²) — which Bäcklund's theorem shows is empty. Two repairs are available, and they are not equivalent.
(a) Keep the space. On J¹(ℝ, ℝ²) the correct statement is that every symmetry of the Cartan distribution is a prolonged point transformation. The Sator correspondence of §5.2 survives untouched; what does not survive is calling the resulting group "genuine contact transformations." It is the prolonged point group — a perfectly good group, and what Chapter 6 has in fact been computing.
(b) Move to the space where the claim is true. Genuine contact transformations exist on contact 3-manifolds. §23.7 shows the dm³ manifold is one, so the Chapter 6 programme is not lost — it is relocated to the space the framework started in.
Both repairs are honest, and (b) is the one this book takes, for the reason in the next section. Chapter 5 is amended to state Bäcklund and to name its own space correctly; the Sator material, which is a mnemonic and was never load-bearing, is unaffected.
The dm³ form is the standard one
The framework's contact form is α = dz − r² dθ on ℝ³, with r² = x² + y². Three facts about it were assumed rather than checked, and one of them was assumed wrongly.
(i) r² dθ = x dy − y dx, so α = dz − x dy + y dx.
(ii) α ∧ dα = −2 dx ∧ dy ∧ dz. The form is contact on all of ℝ³, including the axis r = 0.
(iii) The shear ψ(x,y,z) = (x, y, z + xy) followed by y ↦ y/2 carries α to dz + y dx. Hence (ℝ³, ker α) is contactomorphic to the standard contact ℝ³ — that is, to J¹(ℝ, ℝ) with u = z and p = −y.
Checked
Blocks [1]–[3]. Point (ii) corrects a reading the corpus had carried: in cylindrical coordinates α ∧ dα = −2r dz ∧ dr ∧ dθ, which vanishes at r = 0, and this had been read as a degeneracy on the axis. The vanishing factor is the Jacobian of the polar chart. In Cartesian coordinates the coefficient is the constant −2, and there is no degeneracy anywhere.
Point (iii) is what makes repair (b) available. The dm³ manifold is J¹(ℝ, ℝ) up to contactomorphism; n = 1 is exactly Bäcklund's exceptional case; so genuine contact transformations — the Legendre transformation among them — do act on it. The Chapter 6 programme was being run on the one space where the theorem it needs is false, and it belongs on the space Chapter 3 already built.
What this does to the open question of §22.8
Chapter 22 left one row open: whether the dm³ contact structure is the real locus of a holomorphic contact structure on a projective variety. Point (iii) narrows it substantially. ℙT*ℙ² is a projective variety of dimension 3 carrying a canonical holomorphic contact structure, and in an affine chart that structure is the tautological form du − p dx. By (iii) the dm³ structure is contactomorphic to exactly that. So the structure algebraises.
What remains open, stated narrowly. Algebraising the structure is not algebraising the system. Half of the residual question has since been settled: the attractor is Legendrian. Its tangent is (0, 1, 1) in (r, θ, z), and α(γ̇) = ż − r²θ̇ = 1 − r², which vanishes at r = 1 — so γ* lies inside the contact distribution rather than transverse to it. (Chapter 3 had called it a closed Reeb orbit; the Reeb field of α is ∂z, which is the complementary condition, and Chapter 3 is corrected.) What is still open is algebraic: on the attractor ż = θ̇ = 1, so z = θ and the curve is a helix, whose parametrisation is transcendental in these coordinates even though the circle r = 1 is not. Whether the identification of (iii) carries it to an algebraic Legendrian in ℙT*ℙ², and whether the operator chain is induced by a morphism, is not settled here. ch24-verify.py, block [1].
Honest inventory
| Claim | Status | What it rests on |
|---|---|---|
| π₁ has constant fibre dimension on the smooth locus | Classical | definition of Con(X) |
| dual of Cd is the discriminant, degree 2(d−1) | Classical | GKZ ch. 1; recomputed, block [7] |
| d∨ = d(d−1) − 2δ − 3κ | Classical | Plücker; four cases checked, block [5] |
| dual of the cuspidal cubic is 4a³ + 27b²c | Computed | elimination, block [6] |
| that dual is itself cuspidal, at [0 : 0 : 1] | Computed | singular locus, block [11]; invisible in the chart b = 1 |
| 4a³ + 27b² is dual, discriminant and bifurcation set | Computed | block [12] — the bridge to Ch 24 |
| def(ℙ¹ × ℙ²) = 1 | Classical | rank locus of 2 × 3 matrices, block [8] |
| L*α = −α and L ∘ L = id | Computed | blocks [9], [10] |
| J¹(ℝ,ℝⁿ) is contact iff n = 1 | Classical | corank count, block [4] |
| Theorem 5.1 as printed is false for n ≥ 2 | Correction | Bäcklund — see §23.6 |
| α = dz − r²dθ is contact on all of ℝ³ | Computed | block [2]; corrects the axis reading |
| dm³ form is contactomorphic to J¹(ℝ,ℝ) | Computed | block [3], explicit shear |
| the dm³ structure algebraises | Follows | (iii) plus the canonical form on ℙT*ℙ² |
| the dm³ attractor is Legendrian | Computed | α(γ̇) = 1 − r² = 0 at r = 1; ch24-verify.py [1] |
| the dm³ flow algebraises | OPEN | Legendrian yes; the helix z = θ is transcendental in these coordinates |
Every row marked Computed is regenerated by ch23-verify.py, which sits beside this chapter and exits non-zero on any failure. It caught two errors while the chapter was being written. In §23.2: the resultant Res(f, f′) has degree 2d−1, not 2(d−1), and the discriminant is that divided by the leading coefficient. And in §23.3, the dual of the cuspidal cubic had been written in the chart b = 1 and plotted there, where it is a smooth graph — under a caption asserting that its cusp was visible. The singular point [0 : 0 : 1] is precisely the point that chart omits. Both figures printed above are the corrected ones. A dual curve plotted in a chart that misses its singular locus is the failure mode this chapter is about, arriving from the inside.