Evolute of a Projective Hypersurface

• The evolute of a projective hypersurface is the envelope of its normal lines, determined by a chosen Euclidean structure involving a hyperplane at infinity and a nonsingular quadric.
• In the hypersurface case, the evolute coincides with the branch locus of the normal-line map, capturing classical focal points and centers of curvature through a singular image construction.
• Thom–Boardman singularity theory and precise bundle formulas yield enumerative results, relating the degree and stratified geometry of evolutes to classical curvature and contact phenomena.

Searching arXiv for the specified paper and closely related work on projective hypersurfaces, envelopes, and higher-contact loci. The evolute of a projective hypersurface is a projectively formulated focal locus obtained from the family of its normal lines, once projective space has been equipped with an auxiliary notion of perpendicularity. In the framework developed in "Envelopes and evolutes" (Piene, 11 Aug 2025), the evolute is not an intrinsic construction of projective space alone: it depends on a chosen hyperplane at infinity and a nonsingular quadric in that hyperplane, together forming a projective “Euclidean structure.” For a nonsingular hypersurface $X\subset \mathbb P^n$, the evolute is the envelope of the family of normal lines, equivalently the branch locus of the natural map from the total space of those normal lines to the ambient projective space (Piene, 11 Aug 2025). In higher codimension, the same paper places the construction in the Thom–Boardman hierarchy: for a variety of dimension $r$ in $n$-space, the evolute is the image of the $(n-r)$-th iterated singular locus $\Sigma^{1,\dots,1}$ of the normal-space map (Piene, 11 Aug 2025).

1. Projective definition and basic setup

The modern projective definition begins with a family of linear spaces in projective space. Let $V$ be a complex vector space of dimension $n+1$, so that the ambient space is $\mathbb P(V)\cong \mathbb P^n$. Let $X$ be a nonsingular projective variety of dimension $r$, with a morphism

$r$0

If $r$1 is a rank $r$2 locally free sheaf on $r$3 together with a surjection

$r$4

then one obtains a projective bundle

$r$5

whose fiber over $r$6 is a projective $r$7-space in $r$8, and an induced map

$r$9

The envelope $n$0 of this family is defined to be the branch locus of $n$1, or equivalently the image of the first Thom–Boardman locus

$n$2

(Piene, 11 Aug 2025).

This definition is decisive for the projective theory because it replaces the naive union of members of the family by a singular image construction. The envelope is therefore a discriminantal object: it records where the family fails to vary transversely. In the hypersurface case, this branch-locus viewpoint coincides with the evolute, because the normal spaces are lines and no higher iteration is required (Piene, 11 Aug 2025).

The generality of this setup also clarifies the status of the hypersurface case. For arbitrary codimension, the first branch locus is only the first envelope of the family of normal spaces. The evolute proper lies deeper in the singularity tower. For hypersurfaces, by contrast, $n$3, so $n$4, and the first singular image already is the evolute (Piene, 11 Aug 2025).

2. Projective perpendicularity and normal lines

Projective space has no intrinsic Euclidean orthogonality, so the notion of a normal line cannot be defined without extra structure. The projective theory therefore begins by choosing a hyperplane at infinity

$n$5

where $n$6 is an $n$7-dimensional quotient, together with a nonsingular quadric

$n$8

This pair is called a “Euclidean structure” on $n$9 (Piene, 11 Aug 2025).

The quadric $(n-r)$0 defines a polarity on $(n-r)$1. If $(n-r)$2 is an $(n-r)$3-dimensional linear subspace, then $(n-r)$4 denotes the $(n-r)$5-dimensional polar subspace of $(n-r)$6 with respect to $(n-r)$7. This induced polarity supplies the missing notion of perpendicularity for projective linear spaces and thereby makes projective normal spaces well-defined (Piene, 11 Aug 2025).

Let $(n-r)$8 be a nonsingular variety of dimension $(n-r)$9, assumed to be in general position with respect to $\Sigma^{1,\dots,1}$0 and $\Sigma^{1,\dots,1}$1. Writing

$\Sigma^{1,\dots,1}$2

the Euclidean normal bundle is defined by

$\Sigma^{1,\dots,1}$3

Using the isomorphism $\Sigma^{1,\dots,1}$4 induced by $\Sigma^{1,\dots,1}$5, there is a natural surjective map

$\Sigma^{1,\dots,1}$6

hence a projective bundle $\Sigma^{1,\dots,1}$7 and a map

$\Sigma^{1,\dots,1}$8

(Piene, 11 Aug 2025).

Geometrically, the fiber over $\Sigma^{1,\dots,1}$9 is the projective normal $V$0-space

$V$1

One first intersects the tangent space with the hyperplane at infinity, then takes the polar subspace with respect to $V$2, and finally spans with the point $V$3 itself. For a hypersurface $V$4, the normal $V$5-spaces are lines, so the construction produces a family of normal lines (Piene, 11 Aug 2025).

The dependence on $V$6 is essential. The evolute of a projective hypersurface is therefore not intrinsic to projective space in the absence of this extra structure. A plausible implication is that distinct choices of projective Euclidean structure may lead to distinct evolutes even for the same underlying hypersurface.

3. Evolute as branch locus and Thom–Boardman singularity

For the normal-space map

$V$7

the envelope of the family of normal spaces is the branch locus of $V$8, namely the image of $V$9. The paper then defines the evolute of $n+1$0 to be the image of the $n+1$1-th iterated singular locus

$n+1$2

with $n+1$3 ones (Piene, 11 Aug 2025). In particular, if $n+1$4 is a hypersurface, then $n+1$5, so

$n+1$6

This formulation gives the hypersurface case a particularly transparent interpretation. The total space $n+1$7 may be viewed as the incidence space of points lying on the normal lines of $n+1$8, and the evolute is the locus in the ambient projective space where the projection from this incidence space becomes singular. In geometric language already used in the paper, the evolute is the locus of focal points on the normal lines (Piene, 11 Aug 2025).

The same construction organizes finer strata by iteration. The first singular image gives the envelope; the image of $n+1$9 gives a cuspidal locus; the image of $\mathbb P(V)\cong \mathbb P^n$0 gives a still higher singular stratum. For a surface in $\mathbb P(V)\cong \mathbb P^n$1, these become respectively the evolute surface, its cuspidal curve, and the cusps of that curve (Piene, 11 Aug 2025). This singularity-theoretic reinterpretation is one of the paper’s central conceptual contributions.

The Thom–Boardman viewpoint is made explicit through standard polynomials in the relative Chern classes

$\mathbb P(V)\cong \mathbb P^n$2

For the first three loci, the Thom polynomials are

$\mathbb P(V)\cong \mathbb P^n$3

(Piene, 11 Aug 2025). These formulas permit the systematic computation of classes and degrees of envelopes and higher cuspidal loci.

4. Bundle formulas and enumerative results

A general formula is given for the class and degree of the envelope of a family of linear spaces. Proposition 1.3 in (Piene, 11 Aug 2025) states

$\mathbb P(V)\cong \mathbb P^n$4

and

$\mathbb P(V)\cong \mathbb P^n$5

For hypersurfaces one substitutes $\mathbb P(V)\cong \mathbb P^n$6, the Euclidean normal bundle (Piene, 11 Aug 2025).

If $\mathbb P(V)\cong \mathbb P^n$7 is a smooth hypersurface of degree $\mathbb P(V)\cong \mathbb P^n$8, in general position with respect to the chosen $\mathbb P(V)\cong \mathbb P^n$9 and $X$0, then

$X$1

and the degree of the evolute is

$X$2

(Piene, 11 Aug 2025). The paper states that this recovers Trifogli’s formula. Since for hypersurfaces the evolute is $X$3, this is the degree formula for the evolute of a projective hypersurface.

The quadratic case is singled out explicitly: $X$4 when $X$5 (Piene, 11 Aug 2025). This recovers the classical degrees of the evolute of a plane conic and of a quadric surface.

For smooth surfaces $X$6, the formulas become more detailed because the singularity tower beyond the branch locus is still visible on the evolute. The paper gives

$X$7

$X$8

$X$9

where $r$0 is the evolute surface, $r$1 its cuspidal curve, and $r$2 the cuspidal locus of that cuspidal curve (Piene, 11 Aug 2025). These formulas arise from known Thom polynomials for $r$3, $r$4, and $r$5, together with Chern-class computations for $r$6.

5. Classical geometry, focal interpretation, and special loci

The projective theory is explicitly situated in the historical line from Huygens, Monge, Darboux, and Salmon. For plane curves, the classical evolute is the envelope of normal lines, or equivalently the locus of centers of curvature. The projective hypersurface construction extends this principle by replacing Euclidean orthogonality with polarity relative to the quadric at infinity (Piene, 11 Aug 2025).

For a smooth surface in $r$7, Salmon called the evolute the “surface of centres” or “centro-surface,” since it is the locus of spherical curvature centers, the focal points (Piene, 11 Aug 2025). In this surface case, each normal line generically has two focal points, corresponding to the two principal curvature directions; each normal line is tangent to the evolute at each focal point; and the normal lines are therefore bitangents of the evolute (Piene, 11 Aug 2025). This yields a double-cover interpretation inside $r$8.

The paper also recalls a classical exceptional phenomenon: the two focal points on a normal line may coincide. These are the umbilic points of the surface. Salmon’s formula for their number, ignoring points at infinity, is

$r$9

(Piene, 11 Aug 2025). The corresponding points on the evolute are discussed in connection with singularities of Lagrangian type $r$00 for the map

$r$01

(Piene, 11 Aug 2025). The paper does not fully work this out via Thom polynomials, but it identifies an additional singular stratum beyond the cuspidal curve.

These facts clarify a frequent misconception. The evolute is not merely the union of centers of curvature expressed in Euclidean coordinates, nor is it purely a dual object such as the dual variety. In the projective framework of (Piene, 11 Aug 2025), it is a branch locus or singular image associated with the normal-line congruence. The classical focal interpretation survives, but only after the projective Euclidean structure has been fixed.

6. Related higher-contact loci and adjacent constructions

Two other papers clarify neighboring, but distinct, projective-differential constructions. Atsushi Ikeda’s "The varieties of tangent lines to hypersurfaces in projective spaces" (Ikeda, 2010) does not define the evolute of a projective hypersurface, but it develops the incidence geometry of pointed lines with prescribed order of contact. For a hypersurface $r$02, the basic objects are

$r$03

where $r$04 is the point-line incidence variety (Ikeda, 2010). The case $r$05 gives tangent lines, $r$06 gives lines with at least third-order contact, and higher $r$07 encode higher-contact directions (Ikeda, 2010).

Locally, $r$08 is cut out by vanishing of the first $r$09 coefficients in the Taylor expansion of $r$10 along the line: $r$11 equivalently,

$r$12

as inferred from formula (2.1) in the paper (Ikeda, 2010). For general hypersurfaces and $r$13 prime to $r$14, $r$15 is smooth of dimension

$r$16

(Ikeda, 2010). This does not produce an evolute, but it supplies the incidence-theoretic infrastructure for studying tangent and higher-contact families from which envelope phenomena may be extracted.

A different adjacent construction appears in "The geometry of the flex locus of a hypersurface" (Busé et al., 2018). That paper also does not define an evolute, focal hypersurface, or caustic, but studies the flex locus: the set of points at which there exists a line having order of contact at least $r$17 with the hypersurface. For a hypersurface

$r$18

it introduces the expansion

$r$19

and proves that the flex locus is cut out by a polynomial $r$20 of degree

$r$21

(Busé et al., 2018). For surfaces in $r$22, this yields Salmon’s degree $r$23 for the flecnodal polynomial (Busé et al., 2018). In the generic case, the paper shows that a generic flex point of a generic hypersurface carries a unique flex line; if $r$24, its order of contact is exactly $r$25 (Busé et al., 2018).

These two papers are relevant because evolutes, flex loci, and higher-contact loci all organize failures of generic transversality. Yet they should not be conflated. The evolute in the sense of (Piene, 11 Aug 2025) is built from normal spaces after choosing a projective Euclidean structure. Ikeda’s spaces $r$26 concern tangent and higher-contact lines (Ikeda, 2010), while Busé–D’Andrea–Sombra–Weimann study abnormal osculation and the flex locus via resultants (Busé et al., 2018). A plausible synthesis is that they provide complementary incidence and elimination frameworks around the same broad projective-differential problem: identifying singular directions, singular images, and exceptional contact phenomena on projective hypersurfaces.