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Polar degree of hypersurfaces with 1-dimensional singularities

We prove a formula for the polar degree of projective hypersurfaces in terms of the Milnor data of the singularities, extending to 1-dimensional singularities the Dimca-Papadima result for isolated singularities. We discuss the semi-continuity of the polar degree in deformations, and we classify the homaloidal cubic surfaces with 1-dimensional singular locus. Some open questions are pointed out along the way.


introduction
For any projective hypersurface V ⊂ P n , defined by a homogeneous polynomial f : C n+1 → C of degree d, the polar degree is defined as the topological degree of the gradient map, also known as the Gauss map: (1) grad f : P n \ Sing (V ) → P n .
The polar degree depends only on V and not on the defining polynomial f , as conjectured by Dolgachev [Do] and proved by Dimca and Papadima in [DP]. One therefore denotes it by pol(V ).
The concept of polar degree goes back to 1851 when Hesse studied hypersurfaces with vanishing Hessian [Hes1,Hes2], which is equivalent to pol(V ) = 0, and to Gordon and Noether [GN] (see also §5.3 for more comments).
The gradient maps (1) of polar degree equal to 1 are examples of Cremona transformations since they are birational maps. The corresponding hypersurfaces V were called homaloidal, and Dolgachev [Do] found the list of all (reduced) projective plane curves.
In the beginning of the 2000's, whereas the algebraic approach was dominant before that date, Dimca and Papadima [DP] gave the following topological interpretation: For any projective hypersurface V , if H is a general hyperplane with respect to V , then the homology of the affine part H n−1 (V \ H) is concentrated in dimension n − 1, and: (2) pol(V ) = rank H n−1 (V \ H).
The classification of all homaloidal hypersurfaces with isolated singularities was carried out by Huh [Huh] and confirmed a conjecture stated by Dimca and Papadima [DP, Di] that there are no homaloidal hypersurfaces with isolated singularities besides the smooth quadric and the plane curves found by Dolgachev. Huh [Huh] proves and uses the bound: where the Milnor number µ p (V ∩ H) is > 0 as soon as p ∈ Sing V . This holds at any p such that V is not a cone of apex p.
More recently, the authors together with Steenbrink classified in [SST] the hypersurfaces with isolated singularities and polar degree 2, confirming Huh's conjectural list [Huh]. The finiteness of the range of (n, d) in which there may exist hypersurfaces with isolated singularities and polar degree k > 2 has been also proved in [SST].
Still for isolated singularities, Dimca and Papadima [DP] had shown the formula: which allows to compute the polar degree in terms of the Milnor data of the singular points.
In this paper we consider hypersurfaces with 1-dimensional singular locus. We extend the formula (4) and compute the polar degree from the Milnor data of the singularities (Theorem 2.1): by using the study of the hypersurfaces with 1-dimensional singular locus in [ST5] and in earlier papers, see [Si3]. The notations are explained in §2.
We use the semi-continuity of the polar degree in deformations (Proposition 4.1) in order to compare the polar degree of V with 1-dimensional singularity with its deformation V t (of Yomdin type) to a hypersurface with isolated singularities (Corollary 4.2). Then pol(V t ) may serve as upper bound for the polar degree of V . As a consequence, we derive a Lefschetz type inequality for the slicing with a generic hyperplane H gen : We highlight the concept of special point of a hypersurface introduced in [ST6]. These are points p where the complex link of (V, p) is non-trivial, it appears that they are finitely many, and they provide lower bounds for the polar degree. We show that, in the 1-dimensional singularity case, one can detect them by the Milnor number jump of the transversal singularity type.
We treat cubic surfaces in §5. We compute all polar degrees in a topological way and prove that there are 3 homaloidal cubic surfaces with non-isolated singularities.
We discuss homaloidal hypersurfaces with 1-dimensional singularities and transversal type A 1 , and we state and discuss the question of the existence of hypersurfaces with a 1-dimensional singularities and polar degree equal to zero, in this manner coming back to Hesse's problem cited in the beginning.

Formula for pol(V ) in case of 1-dimensional singularities
Let V ⊂ P n be a projective hypersurface with singular locus Sing (V ) of dimension ≤ 1 and Sing (V ) = Σ c ∪ Σ is , where Σ c is a non-degenerate curve with irreducible components Σ c i , i = 1, . . . , r, and Σ is is the finite set of isolated singularities. Each curve branch Σ c i of Sing (V ) has a generic transversal type, of transversal Milnor fibre F ⋔ i and Milnor number denoted by µ ⋔ i . Each branch Σ c i contains a finite set Q i of points where the transversal type is not the generic one, and which we have called special points (see also §3.2 for a more general definition). We denote by A q the local Milnor fibre of the hypersurface germ (V, q) for q ∈ Q := ∪ r i=1 Q i , and byΣ c i the normalisation of Σ c i . At each point q ∈ Q i there are finitely many locally irreducible branches of the germ (Σ c i , q), we denote by γ i,q their number and let γ i := q∈Q i γ i,q . In other words, γ i is equal to the number of "punctures" inΣ i \Q.
This setting has been studied in [ST5], in particular a formula for χ(V ) has been found. It depends on the isolated singularities of V , on the special nonisolated singularities, on the topology of the curve components of Sing (V ), and on the transversal singularity type of each such curve component.
Under the above notations, our following formula generalises the Dimca-Papadima formula (4) for isolated singularities to the case of a 1-dimensional singular set.
Theorem 2.1. Let V ⊂ P n be a hypersurface of degree d with a 1-dimensional singular set. Then: where g i is the genus of the normalizationΣ c i of Σ c i , and where deg Σ c i denotes the degree of Σ c i as a reduced curve. Proof. The Euler characteristic χ(V ) has been computed in [ST5] by using a local pencil of hypersurfaces V ε : We have introduced in [ST5] the vanishing homology 2 of projective hypersurfaces V with dim Sing (V ) = 1, defined as: and in its study we have established the following Euler characteristic formula for Since V ε is a nonsingular hypersurface in P n , its Euler characteristic is that of the smooth hypersurface of degree d in P n , namely: Let now H be a generic hyperplane with respect to the canonical Whitney stratification of V . Since V ∩ H has only isolated singularities, one has the well-known formula: To prove our claimed formula for pol(V ) we use the above formulas and the relation (2) of [DP] in the form: The above formula is useful as soon as the local topological information is available. We show this is some examples taken from [ST5]. [ST5,Ex 7.1a]: V := {x 2 z + y 2 w = 0} ⊂ P 3 . Then Sing (V ) is a projective line with transversal type A 1 and with two special points of type D ∞ , the Milnor fibre of which is homotopy equivalent to S 2 . Applying formula (5) [ST5, Ex 7.1b]: V := {x 2 z + y 2 w + t 3 = 0} ⊂ P 4 . Then Sing (V ) = P 1 is also a projective line but here with transversal type A 2 and with two special points having Milnor fibre S 3 ∨ S 3 . From formula (5) we get pol(V ) = 16 − 0 − (0 + 2 + 4 − 2) · 2 − 4 = 4.

Special points and lower bounds
If one looks for hypersurfaces with small polar degree, for instance homaloidal, it is useful to have lower bounds for the polar degree in terms of the singularities of V or its dual. An example is Huh's bound (3) for isolated singularities, which has been used in [Huh] and in [SST] to determine all hypersurfaces with isolated singularities which have polar degree 1 or 2. A more general bound has been found in [ST6].
3.1. Quantisation of the polar degree, after [ST6]. Let us fix a Whitney stratification W of V ⊂ P n . This depends only on the reduced structure of V . We consider hyperplanes H which are stratified transversal to all strata of V , except at finitely many points. One says that the hyperplane H = {l = 0} is admissible iff its non-transversality locus consists of isolated points only, and if the affine polar locus Γ(l, f ) has dimension ≤ 1.
If H has an isolated non-transversality at q ∈ V then the linear function l : C n → C defining H near q has a stratified isolated singularity at q. Consequently, it's local Milnor-Lê fibre B ε ∩ V ∩ {l = s}, for some s close enough to l(q), has the homotopy type of a bouquet of spheres of dimension n − 2, cf Lê's results [Lê2]. We denote by α q (V, H) it's Milnor-Lê number. In case H = H gen is a general hyperplane through q, then α q (V, H gen ) is the Milnor number of the complex link of V at q, denoted by α q (V ), and we have Let us denote by α(V, H) the sum q α q (V, H) for all points p of the non-transversality locus of the admissible hyperplane H. We have shown in [ST6]: Theorem 3.1. [ST6,Theorem 5.4] If H is an admissible hyperplane, then there is the following decomposition of the polar degree: where β(V, H) is related to the vanishing cycles of the affine polynomial map which is the restriction of f : C n+1 → C to the affine hyperplane plane l = 1.
Let us consider now the special case of hypersurfaces V with 1-dimensional singular locus. In this case, the numbers α q (V, H) can be described by sectional Milnor numbers, as follows: (i) For isolated singular points p ∈ V we have α p (V, H) = µ p (V ∩ H) by definition.
(ii) For points p on the 1-dimensional singular set, we have the formula proved in [ST2]: where the points p j are the singular points of the restriction f p |Hs in B ε ∩ V ∩ H s , where f p = 0 is a reduced local equation for V . The number α p (V, H) was called the Milnor number jump at p for the family of functions f p |Hs . Even if H 0 may be not the most generic at p, for s = 0 the hyperplane H s is a locally generic slice of any branch Σ c i , and so the above equality reads: where µ ⋔ i is the generic transversal Milnor number and does not depend on the choice of the point of the irreducible component Σ c i . Here the sum is taken over all local branches at p.
3.2. The special points of V . For any singular projective hypersurface V , we say after [ST6] that p ∈ V a special point of V if α p (V ) > 0. It turns out that the set of special points V spec is finite.
In case dim p Sing (V ) = 1, the set of special points of V consists of the isolated singularities of V together with the set Q of points p where the generic transversal Milnor number is jumping (see §2), which is equivalent to the inequality α p (V, H) > 0. Indeed, it is well-known that α p (V, H) = 0 implies that Sing(V ) is smooth at p and that V ∩ H s is a µ-constant local family of hypersurfaces. This is equivalent to A'Campo's "non-splitting principle" [AC].
It is known (see [ST6,Remark 4.3]) that the set of admissible hyperplanes for f containing a fixed point p ∈ Sing V is a Zariski-open subset of the set of all hyperplanes through p. The following useful lower bound then holds in general, as a consequence of Theorem 3.1: Corollary 3.2. [ST6, Corollary 6.6] Let V ⊂ P n be a projective hypersurface which is not a cone of apex p. Then: In particular, if V is not a cone, then: We shall apply this lower bound result in case of homaloidal hypersurfaces, see §5.2.

Semi-continuity of pol(V ) under deformations
The following result is general, it holds for any singular locus and whatever pol(V ) might be, including the case pol(V ) = 0. This could be folklore, but we didn't find a precise reference. Proof. Let b ∈ P n be a regular value for grad f : P n \ Sing V → P n and let (grad f ) −1 (b) = {a 1 , · · · , a k }, k ≥ 0. There exist disjoint compact neighborhoods U i of a i and U ′ of b such that grad f : (U i , a i ) → (U ′ , b) is a diffeomorphism. Next take s so close to 0 such that grad f s |U i are still diffeomorphisms, and that grad f s (U The restriction grad f s : i Z i → W is a diffeomorphism on each component Z i , and has topological degree pol(V ). Moreover b is still a regular value for this restriction, but perhaps not anymore for the full map grad f s : P n \ Sing V s → P n . Arbitrarily close to b there exist points b ′ which are regular values for grad f s . Then the number of counter-images #(grad f s ) −1 (b ′ ) is pol(V s ) and (grad f s ) −1 (b ′ ) contains at least one point in each Z i . This shows the inequality pol(V s ) ≥ pol(V ).
Yomdin type families and polar degree. Let V = {f = 0} have at most 1dimensional singularities, and let us consider the particular deformation 3 f s = f + sl d of degree d, where H = {l = 0} is a hyperplane such that H ∩ V has isolated singularities only. By direct computation, one can show that for generic s = 0 (actually except of a finite number of values of s ∈ C), the singular locus of V s = {f s = 0} is the set H ∩Sing V , and therefore Sing V s consists of isolated singular points. We may then show: Corollary 4.2. If V has at most 1-dimensional singularities, and if H gen is a generic hyperplane, then: whered i := int(Σ c i , H gen ) denotes the global intersection number. Proof. By [Yo], one has the following formula for the local "Yomdin series" g N = g +sl N of a function germ g : (C n , 0) → (C, 0) with 1-dimensional singular locus Σ = ∪ j Σ j , where l is a general linear form, see also [Si3]: is the Milnor number of g N , where b n−1 (g) and b n−2 (g) are the respective Betti numbers of the local Milnor fibre of g, and where µ ⋔ j is the transversal Milnor number of the local branch Σ j . The sum is taken over the branches Σ j of Σ, and d j = int 0 (Σ j , {l = 0}) is the intersection multiplicity of {l = 0} and Σ j with reduced structure.
This formula was proved initially for "high enough N" depending on the polar ratios of the discriminant of the map (l, g). It was shown in [Si2] that actually it holds for N greater or equal to the highest polar ratio. Moreover, from the proof of this formula in [Si2,pag 187], one deduces the following statement: Lemma 4.3. If the polar locus Γ(l, g) is empty, then formula (12) holds for any N ≥ 2.
Let us remark that if Γ(l, g) = ∅ then b n−1 (g) = 0 and b n−2 (g) = µ(g |l=0 ). Thus formula (12) becomes: We will apply Lemma 4.3 and formula (13) at some point p ∈ H gen ∩Σ c i , for the following data: g := f p , where f p = 0 is a local equation of V at p, N = d and g d := f s = f + sl d as defined above.
We remark that the genericity of H gen with respect to SingV implies that the local polar locus Γ p (l, f ) is empty, and therefore Lemma 4.3 holds indeed for our data at p.
Moreover, since H gen is generic, Σ c i is smooth at p ∈ H gen ∩ Σ c i , thus in formula (13) we have a single term in the sum, a single multiplicity d i = 1, a single transversal Milnor number µ ⋔ i , and b n−2 (f p ) = µ ⋔ i . Therefore formula (13) reduces to: Taking the sum over all points p ∈ Sing V s , we get: Inserting this in the Dimca-Papadima formula (4) for V s , we get: By applying now the semi-continuity result Proposition 4.1, we obtain (10).
Next, the Bruce-Wall classification [BW] of irreducible cubics with nonisolated singularities contains: (CN) cone over a nodal curve, (CC) cone over a cuspidal curve both with pol(V ) = 0 because they are cones, and two other cases, for which we use our formula (5) to compute pol(V ): where the singular set is a projective line with two special points of type D ∞ in the first case, and a single special point of type J 2,∞ in the second case 4 . The irreducible surface (E2) with the J 2,∞ point is also mentioned in [CRS,Section 3.2] as Y (1, 2), related to a rational scroll surface and in [CRS,Example 4.7] as a sub-Hankel surface.
Among the reducible cubics there are only the following three cases with non-zero polar degree: (QP) The union of a smooth quadratic with a general hyperplane: pol(V ) = 2, (QT) The union of a smooth quadratic with a tangent hyperplane: pol(V ) = 1, (CP) The union of a quadratic cone and a general hyperplane: pol(V ) = 1.
For these computations one can use the union formula (16) for the union of two hypersurfaces V i ⊂ P n : where H be a generic hyperplane with respect to V . This is a consequence of (2) and of the inclusion-exclusion principle for the Euler-characteristic. It was observed in several papers, e.g. [FM,Cor. 4.3].
All the other reducible cubics are cones and thus have pol(V ) = 0. In particular, there are only three homaloidal cubic surfaces, all with nonisolated singularities.

5.2.
Homaloidal singularities with transversal type A 1 . A necessary condition for V to be homaloidal is that α q (V ) = 1 for any special point p ∈ V , cf Corollary 3.2. This restriction tells for instance that all isolated singularities of V must be of type A k , as observed in [Huh] in case of isolated singularities only, but the argument holds at any isolated singular point, cf Corollary 3.2, whatever the other singularities of V might be.
For instance, a hypersurface V with 1-dimensional singular locus can be viewed (by slicing) as a 1-parameter family of hypersurfaces with isolated singularities. Therefore, a natural classification question is: what are the 1-parameter families of hypersurfaces with isolated singularities with Milnor jumps equal to 1?
It is known that in certain generic transversal types (e.g.Ẽ 6 ,Ẽ 7 , orẼ 8 ) this jump do not exist. Let us look now to the case of the transversal type A 1 , more precisely the subclass of "smooth singular locus and generic transversal type A 1 ".
Proposition 5.1. Let V ⊂ P n be a hypersurface with singular locus Sing (V ) = Σ c ⊔ Σ is such that Σ c is a smooth projective line with transversal type A 1 , and Σ is is a finite set of isolated points.
Then α p (V ) = 1 if and only if the hypersurface germ (V, p) is a J k,∞ singularity, i.e has local equation: y 2 (y − x k ) + z 2 1 + · · · + z 2 n−2 = 0. In particular, if V is homaloidal, then all its special points on Σ c must be of type J k,∞ , and all its isolated singular points of type A k .
Proof. The generic transversal type is A 1 by hypothesis, with transversal Milnor number µ ⋔ = 1. Then by (6) the condition α p (V, H) = 1 is equivalent to µ(V ∩ H, p) = 2, which means that the hypersurface germ (V ∩ H, p) is an A 2 -singularity. One may then apply the local classification of line singularities [Si1] and obtain, firstly that the local function defining (V, p) is R-equivalent to y 2 g(x, y) + z 2 1 + · · · + z 2 n−2 , with g(0, 0) = 0, and secondly that y 2 g(x, y) is R-equivalent to y 2 (y − x k ) for some k. This shows ou first claim.
To show our second claim, we apply Corollary 3.2 as in the first paragraph of this subsection, together with the result that we have just proved. 5.3. On projective cones and other hypersurfaces with polar degree zero. Let V ⊂ P n be defined by f (x 0 , x 1 , · · · , x n ) = g(x 1 , · · · , x n ) = 0. This is one of the possible definitions of a projective cone V over the hypersurface V ′ ⊂ P n−1 given by g(x 1 , · · · , x n ) = 0. The point q = [1 : 0 : · · · : 0] is called the apex of the cone. From the definiton (1) it follows that pol(V ) = 0. Let V ′ have isolated singularities only, with Milnor numbers µ 1 , · · · , µ r . Then Sing (V ) = ∪ r i Σ c i , where each Σ c i is a projective line with transversal type µ i , and such that all these lines intersect only at the apex q.
Let us compute pol(V ) with formula (5). First observe that g i = 0, γ i = 1 and deg Σ c i = 1. There are no isolated critical points. The only special point is the apex q, where a local affine equation is given by g = 0. As computed in [Si3], one has χ(A q ) = (−1) n ((d − 1) n − d i µ ⋔ i ). We therefore get pol(V ) = (d − 1) n − d i µ ⋔ i − (d − 1) n + d i µ ⋔ i = 0, which is indeed what was expected since V is a cone.
What can be said about other hypersurfaces with polar degree zero? Historically, Hesse claimed that hypersurfaces with vanishing Hessian are always projective cones. In 1875 Gordan and Noether [GN] gave several examples with polar degree zero but not cones.
All known examples seem to have a singular locus of dimension at least 2. It follows from the lower bound results [ST6] that the polar degree zero can not occur if V contains isolated singularities. The following question is open: Do there exist hypersurfaces with 1-dimensional critical set and polar degree 0?
We have the following partial result, by [ST6,Corollary 6.9]: Let V be not a cone and pol(V ) = 0. Then V has no special point. In particular V has no isolated singularity (besides its non-isolated singularities).