Schmidt rank and singularities

We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous polynomials of degree d (assuming that the characteristic does not divide d(d-1)). We then use this to relate the Schmidt rank of a homogeneous polynomial (resp., a collection of homogeneous polynomials of the same degree) with the codimension of the singular locus of the corresponding hypersurface (resp., intersection of hypersurfaces). This gives an effective version of Ananyan-Hochster's Theorem A from arXiv:1610.09268.


Introduction
Let k be a field (of any characteristic) and a polylinear form, where V i are finite dimensional vector spaces over k.Equivalently, we view P as a tensor in V * 1 ⊗ . . .⊗ V * d .Definition 1.1.(i) We say that P ≠ 0 has Schmidt rank 1 if there exists a partition [1, d] = I ⊔J into two nonempty parts and polylinear forms P I (v i 1 , . . ., v ir ), P J (v j 1 , . . ., v js ), where v a ∈ V a , I = {i 1 < . . .< i r }, J = {j 1 < . . .< j r }, such that P = P I ⋅ P J .In general the Schmidt rank of P , denoted as rk S (P ), is the smallest number r such that P = ∑ r i=1 P i with P i of Schmidt rank 1.For a collection of tensors P = (P 1 , . . ., P s ) we define the Schmidt rank rk S (P ) as the minimum of Schmidt ranks of nontrivial linear combinations of (P i ).
(ii) Given a collection of nonempty subsets I 1 , . . ., I r ⊂ [1, d] and a collection (P I 1 , . . ., P Ir ), where P I i is a polylinear form on ∏ a∈I i V a , we denote by (P I 1 , . . ., P Ir ) ⊂ V * 1 ⊗ . . .⊗ V * d and call this the tensor ideal generated by P I 1 , . . ., P Ir , the subspace of polylinear forms of the form for some polylinear forms Q J i on ∏ b∈J i V b , where The Schmidt rank of a tensor, along with a set of related notions, such as slice rank, G-rank, analytic rank, as well as the version of Schmidt rank for homogeneous polynomials also known as strength (see below), has been a subject of study in many recent works (see [1], [4], [5], [8], [9] and references therein).One of the goals of this paper is to establish a precise relation (in the case of an algebraically closed base field k) between this notion and the codimension of the singular locus of the corresponding hypersurface, thus giving an effective version of Ananyan-Hochster's theorem [2,Theorem A].
In [10] (where the authors consider the case d = 3), this number is called geometric rank of P .Using [10,Thm. 3.2], one can see that it does not depend on the ordering of the variables v 1 , . . ., v d .
The proof of the following theorem follows closely the proof of a similar result in the case where k = C and P is symmetric, given in [11].We modified the proof so that it would work in arbitrary characteristic and also streamlined some parts of the original argument.The fact that the original proof can be adapted to arbitrary characteristic was also pointed out in [11,Sec. 4].
Theorem 1.2.(i) Let g ′ (P ) denote the codimension in V 2 × . . .× V d of the Zariski closure of the set of k-points in Z P (so g(P ) ≤ g ′ (P ) and g(P ) = g ′ (P ) if k is algebraically closed).Then one has rk S (P ) In the appendix we prove another version of theorem 1.2 with better bounds for d ≥ 6.Even though Schmidt applied the above result to symmetric tensors P corresponding to homogeneous polynomials, we observe that in the symmetric case it is natural to modify the relevant variety Z P , and that this leads to much better estimates on the rank.
Let f be a homogeneous polynomial of degree d on a finite-dimensional k-vector space V .The Schmidt rank of f , denoted as rk S (f ), is the minimal number r such that f = ∑ r i=1 g i h i , where g i and h i are homogeneous polynomials of positive degrees.Note that if rk S (f ) = r then in terminology of [2], f has strength r −1.For a collection f = (f 1 , . . ., f s ), the Schmidt rank rk S (f ) is defined as the minimum of Schmidt ranks of nontrivial linear combinations of f i .
Let H f (x)(⋅, ⋅) denote the Hessian form of f given by the second derivatives of f .It is a symmetric bilinear form on V depending polynomially on a point x ∈ V .The symmetric analog of the variety Z P is the subvariety The symmetric analog of (1.1) is the inequality Similarly for a collection f = (f 1 , . . ., f s ) of homogeneous polynomials of degree d, we define the subvariety Z sym f ⊂ V × V as the set of (v, x) such that the map With the same assumptions as in (i), assume also that k is algebraically closed.Then For k algebraically closed, we prove another version of theorem 1.3 in the appendix with better bounds for d ≥ 6.The invariant g sym (f ) can be viewed as an invariant measuring singularities of the polar map x ↦ (∂ i f (x)) 1≤i≤dim V of f (see Sec. 3.3).We also prove that g sym (f ) is related to the codimension of the singular locus of the hypersurface f = 0. Namely, let us set c(f More generally, for a collection f = (f 1 , . . ., f s ), let us set where V (f ) ⊂ V is the subscheme defined by the ideal (f 1 , . . ., f s ).We also consider the related invariant c ′ (f ) ∶= codim V S(f ), where S(f ) ⊂ V is the locus where the Jacobi matrix of (f 1 , . . ., f s ) has rank < s.It is easy to see that Here is our main result concerning the relation between the Schmidt rank and the codimension of the singular locus.It can be viewed as a more precise version of the corresponding result in [9] in the case of an algebraically closed field of sufficiently large (or zero) characteristic, as well as an effective version of a result of Ananyan and Hochster (see [2, Theorem A(a)]), playing a central role in their proof of Stillman's conjecture.
Theorem 1.4.Assume that char(k) does not divide d.Let c k (f ) be the codimension in V of the Zariski closure of the k-points of Sing(f = 0).
(ii) Assume k is algebraically closed.Then for a collection f = (f 1 , . . ., f s ), we have Combining Theorem 1.4(i) with [2, Theorem A(c)] we get the following result.
Corollary 1.5.Assume k is algebraically closed and char(k) does not divide d!.For i = 2, . . ., d, let W i ⊂ k[V ] i be a subspace of forms of degree i.
Assume that for some m ≥ 1, one has Then every sequence of linearly independent homogeneous forms in W is regular and the corresponding complete intersection subscheme in V satisfies Serre condition R m .
Note that without any assumptions on the characteristic on k we are able to estimate in terms of c(f For a homogeneous polynomial f (x) of degree d on V and a vector v ∈ V , we denote by ∂ v f (x), the derivative of f in the direction v. Our next result concerns ∂ v f for generic v.
Theorem 1.6.Let f be a homogeneous polynomial of degree d ≥ 3. Assume that k is algebraically closed of characteristic not dividing ), and for generic v 1 , . . ., v s ∈ V , the derivatives (∂ v 1 f, . . ., ∂ vs f ) define a (resp., normal) complete intersection of codimension s in V .
In the appendix we prove another version of theorem 1.6 with better bounds for d ≥ 6.In Section 3.4 we will also discuss the relation of the invariant g sym (f ) with the polar map of f and with the Gauss map of the corresponding projective hypersurface.
2. Schmidt rank of polylinear forms 2.1.Elementary observations.First, let us prove (1.1) and its symmetric version (1.2).We denote by k[V ] the space of polynomial functions on a vector space V and by k Proof.(i) If r = rk S (P ) then there exists a decomposition as in Definition 1.1.Swapping some I i with J i if necessary, we can assume that 1 ∈ I i for all i.Then the intersection of r hypersurfaces For a subset of indices We have the following simple observation.
Lemma 2.2.Let V ′ 1 ⊂ V 1 be a subspace of codimension c, and let (ℓ 1 , . . ., ℓ r ) be a basis of the orthogonal to V ′ 1 in V * 1 .Suppose we have tensors In particular, rk S (P ) ≤ rk S (P V ′ 1 ×V 2 ×...×V d ) + c.Proof.This follows immediately from the fact that the tensor ideal (ℓ i i = 1, . . ., c) is exactly the kernel of the restriction map be a morphism of vector bundles on a scheme X.For every r ≥ 0, we have a natural morphism where for a section ψ of V ∨ we denote by Then the image of κ r is contained in ker(f ).
(ii) Assume in addition that V 1 and V 2 are trivial vector bundles and that for some point Proof.(i) This is equivalent to the statement that ι f ∨ φ r+1 κ r (α) = 0 for any local section Choosing a trivialization of the target of s, we can write s as a collection of global sections of V 1 , which has the required properties.
2.3.Higher derivatives.Let V be a finite dimensional vector space, and let k[V ] denote the ring of polynomial functions on V .
For each f ∈ k[V ], each n ≥ 1 and v 0 ∈ V , we define the homogeneous form of degree n on V , f (viewed as a function of v, for fixed v 0 ) with respect to the degree grading on k[V ], so that we have (finite) Taylor's decomposition We refer to f . ., g c be a set of elements in the ideal I X of X, with linearly independent differentials at v 0 .Then for any f ∈ I X and any n ≥ 1, the form f Proof.Without loss of generality we can assume that v 0 = 0. Set A = k[V ], and let Â denote the completion with respect to the ideal of the origin (the ring of formal power series).Then the key point is that I X ⋅ Â is generated by g 1 , . . ., g c .Indeed, this follows from the fact that the local homomorphism of local regular k-algebras A m (g 1 , . . ., g c ) → O X,v 0 (where m is the maximal ideal of v 0 in A) induces an isomorphism on tangent spaces, so it induces an isomorphism of completions.Note that higher derivatives make sense for elements of Â (as components in A n = k[V ] n ), so the assertion follows once we express any element of I X in the form ∑ i g i h i for some h i ∈ Â.
We also need to work with certain polylinear forms of mixed derivatives.Assume that we have a decomposition V = V 1 ⊕. ..⊕V n .Then we have the induced direct sum decomposition In particular, when we apply this to the mth derivative of f at v 0 , we get a polylinear form which we call the (V i 1 , . . ., V im )-mixed derivative of f at v 0 .
Lemma 2.5.In the situation of Lemma 2.4, assume in addition that and any collection of indices belongs to the tensor ideal generated by (g i ) Proof.This follows easily from Lemma 2.4.

Dimension count. Let us change the notation to
We will denote W = W 1 × . . .× W d−2 , and consider the variety Z P ⊂ V × W of all (v, w) such that P (u, v, w) = 0 for all u ∈ U.
Let Z be an irreducible component of the Zariski closure of the set of k-points Z P (k) (with reduced scheme structure), such that codim V ×W Z = g ′ (P ), and let Z W ⊂ W denote the closure of the image of Z under the projection π W ∶ V × W → W (also with reduced scheme structure).Then k-points are dense in Z W .
We can think of P as a linear map from U ⊗ V to the space of polynomial functions on W , hence, it gives a morphism of trivial vector bundles over W , and for w ∈ Z W , π −1 W (w) ∩ Z P can be identified with ker(P W (w)). Let U ⊂ Z W denote the nonempty open subset where P W has maximal rank that we denote by r.Then over U the cokernel of P W is locally free over Z W , hence, the kernel of P W is a subbundle K ⊂ V ⊗ O. Denoting by tot U (K) the total space of the bundle K over U , we have tot Hence, we have dim Step 1. Choosing a general k-point.Shrinking the open subset U ⊂ Z W above, we can assume that U is smooth.Since k-points are dense in Z W we can choose a k-point Step 2. The first set of key tensors.Set Since w 0 is a smooth point of Z W , we can choose c elements g 1 , . . ., g c in the ideal Thus, for each a = 1, . . ., c, and each nonempty subset of indices we can consider the polylinear forms, obtained as mixed derivatives at w 0 , g a,I ∶= g Step 3. Setting up the key identity.Let us set k = dim V − r.Applying Lemma 2.3(ii) to the morphism of trivial vector bundles (2.2) over Z W , we find global sections v 1 (w), . . ., v k (w) ∈ V ⊗ k[Z W ], such that v 1 (w 0 ), . . ., v k (w 0 ) form a basis of S V , and P (u, v i (w), w) = 0 for any u ∈ U and w ∈ Z W , i = 1, . . ., k. Since which we denote in the same way.Now we define a collection of U * -valued polynomials on W , ) will be the key identity that we will use.
Step 4. The second set of key tensors.We will consider certain mixed derivatives of v i (w), viewed as V -valued polynomials on W . Namely, for each Since (v i (w 0 )) form a basis of S V , there exists a unique operator We extend C I in any way to an operator V → Hom(W I , V ), which we still denote by C I .Note that we can also view C I as a linear map For an ordered collection of disjoint subsets I 1 , . . ., I p ⊂ [1, d − 2], we consider the composition We allow the case of an empty collection, i.e., p = 0, in which case we just get the identity map V → V .
Step 5. Differentiating the key identity.For each , which completes w i 1 ⊗ . . .⊗ w ip by the components w 0 j in the factors W j with j ∈ I. Let us prove by induction on p = 0, . . ., d − 2 that for any where on the right we have the tensor ideal generated by the specified elements.Note that all the subsets I t are supposed to be nonempty.
The base of induction p = 0 is clear, since P (u, v, w 0 1 , . . ., w 0 d−2 ) = 0 for any u ∈ S U and v ∈ V .Assume that p > 0 and the assertion holds for p − 1.Let us fix a subset Now let us equate the (W i 1 , . . ., W ip )-mixed derivatives at w 0 of both sides of the key identity (2.4).We get the following equality in U * ⊗ W * I 0 : (2.5) Note that by Lemma 2.5, belong to the tensor ideal generated by g a,I ′ with 1 ≤ a ≤ c and I ′ ⊂ I 0 , I ′ ≠ ∅.Note also that the term in the sum in (2.5) corresponding to J = ∅ has zero restriction to S U .Hence, we get Now the induction assumption implies that P S U ⊗S V ⊗ι(I 0 )W I 0 belongs to the tensor ideal generated by g a,I ′ with I ′ ⊂ I 0 , I ′ ≠ ∅ and by the restrictions of ℓ j ○ C I 1 . . .C Is with s ≥ 1 (where I 1 ⊔ . . .⊔ I s is a proper subset of I 0 ).By Lemma 2.2, adding (ℓ j ) to the generators of the tensor ideal we get the required assertion about P S U ⊗V ⊗ι(I 0 )W I 0 .
Step 6.Conclusion of the proof for a single tensor.Now using the result of the previous step for p = d − 2, we get where θ n is the number of ordered collections of disjoint nonempty subsets I 1 ⊔. ..⊔I p ⊊ [1, n] (with p ≥ 1).By Lemma 2.2, this implies that

Now we recall that
r + c = g ′ (P ) (see (2.3)).Hence, we get Step 7. The case of several tensors.Now assume that k is algebraically closed.Suppose we are given a collection P = (P 1 , . . ., P s ) of polylinear forms on V 1 × . . .× V d .For a nonzero collection of coefficients c = (c 1 , . . ., c s ) in k, we set The key observation is that where we can consider c as points in the projective space P s−1 .As we have already proved, for each c, Similarly, we define an operation for n 1 + . . .np) , by letting f (n 1 ,...,np) to be the component of multidegree (n 1 , . . ., n p ) in f (v 1 + . . .+ v p ).For example, the Hessian symmetric form on V (depending polynomially on x ∈ V ).
We will use two properties of this construction, which are easy to check:

Proof of Theorem 1.3.
It will be convenient to denote one copy of V as X in the product V × V = V × X.In addition, we view Step 1. Dimension count and choosing a general k-point.Let Z be an irreducible component of the Zariski closure of the set of k-points Z sym f (k), such that codim V ×X Z = g ′ sym (f ), and let Z X ⊂ X denote the closure of the image of Z under the projection p 2 ∶ V × X → X.As before, we choose a nonempty smooth open subset U ⊂ Z X over which H f has maximal rank r, so that p −1 2 (U ) ∩ Z is a vector bundle of rank dim V − r over U , in particular, codim X Z X + r = g ′ sym (f ).We choose a k-point x 0 in U ⊂ Z X and set Step 2. The first set of key polynomials.Set c ∶= codim X Z X .Since x 0 is a smooth point of Z X , we can choose c elements g 1 , . . ., g c in the ideal I Z X ⊂ k[X] with linearly independent derivatives at x 0 .Thus, for each a = 1, . . ., c, and for 1 ≤ i ≤ d − 2, we consider the derivatives (g a ) Step 3. Setting up key identity.Let us set k = dim V − r.Applying Lemma 2.3(ii) to the morphism of trivial vector bundles ) form a basis of S, and We lift v i (x) to polynomials in V ⊗k[X], which we denote in the same way.Now we define a collection of U * -valued polynomials on X, Step 4. The second set of key forms.For each 1 ≤ m ≤ d − 2, we consider higher derivatives of v i at x 0 , viewed as V -valued polynomials on X.
)) form a basis of S, there exists a linear operator We extend C m in any way to an operator V → V ⊗ k[X] m , which we still denote by C m .For m 1 + . . .+ m p ≤ d − 2, we consider the composition We allow the case of an empty collection, i.e., p = 0, in which case we just get the identity map V → V .
Finally, we denote by ℓ 1 , . . ., ℓ r ∈ V * a basis in the orthogonal subspace to S. For m 1 + . . .+ m p ≤ d − 2 and for j = 1, . . ., r, we consider the elements Note that for an empty collection, i.e., for p = 0, we just get ℓ j ∈ V * .
Step 5. Differentiating the key identity.Let us prove by induction on p = 0, . . ., d − 2 that one has where on the right we have the ideal generated by the specified elements.
The base of induction p = 0 is clear, since f (1,1,d−2) (u, v, x 0 ) = 0 for any u ∈ S and v ∈ V .Assume that p > 0 and the assertion holds for p − 1.Now let us equate the pth derivatives at x = x 0 of both sides of (3.1).We get the following equality in U * ⊗ k[X] p : The left-hand side belongs to the ideal generated by (g a ) (m) x 0 (x) with 1 ≤ a ≤ c and 1 ≤ m ≤ p.Note also that the term corresponding to q = p in the right-hand side has zero restriction to u ∈ S. Hence, we get Now the induction assumption implies that f (1,1,p,d−2−p) (u, v, x, x 0 ) S×S×X belongs to the ideal generated by (g a ) (m) x 0 (x) for 1 ≤ a ≤ c, 1 ≤ m ≤ p and by the restrictions to S ×X of (ℓ j ○ C m 1 . . .C ms )(v, x) with s ≥ 1, m 1 + . . .+ m s < p, 1 ≤ j ≤ r.By Lemma 2.2, adding (ℓ j ) to the generators of the ideal we get the required assertion about f (1,1,p,d−2−p) (u, v, x, x 0 ) S×V ×X .
Step 6.Conclusion of the proof for a single polynomial.Now using the result of the previous step for p = d − 2, we get , where θ sym n is the number of (m 1 , . . ., m s ), with s ≥ 1, Step 7. The case of several polynomials.Now assume that k is algebraically closed, and we are given a collection f = (f 1 , . . ., f s ) of homogeneous polynomias on V of degree d.For a nonzero collection of coefficients c = (c 1 , . . ., c s ) in k, we set As in the non-symmetric case, the key observation is that where we can consider c as points in the projective space P s−1 .Using the case of a single polynomial, we deduce that as claimed.
3.3.Relation to singularities.Now we will relate g sym (f ) to c(f ), the codimension in V of the singular locus of the hypersurface f = 0.
(ii) Since we are comparing dimensions of algebraic varieties, without loss of generality, we can assume that k is algebraically closed.
By part (i), we have g sym (f ) ≤ c(F ), where F = f (2,g−2) .It is easy to see that if Thus, it remains to check that f (2,d−2) (v, x) is a linear combination of d + 1 (resp., d, if d is odd and char(k) ≠ 2) polynomials of the form f (A i (v, x)), for some linear surjective maps Let us view f (v + x) as a nonhomogeneous function of v, g(v) = g 0 + g 1 + . . .+ g d of degree ≤ d (with coefficients in k[V ]).Now picking any d + 1 distinct elements λ 0 , . . ., λ d ∈ k, we can express g 0 , . . ., g d as linear combinations of g(λ 0 v), . . ., g(λ d v) (since the corresponding linear change is given by the Vandermonde matrix).
In the case when d is odd and char(k) ≠ 2, we can similarly express the components of even degree, (g 2i ) i≤(d−1) 2 as linear combinations of g 0 = g(0) and (g(λ i v) + g(−λ i v)) 2, for 1 ≤ i ≤ (d − 1) 2, where (λ i ) are nonzero constants such that (λ 2 i ) are all distinct.It remains to observe that g 2 = f (2,d−2) and that each g(λv) = f (λv + x) is of the required type.
(iii) This follows from the relation Indeed, this implies that the intersection of Z sym f with the diagonal V ⊂ V × V is exactly the singular locus of f = 0, which gives the claimed inequality.Now let us consider the case of a collection f = (f 1 , . . ., f s ) of homogeneous polynomials on V of degree d.We consider the corresponding family of hypersurfaces in V , f c = 0 parametrized by the projective space P s−1 .It is clear that for the locus S(f ) ⊂ V where the rank of Jacobi matrix of (f 1 , . . ., f s ) is < s, we have Assume in addition that char(k) does not divide d − 1.Then Proof.(i) This follows from Proposition 3.1(i) due to (3.3).
(ii) Since S(f ) has codimension c ′ (f ) in V , it follows that for some a ≤ s − 1, there exists an a-dimensional subvariety X ⊂ P s−1 such that Applying Proposition 3.1(ii) we see that for each c ∈ X, one has this implies the assertion.(iii) If (f 1 , . . ., f s ) define a complete intersection then by the Jacobi criterion of smoothness, we have SingV In particular, we have an inclusion SingV (f ) ⊂ S(f ), so If we assume in addition that char(k) does not divide d − 1 then the intersection of Z sym f with the diagonal V ⊂ V × V is exactly S(f ).Hence, we get which is a locally trivial fibration whose fibers are irreducible of dimension n(s − 1) + s, is exactly the fiber over (v 1 , . . ., v s ) of the projection Z (s) → V s .For generic v 1 , . . ., v s , only the components of Z (s) dominant over V s will play a role, and we deduce that Proof of Theorem 1.6.(i) By Lemma 3.4 with s = 1, c(∂ v f ) ≥ g sym (f ).Hence, by Theorems 1.4(i) and 1.3, = df x .Thus, g sym (f ) measures the degeneracy of this map.
More precisely, for any morphism φ ∶ X → Y between smooth connected varieties, let us define the Thom-Boardman rank 1 of φ, denoted as rk T B (φ), as follows.Consider the subvariety Z φ in the tangent bundle T X of X consisting of (x, v) such that dφ x (v) = 0. Then we set rk T B (φ) = codim T X Z φ .
Note that rk T B (φ) ≤ r, where r is the generic rank of the differential of φ, however, the inequality can be strict.By definition, g sym (f ) = rk T B (φ f ).As is well known, the generic rank of dφ f = H f is related to the dimension of the projective dual variety X * of the projective hypersurface associated with f (more precisely, dim X * +2 is the generic rank of H f over the hypersurface f = 0).However, it is easy to see that g sym (f ) can be much smaller than the generic rank of φ f .For example, if q 1 (x) and q 2 (y) are nondegenerate quadratic forms in two different groups of variables (x 1 , . . ., x n ), (y 1 , . . ., y n ), then rk S (q 1 (x)q 2 (y)) = 1, so g sym (q 1 (x)q 2 (y)) ≤ 4. On the other hand, the generic rank of φ q 1 (x)q 2 (y) is 2n (assuming the characteristic of k is ≠ 2, 3). 1 The name is due to the relation with Thom-Boardman stratification in singularity theory, see [6].
Example 3.5.In the case d = 3, the Schmidt rank of f is equal to its slice rank s(f ), i.e., the minimal s such that there exists a linear subspace L ⊂ V of codimension s contained in (f = 0).Thus, for a cubic form f , assuming that k is algebraically closed of characteristic ≠ 2, 3, we get from Theorem 1.3 and from (1.2) that s(f ) ≤ rk T B (φ f ) ≤ 4s(f ).
If f is a general homogeneous polynomial of degree d then we still have rk S (f ) = s(f ) (see [4]), so for such f , assuming k to be algebraically closed of characteristic not dividing (d − 1)d, we have It seems that the invariant rk T B (φ) deserves to be studied more.For example, we do not know whether it is always true that rk T B (φ) = dim X for a finite morphism φ between smooth projective varieties in characteristic zero.Note the following corollary from Prop.3.1(iii).In particular, if the projective hypersurface associated with f is smooth then Let V f ⊂ PV denote the projective hypersurface associated with f .In [7] the authors consider (for k = C) the closed locus S ≥r ⊂ V f where the co-rank of the Hessian H f is ≥ r.They prove that if V f is smooth then for r(r + 1) ≤ dim V , the subvariety S ≥r (V ) is nonempty and codim V f S ≥r (V ) ≤ r(r + 1) 2.
Using Corollary 3.6 we get the inequality codim V f S ≥r (V ) ≥ r − 1.
If V f is smooth then the projectivization of the restriction of φ f to (f = 0) can be identified with the Gauss map It is easy to check that if char(k) does not divide d(d−1) then for any point x ∈ (f = 0) ⊂ V one has ker(d(φ f ) x ) ⊂ T x (f = 0) and the natural projection ker(d(φ f ) x ) → ker(dγ x ) is an isomorphism.Thus, the above inequalities can be viewed as restrictions on possible degeneracies of the Gauss map of V f (which is finite by a result of Zak in [13]).

A
.L. is supported by the National Science Foundation under Grant No. DMS-1926686 and by the Israel Science Foundation under Grant No. 2112/20.A.P. is partially supported by the NSF grant DMS-2001224, and within the framework of the HSE University Basic Research Program and by the Russian Academic Excellence Project '5-100'.