On singularities of third secant varieties of Veronese embeddings
OON SINGULARITIES OF THIRD SECANT VARIETIES OF VERONESEEMBEDDINGS
KANGJIN HAN
Abstract.
In this paper we study singularities of third secant varieties of Veronese embedding v d ( P n ), which corresponds to the variety of symmetric tensors of border rank at most three in( C n +1 ) ⊗ d . Contents
1. Introduction 12. Singularities of third secant of v d ( P n ) 22.1. Preliminaries 32.2. Outline for the proof of main theorem 62.3. Non-degenerate orbits : n = 2 case 72.4. Degenerate case : binary forms 102.5. Defining equations of Sing( σ ( X )) 12References 121. Introduction
For a projective algebraic variety X ⊂ P W , the k -th secant variety σ k ( X ) is defined by(1.1) σ k ( X ) = (cid:91) x ··· x k ∈ X P (cid:104) x · · · x k (cid:105) ⊂ P W where (cid:104) x · · · x k (cid:105) ⊂ W denotes the linear span of the points x · · · x k and the overline denotes Zariskiclosure. Let V be an ( n + 1)-dimensional complex vector space and W = S d V be the subspace ofsymmetric d -way tensors in V ⊗ d . Equivalently, we can also think of W as the space of homogeneouspolynomials of degree d in n + 1 variables. When X is the Veronese embedding v d ( P V ) of rank onesymmetric d -way tensors over V in P W , then σ k ( X ) is the variety of symmetric d -way tensors ofborder rank at most k (see Subsection 2.1 for terminology and details).If X is an irreducible variety and σ k ( X ) its k -secant variety, then it is well known that(1.2) Sing( σ k ( X )) ⊇ σ k − ( X ) , (e.g. see [˚Ad87, Corollary 1.8]). Equality holds in many basic examples, like determinantal varietiesdefined by minors of a generic matrix, but the strict inequality also holds for some other tensors(e.g. just have a look at [MOZ15, Corollary 7.17] for the case σ ( X ) when X is the Segre embedding P V × · · · × P V r or [AOP12, Figure 1, p.18] for the third secant variety of Grassmannian G (2 , Mathematics Subject Classification.
Key words and phrases. singularity, secant variety, Veronese embedding, Segre embedding.The author was supported by Basic Science Research Program through the National Research Foundation of Korea(grant No. 2012R1A1A2038506), the POSCO Science Fellowship of POSCO TJ Park Foundation, and the DGISTStart-up Fund of the Ministry of Science, ICT and Future Planning (No. 2016010066). a r X i v : . [ m a t h . AG ] J a n herefore, it should be very interesting to compute more cases and to give a general treatmentabout singularities of secant varieties. Further, the knowledge of singular locus is known to bevery crucial to the so-called identifiablity problem , which is to determine uniqueness of a tensordecomposition (see [COV14, Theorem 4.5]). It has recently been paid more attention in thiscontext. In this paper, we deal with the case of third secant variety of Veronese embeddings, σ ( v d ( P V )).From now on, let X be the Veronese variety v d ( P V ) in P S d V = P N with N = dim C S d V − (cid:0) n + dn (cid:1) −
1. One could ask the following problem:
Problem 1.1.
Let V = C n +1 . Determine for which triple ( k, d, n ) it does hold thatSing( σ k ( v d ( P V ))) = σ k − ( v d ( P V ))for every k ≥ , d ≥ n ≥ σ k ( v d ( P V ))) if it is not the case.We’d like to remark here that our question is a set-theoretic one. First, it is classical that theanswer to Problem 1.1 is true for the binary case (i.e. n = 1) (see e.g. [IK99, Theorem 1.45]) andalso for the case of quadratic forms (i.e. d = 2) (see e.g. [IK99, Theorem 1.26]). In the case of k = 2, Kanev proved in [Kan99, Theorem 3.3] that this holds for any d, n . Thus, we only need totake care of the cases of k ≥ , d ≥ n ≥
2. Look at the table in Figure 1.2.
Singularities of third secant of v d ( P n )Choose any form f ∈ S d V . We define the space of essential variables of f to be (cid:104) f (cid:105) := { ∂ ∈ V ∨ | ∂ ( f ) = 0 } ⊥ in V . So, f also belongs to S d (cid:104) f (cid:105) and dim (cid:104) f (cid:105) is the minimal number of variables in which we canexpress f as a homogeneous polynomial of degree d (we also call dim (cid:104) f (cid:105) the number of essentialvariables of f , see e.g. [C05]). Note that dim (cid:104) f (cid:105) = 1 means f ∈ v d ( P V ) by definition. We oftenabuse f to denote the point [ f ] in P S d V represented by it. We say a form f ∈ σ ( X ) \ σ ( X ) to be degenerate if dim (cid:104) f (cid:105) = 2 and non-degenerate otherwise. We denote the locus of degenerate formsby D . We begin this section by stating our main theorem for the cases of k = 3 , d ≥ n ≥ Theorem 2.1 (Singularity of σ ( v d ( P n ))) . Let X be the n -dimensional Veronese variety v d ( P V ) in P N with N = (cid:0) n + dd (cid:1) − . Then, the following holds that the singular locus Sing( σ ( X )) = σ ( X ) as a set for all ( d, n ) with d ≥ and n ≥ unless d = 4 and n ≥ . In the exceptional case d = 4 ,for each n ≥ the singular locus Sing( σ ( v ( P V ))) is D ∪ σ ( v ( P V )) , where D denotes the locusof all the degenerate forms f (i.e. dim (cid:104) f (cid:105) = 2 ) in σ ( v ( P V )) \ σ ( v ( P V )) . ( k , d , n ) Singular locus of σ k ( v d ( P n )) Comment & Reference ( ≥ , ≥ , σ k − Classical - case of binary forms, [IK99, Theorem 1.45]( ≥ , , ≥ σ k − Symmetric matrice case, [IK99, Theorem 1.26](2 , ≥ , ≥ σ [Kan99, Theorem 3.3](3 , , ≥ σ Aronhold case - Thm. 2.10 ( n = 2), Coro. 2.11 ( n ≥ , ≥ , σ Thm. 2.12 + Thm. 2.14(3 , , ≥ D ∪ σ Only exceptional case ( d = 4), Thm. 2.14(3 , ≥ , ≥ σ Thm. 2.1
Figure 1.
Singular locus of σ k ( v d ( P n )). .1. Preliminaries.
For the proof, we recall some preliminaries on (border) ranks and geometryof symmetric tensors and list a few known facts on them for future use.First of all, the scheme-theoretic equations defining σ ( v d ( P V )) come from so-called symmetricflattenings unless d = 3. In the case of d = 3, we need Aronhold’s equation (2.3.1) additionally (seee.g. [LO13]). Consider the polynomial ring S • V = C [ x , . . . , x n ] (we call this ring S ) and consideranother polynomial ring T = S • V ∨ = C [ y , . . . , y n ], where V ∨ is the dual space of V . Define thedifferential action of T on S as follows: for any g ∈ T d − k , f ∈ S d , we set(2.1) g · f = g ( ∂ , ∂ , . . . , ∂ n ) f ∈ S k . Let us take bases for S k and T d − k as X I = 1 i ! · · · i n ! x i · · · x i n n and Y J = y j · · · y j n n , (2.2)with | I | = i + · · · + i n = k and | J | = j + · · · + j n = d − k . For a given f = (cid:80) | I | = d a I · X I in S d ,we have a linear map φ d − k,k ( f ) : T d − k → S k , g (cid:55)→ g · f for any k with 1 ≤ k ≤ d −
1, which can be represented by the following (cid:0) k + nn (cid:1) × (cid:0) d − k + nn (cid:1) -matrix:(2.3) a I,J with a I,J = a I + J , in the bases defined above. We call this the symmetric flattening (or catalecticant ) of f . It is easyto see that the transpose φ d − k,k ( f ) T is equal to φ k,d − k ( f ).Given a homogeneous polynomial f of degree d , the minimum number of linear forms l i neededto write f as a sum of d -th powers is the so-called (Waring) rank of f and denoted by rank( f ).The (Waring) border rank is this notion in the limiting sense. In other words, if there is a family { f (cid:15) | (cid:15) > } of polynomials with constant rank r and lim (cid:15) → f (cid:15) = f , then we say that f has borderrank at most r . The minimum such r is called the border rank of f and denoted by rank( f ). Notethat by definition σ k ( v d ( P V )) is the variety of homogeneous polynomials f of degree d with borderrank rank( f ) ≤ k .It is obvious that if f has (border) rank 1, then any symmetric flattening φ d − k,k ( f ) has rank 1.By subadditivity of matrix rank, we also know that rank φ d − k,k ( f ) ≤ r if rank( f ) ≤ r . So, we couldobtain a set of defining equations coming from minors of the matrix φ d − k,k ( f ) for σ r ( v d ( P V )). For σ ( v d ( P V )) and d ≥
4, it is known that these minors are sufficient to cut it scheme-theoretically in[LO13, Theorem 3.2.1 (1)] as follows:
Proposition 2.2 (Defining equations of σ ( v d ( P n ))) . Let X be the n -dimensional Veronese variety v d ( P V ) in P N with N = (cid:0) n + dn (cid:1) − . For any ( d, n ) with d ≥ , n ≥ , σ ( X ) is defined scheme-theoretically by the × -minors of the two symmetric flattenings φ d − , ( F ) : S d − V ∨ → V and φ d −(cid:98) d (cid:99) , (cid:98) d (cid:99) ( F ) : S d −(cid:98) d (cid:99) V ∨ → S (cid:98) d (cid:99) V , where F is the form (cid:88) I ∈ N n +1 a I · X I of degree d as considering the coefficients a I ’s indeterminate. Since there is a natural SL n +1 ( C )-group action on σ ( X ), we may take the SL n +1 ( C )-orbitsinside σ ( X ) into consideration for the study of singularity. And we could also regard a canonicalrepresentative of each orbit as below.First, suppose f ∈ σ ( X ) \ σ ( X ) is a degenerate form (i.e. dim (cid:104) f (cid:105) = 2). Choose x , x as abasis of (cid:104) f (cid:105) . Then, we recall the following lemma emma 2.3. For any d ≥ and n ≥ , any general degenerate form f ∈ σ ( v d ( P V )) \ σ ( v d ( P V )) can be written as x d + α · x d + β · ( x + x ) d , up to SL n +1 ( C ) -action, for some nonzero α, β ∈ C .Proof. Since dim (cid:104) f (cid:105) = 2, let U := (cid:104) f (cid:105) = C (cid:104) x , x (cid:105) , a subspace of V . For such a f ∈ σ ( v d ( P V )) \ σ ( v d ( P V )), it is easy to see that 3 = rank( f ) ≤ rank( f, U ) , where the latter is the border rank of f being considered as a polynomial in S • U . On the otherhand, we also have rank( f, U ) ≤ rank( f ) = 3, because U ⊂ V implies that σ ( v d ( P U )) is containedin σ ( v d ( P V )). Since rank( f, U ) and rank( f, U ) coincide for a general f in the rational normalcurve case (see e.g. [CG01]), we have rank( f, U ) = 3. Thus, for some nonzero λ, µ ∈ C we canwrite f as f ( x , x ) = ( a x + a x ) d + ( b x + b x ) d + { λ ( a x + a x ) + µ ( b x + b x ) } d = X d + ( λµ ) d · X d + λ d · ( X + X ) d , by some scaling and using a SL n +1 ( C )-change of coordinates, which proves our assertion. (cid:3) Remark 2.4.
There are some remarks related to Lemma 2.3 as follows:(a) Note that there does not exist a degenerate form corresponding to an orbit in σ ( v d ( P V )) \ σ ( v d ( P V )) if d ≤
3. In this case, if f is degenerate, then f always belongs to σ ( v d ( P V )),for the φ d − , ( f ) have at most two nonzero rows and all the 3 × φ d − , ( f ) vanish.(b) In fact, in d = 4 case, Lemma 2.3 holds for all degenerate form f ∈ σ ( v ( P V )) \ σ ( v ( P V )),because there exist only rank 3 forms in σ ( v ( P )) \ σ ( v ( P )) (see [CG01] and also [LT10,Section 4]).Now, let’s put main types of canonical representatives for SL n +1 ( C )-orbits together as follows: Theorem 2.5.
There are 4 types of homogeneous forms representing SL n +1 ( C ) -orbits in σ ( v d ( P V )) \ σ ( v d ( P V )) ; (1) x d + x d + x d (2) x d − x + x d (3) x d − x + x d − x (4) x d + αx d + β ( x + x ) d (for some nonzero α, β ∈ C ) .The first three types correspond to all the three non-degenerate orbits. And the last ‘binary’ typecorresponds to a general point of D , the locus of all the degenerate forms, which appears only if d ≥ .Proof. It is straightforward from [LT10, Theorem 10.2], Lemma 2.3 and Remark 2.4. (cid:3)
Let us introduce more basic terms and facts. Let Z ⊂ P W be a variety and ˆ Z be its affinecone in W . Consider a (closed) point p ∈ ˆ Z and say [ p ] the corresponding point in P W . Wedenote the affine tangent space to Z at [ p ] in W by ˆ T [ p ] Z and we define the (affine) conormal spaceto Z at [ p ], ˆ N ∨ [ p ] Z as the annihilator ( ˆ T [ p ] Z ) ⊥ ⊂ W ∨ . Since dim ˆ N ∨ [ p ] Z + dim ˆ T [ p ] Z = dim W anddim Z ≤ dim ˆ T [ p ] Z −
1, we get that dim ˆ N ∨ [ p ] Z ≤ codim( Z, P W ) and the equality holds if and onlyif Z is smooth at [ p ]. This conormal space is quite useful to study the tangent space of Z .Let us recall the apolar ideal f ⊥ ⊂ T . For any given form f ∈ S d V , we call ∂ ∈ T t apolar to f ifthe differentiation ∂ ( f ) gives zero (i.e. ∂ ∈ ker φ t,d − t ( f )). And we define the apolar ideal f ⊥ ⊂ T as f ⊥ := { ∂ ∈ T | ∂ ( f ) = 0 } . t is straightforward to see that f ⊥ is indeed an ideal of T . Moreover, it is well-known that thequotient ring T f := T /f ⊥ is an Artinian Gorenstein algebra with socle degree d (see e.g. [IK99]).In our case, we have a nice description of the conormal space in terms of this apolar ideal asfollows: Proposition 2.6.
Let X be the n -dimensional Veronese variety v d ( P V ) as above and f be any formin S d V . Suppose that f corresponds to a (closed) point of σ ( X ) \ σ ( X ) and that rank φ d − , ( f ) =3 , rank φ d −(cid:98) d (cid:99) , (cid:98) d (cid:99) ( f ) = 3 . Then, for any ( d, n ) with d ≥ , n ≥ we have (2.4) ˆ N ∨ f σ ( X ) = ( f ⊥ ) · ( f ⊥ ) d − + ( f ⊥ ) (cid:98) d (cid:99) · ( f ⊥ ) d −(cid:98) d (cid:99) , where the sum is taken as a C -subspace in T d = S d V ∨ .Proof. First, recall that φ d − k,k ( f ) T = φ k,d − k ( f ). We also note thatker φ d − k,k ( f ) = ( f ) ⊥ d − k and (im φ d − k,k ( f )) ⊥ = ker( φ d − k,k ( f ) T ) = ker φ k,d − k ( f ) = ( f ) ⊥ k . Whenever rank φ d − , ( f ) = 3 and rank φ d −(cid:98) d (cid:99) , (cid:98) d (cid:99) ( f ) = 3, we have(2.5) ˆ N ∨ f σ ( X ) = (cid:104) ker φ d − , ( f ) · (im φ d − , ( f )) ⊥ (cid:105) + (cid:104) ker φ d −(cid:98) d (cid:99) , (cid:98) d (cid:99) ( f ) · (im φ d −(cid:98) d (cid:99) , (cid:98) d (cid:99) ( f )) ⊥ (cid:105) (see [LO13, Proposition 2.5.1]), which proves the proposition. (cid:3) Remark 2.7.
Note that, in case of n = 2 or dim (cid:104) f (cid:105) = 2 (i.e. degenerate form), to computeconormal space ˆ N ∨ f σ ( X ) we only need to consider the symmetric flattening φ d −(cid:98) d (cid:99) , (cid:98) d (cid:99) so that wehave(2.6) ˆ N ∨ f σ ( X ) = ( f ⊥ ) (cid:98) d (cid:99) · ( f ⊥ ) d −(cid:98) d (cid:99) . For n = 2 case, φ d − , ( f ) has only 3 rows, there is no non-trivial 4 × σ ( X ) at f . In case of dim (cid:104) f (cid:105) = 2, we may consider f ∈ C [ x , x ] d and choose bases as (2.2).Then, we could write the matrix of φ d − , and its evaluation at f , φ d − , ( f ) as φ d − , = y d − y d − y · · · y d − n x x x a I ... x n , φ d − , ( f ) = y d − y d − y · · · y d − n x ∗ ∗ · · · ∗ x ∗ ∗ · · · ∗ x · · · x n · · · . So, each 4 × φ d − , (say D ( φ d − , )) has at most rank 2 at f . Hence, we see that all thepartial derivatives in the Jacobian ∂D ( φ d − , ) ∂a I ( f ) = 0for each index I with | I | = d and D ( φ d − , ) doesn’t contribute to span the conormal space of σ ( X ) at f , because at least one row of D ( φ d − , ) (say ( a I a J a K a L )) vanishes at f and theLaplace expansion of D ( φ d − , ) along this row D ( φ d − , ) = ± (cid:18) a I · D I ( φ d − , ) − a J · D J ( φ d − , ) + a K · D K ( φ d − , ) − a L · D L ( φ d − , ) (cid:19) uarantees all the partials of D ( φ d − , ) become zero at f as follows: for example, we see that ± ∂D ( φ d − , ) ∂a I ( f ) = D I ( φ d − , )( f ) + a I ( f ) · ∂D I ( φ d − , ) ∂a I ( f ) − a J ( f ) · ∂D J ( φ d − , ) ∂a I ( f )+ a K ( f ) · ∂D K ( φ d − , ) ∂a I ( f ) − a L ( f ) · ∂D L ( φ d − , ) ∂a I ( f ) = 0 , where a I ( f ) = a J ( f ) = a K ( f ) = a L ( f ) = 0 and D I ( φ d − , )( f ) = 0 because of rank D I ( φ d − , )is at most 2 at f .2.2. Outline for the proof of main theorem.
In this subsection we outline the proof of ourmain theorem (Theorem 2.1).For the locus of non-degenerate orbits in σ ( X ) \ σ ( X ), we may consider a useful reductionmethod through the following arguments: Lemma 2.8.
For every f ∈ σ ( v d ( P n )) ( d, n ≥ ), there exists a linear P = P U ⊂ P n = P V suchthat f ∈ σ ( v d ( P U )) . In particular, for every f ∈ σ ( v d ( P n )) \ σ ( v d ( P n )) , ≤ dim (cid:104) f (cid:105) ≤ .Proof. It is enough to show that dim (cid:104) f (cid:105) ≤
3. When f ∈ σ ( v d ( P n )) (i.e. border rank ≤ φ d − , : S d − C n +1 ∨ → C n +1 has dimension ≤ U , i.e. dim (cid:104) f (cid:105) ≤ (cid:3) Recall that we denote the locus of degenerate forms in σ ( X ) \ σ ( X ) by D (see the paragraphbefore Theorem 2.1 for notation). Then, by Lemma 2.8, we have an obvious corollary for non-degnerate orbits as follows: Corollary 2.9.
For each f ∈ σ ( v d ( P n )) \ ( D ∪ σ ( v d ( P n ))) , There exists a unique 3-dimensionalsubspace U such that f ∈ σ ( v d ( P U )) .Proof. For any non-degenerate form f , which correspond to three orbits in Theorem 2.5, the di-mension of (cid:104) f (cid:105) is exactly 3 and the subspace U = (cid:104) f (cid:105) is uniquely determined as the image of theflattening φ d − , . (cid:3) For the proof of the main theorem, we treat the case of non-degenerate forms and the case ofdegenerate ones separately:
Proof of Theorem 2.1.
Let our P n = P V with V = C (cid:104) x , x , · · · , x n (cid:105) and its dual V ∨ = C (cid:104) y , y , · · · , y n (cid:105) .First, for the locus of non-degenerate forms, we claim that one may reduce the problem to the caseof n = 2. Construct the following map σ ( v d ( P n )) \ ( D ∪ σ ( v d ( P n ))) π −→ G (2 , P n ) . This map is well defined by Corollary 2.9 and for each 2-dimensional P U ⊂ P n , the fiber π − ( P U )is isomorphic to σ ( v d ( P U )) \ ( D ∪ σ ( v d ( P U ))). So, if we prove our theorem for the case n = 2,then the fibers of π are all isomorphic and smooth. Hence π becomes a fibration over a smoothvariety with smooth fibers. This shows that its domain σ ( v d ( P n )) \ ( D ∪ σ ( v d ( P n ))) is smooth,so proving our assertion.Thus, in subsection 2.3 we investigate the non-degenerate orbits with condition n = 2 and provethat there are no more singularity than σ ( v d ( P n )) by Corollary 2.11 ( d = 3) and Theorem 2.12( d ≥ D , in subsection 2.4 we directly compute the dimension ofconormal space ˆ N ∨ σ ( v d ( P n )) using Proposition 2.6 and show that D happens to be the extrasingular locus only when d = 4 , n ≥ (cid:3) .3. Non-degenerate orbits : n = 2 case. Aronhold case ( d = 3 ). Here, we settle the equality in Sing( σ ( v d ( P V ))) ⊇ σ ( v d ( P V )) inour first case d = 3 , dim V = 3 (i.e. n = 2). Note that the equation for the hypersurface σ ( v ( P )) ⊂ P is given by the classical Aronhold invariant (see e.g. [Ott09, LO13]). Map S V → ( V ⊗ Λ V ) ⊗ ( V ⊗ V ∗ ), by first embedding S V ⊂ V ⊗ V ⊗ V , then tensoring with Id V ∈ V ⊗ V ∗ ,and then skew-symmetrizing. Thus, F ∈ S V gives rise to an element of C ⊗ C . In suitable bases,if we write F = φ x + φ x + φ x + 3 φ x x + 3 φ x x + 3 φ x x + 3 φ x x + 3 φ x x + 3 φ x x + 6 φ x x x , then the corresponding skew-symmetric matrix is: φ φ φ − φ − φ − φ φ φ φ − φ − φ − φ φ φ φ − φ − φ − φ − φ − φ − φ φ φ φ − φ − φ − φ φ φ φ − φ − φ − φ φ φ φ φ φ φ − φ − φ − φ φ φ φ − φ − φ − φ φ φ φ − φ − φ − φ . All the principal Pfaffians of size 8 of the this matrix coincide with one another (up to scaling),this quartic equation gives the classical Aronhold invariant A ( F ) as follows: A ( F ) = φ − φ φ φ + φ φ + φ φ φ φ − φ φ φ − φ φ φ − φ φ φ φ + 3 φ φ φ φ + φ φ − φ φ φ + 3 φ φ φ φ − φ φ φ − φ φ φ φ − φ φ φ + φ φ φ φ − φ φ φ φ + φ φ φ φ + φ φ − φ φ φ − φ φ φ + φ φ φ φ + φ φ φ φ − φ φ φ φ − φ φ φ + φ φ φ φ . Theorem 2.10.
The singular locus
Sing( σ ( v ( P ))) coincides with σ ( v ( P )) set-theoretically.Proof. We know Sing( σ ( v ( P ))) ⊇ σ ( v ( P )). It is also well-known that the defining equationsof σ ( v ( P )) are given by 3-minors of 3 by 6 catalecticant matrix φ , (e.g. [Kan99]), which are (cid:0) (cid:1) = 20 cubics cutting out degree 15 and codimension 4 variety.On the other hand, the Jacobian of A ( F ) gives 10 cubic equations, which cut out the singularlocus Sing( σ ( v ( P ))), such as g = φ φ φ − φ φ − φ φ φ + φ φ φ + φ φ φ − φ φ g = − φ φ φ + 2 φ φ + φ φ φ − φ φ φ + φ φ φ − φ φ φ − φ φ + 3 φ φ φ − φ φ g = φ φ φ − φ φ φ − φ φ φ + 2 φ φ − φ φ + 3 φ φ φ − φ φ φ − φ φ + φ φ φ = φ φ φ − φ φ + φ φ φ − φ φ φ − φ φ φ + 3 φ φ φ − φ φ φ + 2 φ φ − φ φ g = φ φ φ − φ φ φ − φ φ φ + 4 φ φ φ − φ φ φ − φ φ φ φ φ φ − φ φ φ + 4 φ φ φ − φ g = φ φ φ − φ φ − φ φ φ + 3 φ φ φ − φ φ φ − φ φ φ + φ φ φ + 2 φ φ − φ φ g = φ φ φ − φ φ φ − φ φ φ + φ φ + φ φ − φ φ φ g = φ φ φ + φ φ φ − φ φ φ − φ φ − φ φ φ + 3 φ φ φ + 2 φ φ − φ φ φ − φ φ g = − φ φ φ + φ φ φ + φ φ φ + 2 φ φ − φ φ φ − φ φ φ − φ φ − φ φ + 3 φ φ φ g = φ φ φ − φ φ φ − φ φ + φ φ φ + φ φ φ − φ φ . One can compute the Hilbert polynomial of the singular locus by these ten cubics (e.g. [M2]) asfollows: H( t ) = 155! t + 158 t − t + 818 t − t + 2 , which shows that Sing( σ ( v ( P ))) has also codimension 4 in P and degree 15. This gives theequality Sing( σ ( v ( P ))) = σ ( v ( P )) as a set. (cid:3) Thus, we also have an immediate corollary as follows:
Corollary 2.11 ( d = 3 case) . For every n ≥ and d = 3 , σ ( v ( P n )) \ σ ( v ( P n )) is smooth.Proof. By Remark 2.4 (a), there is no degenerate orbit in this case. Thus, it comes directly fromthe result on the Aronhold hypersurface (i.e. n = 2 case, Theorem 2.10) and using the fibrationargument in the proof of Theorem 2.1 for any n ≥ (cid:3) Cases of non-degenerate orbits ( d ≥ ). Here is the theorem for non-degenerate orbits forany d ≥ n = 2: Theorem 2.12 (Non-degenerate locus) . For every d ≥ and n = 2 , σ ( v d ( P n )) \ ( D ∪ σ ( v d ( P n ))) is smooth.Proof. We are enough to consider three different cases according to Theorem 2.5. It is well-knownthat dim σ ( v d ( P )) is 3 · d ≥ f = x d + x d + x d (Fermat-type). It is well-known that this Fermat-type f becomes analmost transitive SL ( C )-orbit, which corresponds to a general point of σ ( v d ( P )), Thus, σ ( v d ( P ))is smooth at f .Case (ii) f = x d − x + x d (Unmixed-type). Say X = v d ( P ). By Remark 2.7 (i.e. n = 2 case),we just need to consider ( f ⊥ ) (cid:98) d (cid:99) · ( f ⊥ ) d −(cid:98) d (cid:99) as (2.6) to compute dim ˆ N ∨ f σ ( X ). Say s := (cid:98) d (cid:99) . For d ≥
4, we have 2 ≤ s ≤ d − s ≤ d −
2. Note that dim ˆ N ∨ f σ ( X ) ≤ codim( σ ( X ) , P U ) = (cid:0) d +22 (cid:1) − N ∨ f σ ( X ) ≥ (cid:0) d +22 (cid:1) − f .Since the summands of f separate the variables (i.e. unmixed-type), we could see that theapolar ideal f ⊥ is generated as f ⊥ = (cid:18) { Q = y y , Q = y , Q = y y } (cid:91) { other generators in degree ≥ d } (cid:19) . o, we have( f ⊥ ) s = { h · Q i | ∀ h ∈ T s − , i = 1 , , } and ( f ⊥ ) d − s = { h (cid:48) · Q i | ∀ h (cid:48) ∈ T d − s − , i = 1 , , }⇒ ˆ N ∨ f σ ( X ) = ( f ⊥ ) s · ( f ⊥ ) d − s = { h (cid:48)(cid:48) · Q i Q j | ∀ h (cid:48)(cid:48) ∈ T d − , i, j = 1 , , } . Thus, if we denote the ideal ( Q , Q , Q ) by I , then dim ˆ N ∨ f σ ( X ) is equal to the value of Hilbertfunction H ( I , t ) at t = d (concentrating only on degree d -part of ( f ⊥ ) s · ( f ⊥ ) d − s , other generatorsin degree ≥ d do not affect the dimension computation). But, it is easy to see that I has a minimalfree resolution as 0 → T ( − → T ( − → T ( − → I → , which shows the Hilbert function of I can be computed as H ( I , d ) = 6 (cid:18) d − (cid:19) − (cid:18) d − (cid:19) + (cid:18) d − (cid:19) = d ≤ (cid:0) d +22 (cid:1) − d ≥ . This implies that dim ˆ N ∨ f σ ( X ) = (cid:0) d +22 (cid:1) − d ≥
4, which means that our σ ( X ) is smoothat f (see also Figure 2). P ji kd − s P ji ks P + P ji kd Figure 2.
Case of f = x d − x + x d . P is the lattice polytope in R ≥ con-sisting of exponent vectors ( i, j, k ) of the monomials y i y k y k in ( f ⊥ ) d − s and P is the one corresponding to ( f ⊥ ) s . P + P is the Minkowski sum of twopolytopes whose lattice points are exactly the exponent vectors of ˆ N ∨ f σ ( X ) =( f ⊥ ) d − s · ( f ⊥ ) s , which contains all the monomial of T d but 9 monomials y d , y d − y , y d − y , y d − y , y d − y , y y d − , y d , y y d − , y d − y y . This also showsdim ˆ N ∨ f σ ( X ) = (cid:0) d +2 d (cid:1) − f = x d − x + x d − x (Mixed-type). In this case, we similarly use a computation ofdim ˆ N ∨ f σ ( X ) via ( f ⊥ ) s · ( f ⊥ ) d − s to show the smoothness of f (recall s := (cid:98) d (cid:99) and 2 ≤ s ≤ d − s ≤ d − Q := y y − d − y ∈ T . We easily see that f ⊥ = (cid:18) { Q , Q = y y , Q = y } (cid:91) { other generators in degree ≥ d − } (cid:19) . et I be the ideal generated by three quadrics Q , Q , Q . By the same reasoning as (ii), we havedim ˆ N ∨ f σ ( X ) = dim( f ⊥ ) s · ( f ⊥ ) d − s = H ( I , d ) = d ≤ (cid:0) d +22 (cid:1) − d ≥ , because in this case I also has the same minimal free resolution 0 → T ( − → T ( − → T ( − → I →
0. Hence, we obtain the smoothness of σ ( X ) at f (see also Figure 3). P jk id − s P jk is P + P jk id Figure 3.
Case of f = x d − x + x d − x . P (resp. P ) is the lattice polytopeconsisting of exponent vectors ( i, j, k ) of the monomials y i y k y k in ( f ⊥ ) d − s (resp.in ( f ⊥ ) s ). A dashed line means an equivalent relation between monomials givenby the multiples of Q in P and P and by those of Q i Q j ’s in P + P . Thequotient space T d / ( f ⊥ ) d − s · ( f ⊥ ) s can be represented by 9 circle monomials in P + P y d , y d − y , y d − y , y d − y , y d − y , y d − y , y d − y y , y d − y y , y d − y y modulo dashed relations, which says dim ˆ N ∨ f σ ( X ) = (cid:0) d +2 d (cid:1) −
9, the non-singularity at f . (cid:3) Remark 2.13.
From the viewpoint of apolarity, the three cases in Theorem 2.12 can be explainedgeometrically as follows: if we consider the base locus of the ideal I , which is generated by thethree quadrics in each apolar ideal f ⊥ i , then case (i) corresponds to three distinct points, case (ii)to one reduced point and one non-reduced of length 2, and case (iii) to one non-reduced point oflength 3 (not lying on a line).2.4. Degenerate case : binary forms.
Since there is no degenerate form for d = 3 (see Remark2.4 (a)), it is enough to consider the smoothness of the degenerate locus for d ≥ Theorem 2.14 (Degenerate locus) . Let D be the locus of all the degenerate forms in σ ( v d ( P n )) \ σ ( v d ( P n )) . Then, for any d ≥ , n ≥ , σ ( v d ( P n )) is singular on D if and only if d = 4 and n ≥ .Proof. Let f D be any form belong to D . For this degenerate case, by Remark 2.7, we haveˆ N ∨ f D σ ( X ) = ( f ⊥ D ) (cid:98) d (cid:99) · ( f ⊥ D ) d −(cid:98) d (cid:99) . First of all, let us consider f D as a polynomial in C [ x , x ] (i.e. f D = f D ( x , x )). Then, bythe Hilbert-Burch Theorem (see e.g. [IK99, Theorem 1.54]) we know that T /f ⊥ D is an ArtinianGorenstein algebra with socle degree d and that f ⊥ D is a complete intersection of two homoge-neous polynomials F, G of each degree a and b with a + b = d + 2 as an ideal of C [ y , y ]. Since ank φ d − , ( f D ) = 3, there is 1-dimensional kernel of φ d − , ( f D ) in C [ y , y ] , which gives one cubic generator F in f ⊥ D .When f D is general, f D = x d + αx d + β ( x + x ) d for some α, β ∈ C ∗ by Lemma 2.3, so wehave F = y y − y y . Even for the case f D being not general, we have F = y y up to changeof coordinates, because the apolar ideal of this non-general f D corresponds to the case with onemultiple root on P (see [CG01] and also [LT10, Section 4]).Therefore, we obtain that f ⊥ D = (cid:0) F = y y − y y or y y , G (cid:1) for some polynomial G of degree ( d − f ⊥ D as an ideal in T = C [ y , y , . . . , y n ] has its degree parts ( f ⊥ D ) (cid:98) d (cid:99) and ( f ⊥ D ) d −(cid:98) d (cid:99) , bothof which are generated by F, y , . . . , y n , since d ≥ (cid:98) d (cid:99) , d − (cid:98) d (cid:99) < d − d = 4 case (i.e. (cid:98) d (cid:99) = 2) : In this case, we haveˆ N ∨ f D σ ( X ) = ( f ⊥ D ) · ( f ⊥ D ) = ( y , . . . , y n ) · ( y , . . . , y n ) = ( { y i y j | ≤ i, j ≤ n } ) . So, we getdim ˆ N ∨ f D σ ( X ) = dim T − dim (cid:10) y , y y , · · · , y (cid:11) − dim (cid:10) { y · (cid:96), y y · (cid:96), y y · (cid:96), y · (cid:96) | (cid:96) = y , . . . , y n } (cid:11) = (cid:18) n (cid:19) − − n − . This shows us that σ ( X ) is singular at f D if and only if n ≥
3, because the expected codimensionis (cid:0) n (cid:1) − n − d = 5 case (i.e. (cid:98) d (cid:99) = 2) : Recall that F is y y − y y or y y , the cubic generator of f ⊥ D .Then, ˆ N ∨ f D σ ( X ) = ( f ⊥ D ) · ( f ⊥ D ) = ( y , . . . , y n ) · ( F, y , . . . , y n ) . dim ˆ N ∨ f D σ ( X ) = dim T − dim (cid:10) y , y y , · · · , y (cid:11) − dim (cid:28) { y · (cid:96), y y · (cid:96), y y · (cid:96), y y · (cid:96), y · (cid:96) } \ { y F · (cid:96), y F · (cid:96) | (cid:96) = y , . . . , y n } (cid:29) = (cid:18) n (cid:19) − − n −
1) = expected codim( σ ( X ) , P S V ) , which gives that σ ( X ) is smooth at f D in this case.ii) d ≥ N ∨ f D σ ( X ) = ( f ⊥ D ) (cid:98) d (cid:99) · ( f ⊥ D ) d −(cid:98) d (cid:99) = ( F, y , . . . , y n ) (cid:98) d (cid:99) · ( F, y , . . . , y n ) d −(cid:98) d (cid:99) . dim ˆ N ∨ f D σ ( X ) = dim T d − dim (cid:28) { y d − · (cid:96), y d − y · (cid:96), . . . , y d − · (cid:96) } \ { y d − F · (cid:96), . . . , y d − F · (cid:96) | (cid:96) = y , . . . , y n } (cid:29) − dim (cid:18) { y d , y d − y , · · · , y d } \ { y d − · F , y d − y · F , . . . , y d − · F } (cid:19) = (cid:18) d + nd (cid:19) − (cid:8) d − ( d − (cid:9) ( n − − (cid:8) ( d + 1) − ( d − (cid:9) = (cid:18) d + nd (cid:19) − n − − σ ( X ) , P S d V ) , which implies that σ ( X ) is also smooth at f D . (cid:3) .5. Defining equations of
Sing( σ ( X )) . As an immediate corollary of Theorem 2.1, we obtaindefining equations of the singular locus in our third secant of Veronese embedding σ ( X ). Corollary 2.15.
Let X be the n -dimensional Veronese embedding as above. The singular locus of σ ( X ) is cut out by × -minors of the two symmetric flattenings φ d − , and φ d − , unless d = 4 and n ≥ case, in which the (set-theoretic) defining ideal of the locus is the intersection of theideal generated by the previous × -minors and the ideal generated by × -minors of φ d − , and × -minors of φ d −(cid:98) d (cid:99) , (cid:98) d (cid:99) .Proof. It is well-known that σ ( X ) is cut out by 3 × φ d − , and φ d − , (see[Kan99, Theorem 3.3]). It is also easy to see that D , the locus of degenerate forms inside σ ( X ),is cut out by 3 × φ d − , and 4 × φ d −(cid:98) d (cid:99) , (cid:98) d (cid:99) by the argument in Remark2.7 and Proposition 2.2. Thus, using these two facts the conclusion is straightforward by Theorem2.1. (cid:3) Acknowledgements . The author would like to express his deep gratitude to Giorgio Ottaviani forintroducing the problem, giving many helpful suggestions to him, and encouraging him to completethis work. He also gives thanks to Luca Chiantini and Luke Oeding for useful conversations withthem and anonymous referees for their appropriate and accurate comments.
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School of Undergraduate Studies, Daegu-Gyeongbuk Institute of Science & Technology (DGIST),333 Techno jungang-daero, Hyeonpung-myeon, Dalseong-gun Daegu 42988, Republic of Korea
E-mail address : [email protected]@dgist.ac.kr