Strength and slice rank of forms are generically equal
Edoardo Ballico, Arthur Bik, Alessandro Oneto, Emanuele Ventura
aa r X i v : . [ m a t h . AG ] F e b STRENGTH AND SLICE RANK OF FORMS ARE GENERICALLY EQUAL
EDOARDO BALLICO, ARTHUR BIK, ALESSANDRO ONETO, AND EMANUELE VENTURA
Abstract.
We prove that strength and slice rank of homogeneous polynomials of degree d ≥ ≤ d ≤ d = 9. Introduction
Ananyan and Hochster [AH20a] introduced the notion of strength of a polynomial to solve a famous conjectureby Stillman on the existence of a uniform bound, independent on the number of variables, for the projectivedimension of a homogeneous ideal of a polynomial ring. Recently, polynomial strength and related questionshave been intensively investigated [AH20b, BB+21, BV20, BDE19, DES17, ESS20, KZ18].Let k be an algebraically closed field of characteristic zero, let n ≥ S = L d ≥ S d := k [ x , . . . , x n ]be the standard graded polynomial ring in n + 1 variables over k . So the elements of S d are homogeneouspolynomials, also called forms , of degree d . Fix an integer d ≥ f ∈ S d be a degree- d form. Definition 1.1.
The strength of f is the minimal integer r ≥ f = g · h + . . . + g r · h r where g , h , . . . , g r , h r are forms of positive degree. We denote it by str( f ).Computing the strength of a given polynomial is a very difficult task. Hence, a natural problem is todetermine the strength of a general homogeneous polynomial. In [BO20], A.B. and A.O. noticed that aconjectural answer to this problem was implicitly given in [CG+19, Remark 7.7] where the authors studydimensions of secant varieties of the varieties of reducible forms. In particular, it was conjectured that thestrength of a general form coincides with its slice rank; see [BO20, Conjecture 1.1]. Recall that the value ofthe slice rank of a general form is classically known; see Remark 1.5. Definition 1.2.
The slice rank of f is the minimal integer r ≥ f = ℓ · h + . . . + ℓ r · h r where ℓ , . . . , ℓ r are linear forms and h , . . . , h r are forms of degree d −
1. We denote it by sl . rk( f ). Conjecture 1.3 ([BO20, Conjecture 1.1]) . The strength and the slice rank of a general form in S d are equal.So far, this conjecture has been established in the following cases: when the degree d is larger than n + [Sza96], when twice the general slice rank is at most n + 2 [CCG08] and for d ≤ d = 9 [BO20].The aim of this paper is to establish Conjecture 1.3, thereby determining the strength of a general form, byproving the stronger conjecture from [CG+19, Remark 7.7] which we also state below. Geometric formulation of the problem.
For an integer 1 ≤ j ≤ d/
2, we consider the variety of formswith a degree- j factor X j := { [ g · h ] | g ∈ P S j , h ∈ P S d − j } ⊆ P S d . The union of these varieties is the variety of reducible forms X red := S ⌊ d/ ⌋ j =1 X j . For an integer r ≥
1, the r th secant variety of X red is theZariski-closure σ r ( X red ) := { [ f ] ∈ P S d | f = f + . . . + f r , [ f ] , . . . , [ f r ] ∈ X red } of the union of all linear spaces spanned by r points on X red . Since X red is reducible, we can describe its r thsecant variety as σ r ( X red ) = [ ≤ a ,...,a r ≤⌊ d/ ⌋ J a ,...,a r where J a ,...,a r := J ( X a , . . . , X a r ) = { [ f ] ∈ P S d | f = f + . . . + f r , [ f ] ∈ X a , . . . , [ f r ] ∈ X a r } is the join of the varieties X a , . . . , X a r . Now, the general slice rank and strength aresl . rk ◦ d,n := min { r ∈ Z ≥ | σ r ( X ) = P S d } and str ◦ d,n := min { r ∈ Z ≥ | σ r ( X red ) = P S d } . So Conjecture 1.3 is implied by the following stronger conjecture.
Conjecture 1.4 ([CG+19, Remark 7.7]) . For each integer r ≥
1, we have dim σ r ( X red ) = dim σ r ( X ). Remark 1.5.
Recall that the value of the general slice rank is classically known as it equals the minimalcodimension of a linear space contained in a general hypersurface. If d ≥
3, then we havesl . rk ◦ d,n := min (cid:26) r ∈ Z ≥ (cid:12)(cid:12)(cid:12)(cid:12) r ( n + 1 − r ) ≥ (cid:18) n − r + dd (cid:19)(cid:27) and codim P S d σ r ( X ) = (cid:18) n − r + dd (cid:19) − r ( n + 1 − r )for all integers 1 ≤ r < sl . rk ◦ d,n by [Har92, Theorem 12.8]. Note that sl . rk ◦ d,n ≤ n . So we can (and oftenwill) relax the assumption r < sl . rk ◦ d,n to r < n . ♣ The classical approach to computing dimensions of secant and join varieties is via
Terracini’s Lemma [Ter11]which asserts that, if Y , . . . , Y r ⊆ P N are projective varieties, q ∈ Y , . . . , q r ∈ Y r are general points and p ∈ h q , . . . , q r i is general, then T p σ r J ( Y , . . . , Y r ) = h T q Y , . . . , T q r Y r i ;see e.g. [BC+18, Lemma 1] for a recent presentation. By direct computation, it is easy to observe that thetangent space to X a at a general point [ g · h ], with deg( g ) = a and deg( h ) = d − a , is given by P ( g, h ) d where( g, h ) d := ( g, h ) ∩ S d is the degree- d homogeneous part of the ideal generated by g and h . Therefore(1) dim J a ,...,a r = dim( g , h , . . . , g r , h r ) d − , where g i , h i are general forms with deg( g i ) = a i and deg( h i ) = d − a i . The codimensions of the homogeneousparts of a homogeneous ideal are encoded in its Hilbert function , whose generating power series is called the
Hilbert series . These are among the most studied algebraic invariants of a homogeneous ideal. The Hilbertseries of an ideal generated by general forms is prescribed by Fr¨oberg’s famous conjecture; see [Fr¨o85]. In[CCG08, Theorem 5.1], the authors used the known cases of Fr¨oberg’s conjecture to deduce the integers d, n, r, a , . . . , a r with 2 r ≤ n + 2 for which J a ,...,a r = P S d . Similarly, in [CG+19, Theorem 7.4], the authorsshowed that Conjecture 1.4 holds if 2 r ≤ n + 1. The strength of the general form corresponds to the minimalcodimension of a complete intersection inside a general hypersurface. This is the perspective of [Sza96,Corollary A], where the author shows that Conjecture 1.3 holds if d ≥ n + .In [BO20], A.B. and A.O. proved the following results. Theorem 1.6.
Let d ∈ { , , , , , } and n, r ≥ be integers such that r < sl . rk ◦ d,n . Then Conjecture 1.4holds. Furthermore, unless ( d, n, r ) = (4 , , , the subvariety σ r ( X ) is the unique component of σ r ( X red ) ofmaximal dimension. If ( d, n, r ) = (4 , , , the codimensions of σ r ( X ) , J ( X , X ) and σ r ( X ) each equal . Corollary 1.7.
When d ≤ and d = 9 , the general form of S d has strength equal to its slice rank. TRENGTH AND SLICE RANK OF FORMS ARE GENERICALLY EQUAL 3
The main results of this paper are the following complementing theorem and corollary.
Theorem 1.8.
Let d ≥ and n, r ≥ be integers such that r < sl . rk ◦ d,n . Then Conjecture 1.4 holds.Furthermore, the subvariety σ r ( X ) is the unique component of σ r ( X red ) of maximal dimension. Corollary 1.9.
The general form of S d has strength equal to its slice rank. Structure of the paper.
In Section 2, we find a numerical upper bound for the dimension of J a ,...,a r ,which is an equality for a = . . . = a r = 1 and r < sl . rk ◦ d,n . In Section 3, we study this upper bound as a , . . . , a r vary and prove the main result. Acknowledgements.
E.B. is partially supported by MIUR and GNSAGA of INdAM (Italy). E.V. issupported by Vici Grant 639.033.514 of Jan Draisma from the Netherlands Organisation for Scientific Re-search. 2.
An upper bound on the dimensions
Let d, n ≥ r < n and a , . . . , a r ≤ d/ J ◦ a ,...,a r := ( f ∈ P S d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f = r X i =1 g i · h i , g i ∈ S a i , h i ∈ S d − a i , ( g , . . . , g r ) is a complete intersection ) of J a ,...,a r . Let CI n ( a , . . . , a r ) be the set of complete intersections in P n of codimension r defined by theintersection of hypersurfaces of degrees a , . . . , a r .In order to give an upper bound on the dimension of J a ,...,a r , we first observe that the subset J ◦ a ,...,a r is denseand then we bound the dimension of this subset by parametrizing it via the space of complete intersectionsCI n ( a , . . . , a n ) whose dimension can be computed explicitely. Lemma 2.1.
The subset J ◦ a ,...,a r is dense in J a ,...,a r .Proof. Let [ f ] ∈ P S d be a form which admits a strength decomposition f = P ri =1 g i h i with deg( g i ) = a i . Itis enough to show that f ∈ J ◦ a ,...,a r . Consider general forms ( u , . . . , u r ) ∈ S a × · · · × S a r . By generality,since r ≤ n , the u i ’s form a regular sequence. Since being a regular sequence is an open condition in theZariski topology, there exists an ε > su + g , . . . , su r + g r ) ∈ S a × · · · × S a r is a regular sequence for all s ∈ (0 , ε ] ∩ Q . For s ∈ (0 , ε ] ∩ Q , define f s := P ri =1 ( g i + su i ) h i ∈ J ◦ a ,...,a r . Then lim s → f s = f and hence f ∈ J ◦ a ,...,a r . (cid:3) Lemma 2.2.
We have dim J a ,...,a r ≤ dim CI n ( a , . . . , a r ) + (cid:0) n + dd (cid:1) − coeff d (cid:16) Q ri =1 (1 − t ai )(1 − t ) n +1 (cid:17) − .Proof. If I = ( g , . . . , g r ) ⊆ S is an ideal defined by a regular sequence of degrees a , . . . , a r , thendim ( S /I ) d = coeff d (cid:18) Q ri =1 (1 − t a i )(1 − t ) n +1 (cid:19) . Hence dim ( g , . . . , g r ) d = (cid:18) n + dd (cid:19) − coeff d (cid:18) Q ri =1 (1 − t a i )(1 − t ) n +1 (cid:19) =: N + 1 . From Lemma 2.1, we derive that dim J ◦ a ,...,a r = dim J a ,...,a r . Now, let E be the projective bundle onCI n ( a , . . . , a r ) whose fiber at a point Y ∈ CI n ( a , . . . , a r ) is the projective space P ( I Y ) d ∼ = P N . Thendim E = dim CI n ( a , . . . , a r ) + N. We consider the morphism E −→ J ◦ a ,...,a r given by ( Y, f ) f . This map is surjective by definition of E and J ◦ a ,...,a r . Thus dim J a ,...,a r = dim J ◦ a ,...,a r ≤ dim E = dim CI n ( a , . . . , a r ) + N, which gives the desired upper bound. (cid:3) EDOARDO BALLICO, ARTHUR BIK, ALESSANDRO ONETO, AND EMANUELE VENTURA
Now, we compute the dimension of CI n ( a , . . . , a r ). Remark 2.3.
The Hilbert polynomial P a ,...,a r ( t ) of a complete intersection is uniquely determined by thedegrees defining it since it is computed from the Koszul complex. In [Ser06, Section 4.6.1], it is shownthat CI n ( a , . . . , a r ) is parametrized by a Zariski-open subset of Hilb P a ,...,ar ( t ) ( P n ). The latter is smooth at[ Y ] ∈ CI n ( a , . . . , a r ) and [Ser06, Theorem 4.3.5] yields T [ Y ] Hilb P a ,...,ar ( t ) ( P n ) = H ( N Y/ P n ) . So dim CI n ( a , . . . , a r ) = h ( N Y/ P n ), i.e., the dimension of the space of global sections of the normal bundleof Y . ♣ Proposition 2.4.
We have dim CI n ( a , . . . , a r ) = r X i =1 coeff a i (cid:18) Q ri =1 (1 − t a i )(1 − t ) n +1 (cid:19) . Proof.
Let Y ∈ CI n ( a , . . . , a r ) be a general point. By Remark 2.3, dim CI n ( a , . . . , a r ) = h ( N Y/ P n ). Since Y is a complete intersection, its normal bundle is N Y/ P n = L ri =1 O Y ( a i ). Hence, the statement follows fromthe following equality: h ( O Y ( a i )) = coeff a i (cid:18) Q ri =1 (1 − t a i )(1 − t ) n +1 (cid:19) . To see that this equality holds, first notice that Y is projectively normal [Har77, Exercise II.8.4], becauseit is a smooth complete intersection, by the generality assumption. So, for all k ≥
0, the restriction map H ( O P n ( k )) → H ( O Y ( k )) is surjective. From the long exact sequence in cohomology of the short exactsequence 0 → I Y ( k ) → O P n ( k ) → O Y ( k ) → , one has h ( I Y ( k )) = 0 for all k ≥
0. Since HF S /I Y ( d ) = coeff d (cid:16) Q ri =1 (1 − t ai )(1 − t ) n +1 (cid:17) , where I Y is the homogeneousideal of Y , the claimed equality follows. (cid:3) Lemma 2.5.
For integers e ≥ and b , . . . , b s ≥ , we have the following identity: coeff e (cid:18) Q si =1 (1 − t b i )(1 − t ) n +1 (cid:19) = X I ⊆{ ,...,s } ( − I (cid:18) n + e − P i ∈ I b i n (cid:19) . Here (cid:0) ab (cid:1) = 0 whenever a < b .Proof. Left to the reader. (cid:3)
Theorem 2.6.
Let r < n and a , . . . , a r ≤ d/ be positive integers and take ℓ d/ := { i | a i = d/ } . Then dim J a ,...,a r ≤ (cid:18) n + dd (cid:19) − coeff d (cid:18) Q ri =1 (1 − t a i )(1 − t d − a i )(1 − t ) n +1 (cid:19) + (cid:18) ℓ d/ (cid:19) − . When d ≥ , a = . . . = a r = 1 and r < sl . rk ◦ d,n , equality holds.Proof. First, we consider the case where d ≥ a = . . . = a r = 1. In this case, by (1), it is enough tocompute the codimension of ( ℓ , . . . , ℓ r , g , . . . , g r ) d which corresponds todim S d / ( ℓ , . . . , ℓ r , g , . . . , g r ) d = dim S ′ d / ( g , . . . , g r ) d where S ′ ∼ = S / ( ℓ , . . . , ℓ r ) is a polynomial ring in n + 1 − r variables and g i is the class of g i in S ′ . Since the g i are general of degree d −
1, the latter dimension is obtained by [HL87, Theorem 1] which states thatcodim P S d J a ,...,a r = coeff d (cid:18) (1 − t d − ) r (1 − t ) n +1 − r (cid:19) . TRENGTH AND SLICE RANK OF FORMS ARE GENERICALLY EQUAL 5
For the first statement, by Lemma 2.2 and Proposition 2.4, it is enough to prove that r X j =1 coeff a j (cid:18) Q ri =1 (1 − t a i )(1 − t ) n +1 (cid:19) + (cid:18) n + dd (cid:19) − coeff d (cid:18) Q ri =1 (1 − t a i )(1 − t ) n +1 (cid:19) − (cid:18) n + dd (cid:19) − coeff d (cid:18) Q ri =1 (1 − t a i )(1 − t d − a i )(1 − t ) n +1 (cid:19) + (cid:18) ℓ d/ (cid:19) − d (cid:18) Q ri =1 (1 − t a i )(1 − t d − a i )(1 − t ) n +1 (cid:19) = coeff d (cid:18) Q ri =1 (1 − t a i )(1 − t ) n +1 (cid:19) − r X j =1 coeff a j (cid:18) Q ri =1 (1 − t a i )(1 − t ) n +1 (cid:19) + (cid:18) ℓ d/ (cid:19) . We analyze both sides of this equality. For the left hand side, we use Lemma 2.5 with e = d , s = 2 r and( b i , b r + i ) = ( a i , d − a i ) for i = 1 , . . . , r . Since a i ≤ d/ i , the summand corresponding to subset I ⊆ { , . . . , r } is zero whenever the intersection I ∩ { r + 1 , . . . , r } has more than two elements. Theremaining summands correspond to subsets I such that I ⊆ { , . . . , r } , I = I ′ ∪ { r + j } for I ′ ⊆ { , . . . , r } and j ∈ { , . . . , r } or I = I ′ ∪ { r + j, r + k } for I ′ ⊆ { , . . . , r } and distinct j, k ∈ { , . . . , r } . In the last case,the summand is zero unless a j = a k = d/ I ′ = ∅ . So we get X I ⊆{ ,...,r } ( − I (cid:18) n + d − P i ∈ I a i n (cid:19) + r X j =1 X I ′ ⊆{ ,...,r } ( − I ′ +1 (cid:18) n + a j − P i ∈ I ′ a i n (cid:19) + (cid:18) ℓ d/ (cid:19) . For the right hand side of (2), we use Lemma 2.5 with s = r and b i = a i for i = 1 , . . . , r and varying e . Weget X I ⊆{ ,...,r } ( − I (cid:18) n + d − P i ∈ I a i n (cid:19) − r X j =1 X I ⊆{ ,...,r } ( − I (cid:18) n + a j − P i ∈ I a i n (cid:19) + (cid:18) ℓ d/ (cid:19) . Hence (2) holds. (cid:3) Numerical computations
Fix an integer d ≥
5. Let n, r ≥ ≤ a , . . . , a r ≤ d/ r < sl . rk ◦ d,n . Our goal isto prove that dim J a ,...,a r ≤ dim σ r ( X )holds, and that we have equality if and only if a = . . . = a r = 1. Write ℓ j := { i ∈ { , . . . , r } | a i = j } forall j ∈ R . By Theorem 2.6, it suffices to prove that, for fixed n, r , the value of F ( a , . . . , a r ) := coeff d (cid:18) Q ri =1 (1 − t a i )(1 − t d − a i )(1 − t ) n +1 (cid:19) − (cid:18) ℓ d/ (cid:19) = coeff d Q ri =1 (1 − t a i )(1 − t ) n +1 − r X i =1 t d − a i !! is minimal exactly when a = . . . = a r = 1. We first prove that F ( a , . . . , a r ) goes down when replacing all a i > a , . . . , a r ∈ { , } . Take ϑ := max { a , . . . , a r } ≤ d/ The case ϑ > . Write P k := 1 + t + . . . + t k for k ≥ P ∞ := 1 / (1 − t ). Lemma 3.1.
Let s, ℓ, k , . . . , k s ≥ be integers. Then the coefficients of the power series P ℓ +1 ∞ P k · · · P k s form a weakly increasing series.Proof. We have P ℓ +1 ∞ = ∞ X k =0 (cid:18) ℓ + kk (cid:19) t k and so the lemma holds when s = 0. When f is a series whose coefficients increase weakly and k ≥ f P k . Hence the lemma holds for all s using induction. (cid:3) We will often apply the next lemma with g = P a and h = P b , where a ≥ b ≥ Lemma 3.2.
Let f, g, h be series whose coefficients are all nonnegative and suppose that coeff k ( g ) ≥ coeff k ( h ) for all k ≥ . Then coeff k ( f g ) ≥ coeff k ( f h ) for all k ≥ . EDOARDO BALLICO, ARTHUR BIK, ALESSANDRO ONETO, AND EMANUELE VENTURA
Theorem 3.3.
Assume that a r = ϑ > . Then F ( a , . . . , a r ) > F ( a , . . . , a r − , a r − .Proof. Take f := Q r − i =1 (1 − t a i )(1 − t ) n . Then we have F ( a , . . . , a r ) = coeff d Q ri =1 (1 − t a i )(1 − t ) n +1 − r X i =1 t d − a i !! = coeff d f P ϑ − − r X i =1 t d − a i !! and similarly F ( a , . . . , a r − , a r −
1) = coeff d f P ϑ − − r X i =1 t d − a i ! + t d − ϑ (1 − t ) !! . We need to show that the differencecoeff d f P ϑ − − r X i =1 t d − a i !! − coeff d f P ϑ − − r X i =1 t d − a i ! + t d − ϑ (1 − t ) !! is positive. This difference equalscoeff d f t ϑ − − r X i =1 t d − a i ! − t d − ϑ + t d − !! = coeff d − ϑ +1 ( f ) − ℓ ϑ − − ( ℓ ϑ −
1) coeff ( f ) − coeff ϑ ( f )= coeff d − ϑ +1 ( f (1 − t d − ϑ +1 )) − ℓ ϑ − − ( ℓ ϑ − n − ℓ ) . Take g := P n − r ∞ P d − ϑ r − Y i =1 P a i − = Q r − i =1 (1 − t a i )(1 − t ) n (1 − t d − ϑ +1 ) = f (1 − t d − ϑ +1 ) . By Lemma 3.1, the coefficients of g are weakly increasing. Socoeff d − ϑ +1 ( g ) ≥ coeff ϑ +1 ( g ) . Write m = n − ℓ . As ℓ + . . . + ℓ ϑ = r < sl . rk ◦ d,n ≤ n , we have m > ℓ + . . . + ℓ ϑ . Note thatcoeff ϑ +1 ( g ) ≥ coeff ϑ +1 (cid:16) P n − r ∞ P d − ϑ P r − ℓ − ℓ ϑ P ℓ ϑ − ϑ − (cid:17) ≥ coeff ϑ +1 (cid:16) P ∞ P m − ℓ ϑ − P ℓ ϑ − ϑ − (cid:17) = coeff ϑ +1 (cid:16) P ℓ ϑ ∞ P m − ℓ ϑ − (1 − t ϑ ) ℓ ϑ − (cid:17) = coeff ϑ +1 (cid:16) P ℓ ϑ ∞ P m − ℓ ϑ − (cid:17) − ( ℓ ϑ − m − ≥ coeff (cid:16) P ℓ ϑ ∞ P m − ℓ ϑ − (cid:17) − ( ℓ ϑ − m − (cid:16) P ℓ ϑ ∞ P m − ℓ ϑ − (cid:17) > ℓ ϑ − + ( ℓ ϑ − m − ℓ ϑ − ≥ ℓ ϑ ≥ m > ℓ ϑ − + ℓ ϑ . Note that ℓ ϑ − + ( ℓ ϑ − m − ≤ ( m − ℓ ϑ −
1) + ( ℓ ϑ − m −
1) = 2 ℓ ϑ ( m − − m. We have coeff (cid:16) P ℓ ϑ ∞ P m − ℓ ϑ − (cid:17) ≥ coeff (cid:0) P ∞ P m − (cid:1) = X k =0 (cid:18) m − k (cid:19) which is strictly greater than 2( m − m − − m ≥ ℓ ϑ ( m − − m for m ≥
10. This leaves the case m ≤ (cid:3) By Theorem 3.3, it suffices to focus on the cases where a , . . . , a r ∈ { , } . In these cases, we will regard F ( a , . . . , a r ) as a function A ℓ ,ℓ (defined below) depending only on ℓ and ℓ . TRENGTH AND SLICE RANK OF FORMS ARE GENERICALLY EQUAL 7
The case ϑ = 2 . Recall that d ≥
5. We define A ℓ ,ℓ := coeff d (cid:18) (1 − t ) ℓ (1 − t ) ℓ (1 − t ) n +1 (cid:0) − ℓ t d − − ℓ t d − (cid:1)(cid:19) for ℓ , ℓ ≥ B ℓ ,ℓ := A ℓ − ,ℓ +1 − A ℓ ,ℓ for ℓ ≥ ℓ ≥ C ℓ ,ℓ := B ℓ − ,ℓ +1 − B ℓ ,ℓ for ℓ ≥ ℓ ≥ D ℓ ,ℓ := C ℓ − ,ℓ +1 − C ℓ ,ℓ for ℓ ≥ ℓ ≥ E ℓ ,ℓ := D ℓ − ,ℓ +1 − D ℓ ,ℓ for ℓ ≥ ℓ ≥ Theorem 3.4.
We have A ℓ ,ℓ > A ℓ + ℓ , for all integers ℓ ≥ and ℓ ≥ such that ℓ + ℓ < sl . rk ◦ d,n . We write m = n − ℓ and we assume that ℓ + ℓ < n . So ℓ < m . In particular, we have m ≥ Lemma 3.5.
Let ℓ , ℓ ≥ be integers such that ℓ + ℓ < n . (a) We have A ℓ ,ℓ = coeff d ( P m +1 − ℓ ∞ P ℓ ) − ℓ (cid:18) m + 22 (cid:19) − ℓ ( m + 1) + ℓ . (b) When ℓ ≥ , we have B ℓ ,ℓ = coeff d − ( P m +1 − ℓ ∞ P ℓ ) − (cid:18) m + 22 (cid:19) − ℓ m − ℓ + 1 . (c) When ℓ ≥ , we have C ℓ ,ℓ = coeff d − ( P m +1 − ℓ ∞ P ℓ ) − m + 1) − ℓ . (d) When ℓ ≥ , we have D ℓ ,ℓ = coeff d − ( P m +1 − ℓ ∞ P ℓ ) − . (e) When ℓ ≥ , we have E ℓ ,ℓ = coeff d − ( P m +1 − ℓ ∞ P ℓ ) . Proof.
These calculations are straightforward. (cid:3)
Lemma 3.6.
Let ℓ ≥ and ℓ ≥ be integers such that ℓ + ℓ < n . (a) When ℓ < sl . rk ◦ d,n , we have B ℓ , > . (b) When ℓ ≥ , we have C ℓ ,ℓ ≥ . (c) When ℓ ≥ , we have D ℓ ,ℓ ≥ . (d) When ℓ ≥ , we have E ℓ ,ℓ ≥ .Proof. We prove the parts of the lemma in reverse order.(d). We have E ℓ ,ℓ = coeff d − ( P m +1 − ℓ ∞ P ℓ ) ≥ coeff ( P m +11 ) = m + 1 ≥ . (c). By (d), we have D ℓ ,ℓ ≥ D ℓ + ℓ , . So we may assume that ℓ = 0. Now, we have D ℓ , = coeff d − ( P m +1 ∞ ) − (cid:18) m + d − d − (cid:19) − ≥ (cid:18) d − d − (cid:19) − d − − ≥ . (b). By (c), we have C ℓ ,ℓ ≥ C ℓ + ℓ , . So we may assume that ℓ = 0. Now, we have( m + 1) ≥ m + d − · · · ( m + 2)( d − − ≥ (1 + d − · · · (1 + 2)( d − − d − − ≥ EDOARDO BALLICO, ARTHUR BIK, ALESSANDRO ONETO, AND EMANUELE VENTURA C ℓ , = coeff d − ( P m +1 ∞ ) − m + 1) = (cid:18) m + d − d − (cid:19) − m + 1) = ( m + 1) (cid:18) ( m + d − · · · ( m + 2)( d − − (cid:19) ≥ . (a). By (b), B ℓ ,ℓ ≥ B ℓ + ℓ , . So we may assume ℓ = 0. Since ℓ < sl . rk ◦ d,n , we have ℓ ( m + 1) < (cid:0) m + dd (cid:1) .So d ! ℓ < ( m + d ) · · · ( m + 2). We get d ! B ℓ , = d ! (cid:18) coeff d − ( P m +1 ∞ ) − (cid:18) m + 22 (cid:19) − ℓ + 1 (cid:19) = d ! (cid:18)(cid:18) m + d − d − (cid:19) − m ( m + 3)2 (cid:19) − d ! ℓ > d ! (cid:18)(cid:18) m + d − d − (cid:19) − m ( m + 3)2 (cid:19) − ( m + d ) · · · ( m + 2)= d ( m + d − · · · ( m + 1) − d !2 m ( m + 3) − ( m + d ) · · · ( m + 2)= ( m + d − · · · ( m + 2) ( d ( m + 1) − ( m + d )) − d !2 m ( m + 3)= ( m + d − · · · ( m + 2)( d − m − d !2 m ( m + 3)= m (cid:18) ( m + d − · · · ( m + 2)( d − − d !2 ( m + 3) (cid:19) . So it suffices to prove that c + c m + . . . + c d − m d − := ( m + d − · · · ( m + 2)( d − − d !2 ( m + 3) ≥ c = ( d −
1) coeff (( m + d − · · · ( m + 2)) − d !2= ( d − d − X i =2 ( d − i − d !2= ( d − d − X i =2 d − i − d ! ≥ ( d − (cid:18) d −
12 + d − d − − d (cid:19) > c i > i = 2 , . . . , d −
3. Hence c + c m + . . . + c d − m d − ≥ c + c + . . . + c d − = (1 + d − · · · (1 + 2)( d − − d !2 (1 + 3)= d !2 ( d − − d !2 · d !2 ( d − ≥ . This finishes the proof. (cid:3)
Theorem 3.4 now follows easily.
Proof of Theorem 3.4.
By parts (a) and (b) of Lemma 3.6, we have A ℓ ,ℓ − A ℓ +1 ,ℓ − = B ℓ +1 ,ℓ − ≥ B ℓ + ℓ , > . Repeating this, we find that A ℓ ,ℓ > A ℓ +1 ,ℓ − > · · · > A ℓ + ℓ , as desired. (cid:3) TRENGTH AND SLICE RANK OF FORMS ARE GENERICALLY EQUAL 9
The conclusion of the proof.
Proof of Theorem 1.8.
Let d ≥ n, r ≥ ≤ a , . . . , a r ≤ d/ r < sl . rk ◦ d,n . Weneed to show that dim J a ,...,a r ≤ dim σ r ( X )holds, and that we have equality only for a = . . . = a r = 1. By Theorem 2.6, it suffices to prove that F ( a , . . . , a r )is minimal exactly when a = . . . = a r = 1. By Theorem 3.3, it suffices to do this in the case where a , . . . , a r ∈ { , } . Here, we have F ( a , . . . , a r ) = A ℓ ,ℓ and so the statement holds by Theorem 3.4. (cid:3) References [AH20a] T. Ananyan and M. Hochster,
Small subalgebras of polynomial rings and Stillman’s conjecture , J. Amer. Math. Soc.33 (2020), pp. 291–309.[AH20b] T. Ananyan and M. Hochster,
Strength conditions, small subalgebras, and Stillman bounds in degree ≤
4, Transactionsof the American Mathematical Society, (373):4757–4806, 2020.[BB+21] E. Ballico, A. Bik, A. Oneto, and E. Ventura,
The set of forms with bounded strength is not closed , arXiv:2012.01237 .[BV20] E. Ballico and E. Ventura, The strength for line bundles , Math. Scand., to appear, 2020, arXiv:2004.01586 .[BC+18] A. Bernardi, E. Carlini, M. V. Catalisano, A. Gimigliano, and A. Oneto,
The hitchhiker guide to: Secant varietiesand tensor decomposition.
Mathematics 6.12 (2018): 314.[BDE19] A. Bik, J. Draisma, and R. H. Eggermont,
Polynomials and tensors of bounded strength , Communications in Contem-porary Mathematics, 21(07):1850062, 2019.[BO20] A. Bik and A. Oneto,
On the strength of general polynomials , arXiv:2005.08617 .[CCG08] E. Carlini, L. Chiantini, and A. V. Geramita, Complete intersections on general hypersurfaces , Michigan Math. J. 57(2008), pp. 121–136.[CG+19] M.V. Catalisano, A. V. Geramita, A. Gimigliano, B. Harbourne, J. Migliore, U. Nagel, and Y. S. Shin,
Secant varietiesof the varieties of reducible hypersurfaces in P n , J. Algebra 528 (2019), pp. 381–438.[DES17] H. Derksen, R. H. Eggermont, and A. Snowden, Topological noetherianity for cubic polynomials , Algebra & NumberTheory, 11(9):2197–2212, 2017.[ESS20] D. Erman, S. V. Sam, and A. Snowden,
Strength and Hartshorne’s conjecture in high degree , Mathematische Zeitschrift(2020): 1-5.[Fr¨o85] R. Fr¨oberg,
An inequality for Hilbert series of graded algebras , Math. Scand. 56 (1985), no. 2, pp. 117–144.[Har92] J. Harris,
Algebraic Geometry: A First Course . GTM 133, Springer-Verlag, 1992.[Har77] R. Hartshorne,
Algebraic Geometry , Springer-Verlag, New York, 1977.[HL87] M. Hochster and D. Laksov.
The linear syzygies of generic forms , Communications in Algebra 15.1-2 (1987): 227-239.[KZ18] D. Kazhdan and T. Ziegler.
On ranks of polynomials , Algebras and Representation Theory 21.5 (2018): 1017-1021.[Ser06] E. Sernesi,
Deformations of Algebraic Schemes , Grundlehren der mathematischen Wissenschaften 334, Springer-Verlag, Berlin Heidelberg, 2006.[Sza96] E. Szab´o,
Complete intersection subvarieties of general hypersurfaces , Pacific J. of Math. 175 (1996), no. 1, pp. 271–294.[Ter11] A. Terracini,
Sulle V k per cui la variet`a degli S h ( h + 1) -seganti ha dimensione minore dell’ordinario. Rend. Circ.Mat. Palermo 31 (1911), no. 1, pp. 392–396.(E. Ballico, A. Oneto)
Universita di Trento, Via Sommarive, 14 - 38123 Povo (Trento), Italy
Email address : [email protected], [email protected] (A. Bik) MPI for Mathematics in the Sciences, Leipzig, Germany
Email address : [email protected] (E. Ventura) Universit¨at Bern, Mathematisches Institut, Sidlerstrasse 5, 3012 Bern, Switzerland
Email address ::