Higher secant varieties of classical projective varieties

Transcription

1 Higher secant varieties of classical projective varieties Maria Virginia Catalisano DIME Università degli Studi di Genova Research Station on Commutative Algebra Yangpyeong - June 13-17, 2016 Maria Virginia Catalisano Secant Varieties

2 The classic Waring s problem (1770) Lagrange every natural number is the sum of at most 4 squares. Waring in Meditationes Algebricae. every natural number is the sum of at most 9 cubes; every natural number is the sum of at most 19 forth powers. Waring proposed a generalization of Lagrange s four-square theorem every natural number is the sum of at most g(d) d th powers of natural numbers. So g(2) = 4, g(3) = 9, g(4) = 19. 2/103 M.V. Catalisano Secant Varieties

3 The classic Waring s problem Only in 1909 Hilbert proved that g(d) exists for all d. Note that: g(3) = 9 but only 23 e 239 actually need 9 cubes, while 8 cubes suffice for all other natural numbers. Even more is true, and only few natural numbers require 8 cubes and 7 cubes are enough for all large enough natural numbers. 3/103 M.V. Catalisano Secant Varieties

4 The Waring Problem for Polynomials Given a degree d form F in n variables, the Waring Problem for Polynomials asks for the least value of s for which there exist linear forms L 1,..., L s such that F = L d Ld s. This value of s is called the Waring rank of F and will be denoted by rk(f), so rk(f) = min{s F = L d Ld s }. 4/103 M.V. Catalisano Secant Varieties

5 Veronese Variety A Veronese Variety is the image of the embedding ν d : P n P N N = ( ) n+d n 1 via the forms of degree d of the homogeneous coordinate ring R = C[x 0,..., x n ]. 5/103 M.V. Catalisano Secant Varieties

6 Example ν 2 : P 2 P 5 (a, b, c) (a 2, ab, ac, b 2, bc, c 2 ) 6/103 M.V. Catalisano Secant Varieties

7 Veronese Variety A Veronese Variety is the image of the embedding P n = P(R 1 ) P N = P(R d ) [L] [L d ] 7/103 M.V. Catalisano Secant Varieties

8 Example ν 2 : P 2 P 5 [ax +by +cz] [a 2 x 2 +2abxy +2acxz +b 2 y 2 +2bcyz +c 2 z 2 ]. Hence, if we use as a basis for R d the monomial of degree d and order them lexicographically and write things with respect to coordinates, we get that the map is defined by (a, b, c) (a 2, 2ab, 2ac, b 2, 2bc, c 2 ). 8/103 M.V. Catalisano Secant Varieties

9 Veronese Variety It is clear that the two methods of describing the Veronese embeddings are equivalent in characteristic zero. The second method gives us a bridge between the Waring problem for forms and the higher secant varieties of the Veronese variety. Recall that the Waring rank of F = min{s F = L d Ld s }. Example 4xy = (x + y) 2 + (ix iy) 2 (0, 4, 0) = (1, 2, 1) + ( 1, 2, 1) 9/103 M.V. Catalisano Secant Varieties

10 HIGHER SECANT VARIETIES 10/103 M.V. Catalisano Secant Varieties

11 Definition of higher secant variety Let X P N be a reduced, irreducible, non-degenerate projective variety, i.e., X arises as the set of zeros of a prime homogeneous ideal which does not contain any linear forms (i.e. X does not lie in any projective linear subspace of P N ). The s th secant variety of X, denoted by σ s (X), is the closure of the union of all linear spaces spanned by s linearly independent points of X, that is σ s (X) := P i X < P 1,..., P s > When s = 2 we refer to σ s (X) as the secant line variety, when s = 3 the secant plane variety and so on. 11/103 M.V. Catalisano Secant Varieties

12 General forms Example. If F is a general form of degree d in n + 1 variables, then the rank of F is the minimum s such that σ s (X) fiills the ambient space, where X is the Veronese variety ν d (P n ). 12/103 M.V. Catalisano Secant Varieties

13 Problem Important questions to answer about σ s (X). What is the minimal free resolution of the ideal of σ s (X)? What is the degree of σ s (X)? What are the equations defining σ s (X)? What is the dimension of σ s (X)? 13/103 M.V. Catalisano Secant Varieties

14 Dimension of Secant Variety By a parameter count we get exp dim σ s (X) = min{s dim X + s 1; N} If σ s (X) does not have the expected dimension, we say that X is defective. Note that, for s(dim X + 1) N + 1 we expect σ s (X) to fill P N, and it is not a coincidence that: dim X + 1 is the degree of a double point on X. 14/103 M.V. Catalisano Secant Varieties

15 Motivation Strong motivation for studying the secant varieties connections to questions in Representation Theory, Coding Theory, Algebraic Complexity Theory, Statistics, Phylogenetics, Data Analysis, Electrical Engineering (Antenna Array Processing and Telecommunications), etc 15/103 M.V. Catalisano Secant Varieties

16 Secant Varieties and Tensor decomposition Connection with the problem of how to minimally represent certain kinds of tensors as a sum of decomposable ones: Rank of Tensors 16/103 M.V. Catalisano Secant Varieties

17 Secant Varieties of Segre-Veronese Varieties and Tensor decomposition Example: The Segre-Veronese Variety is the image of the mapping ν (d1,...,d t ) : P(V 1 )... P(V t ) P(S d 1 V 1... S d t V t ) [v 1 ]... [v t ] [v d v d t t ], hence parameterizes partially symmetric tensors. 17/103 M.V. Catalisano Secant Varieties

18 Segre Variety A Segre Variety is the image of the embedding ν (1,...,1) : P n 1... P n t P N N = Π(n i + 1) 1 via the forms of multi-degree (1,..., 1) of the multi-graded homogeneous coordinate ring C[x 0,1,..., x n1,1; x 0,2,..., x n2,2;...; x 0,t,..., x nt,t]. 18/103 M.V. Catalisano Secant Varieties

19 Example ν (1,1,1) : P 1 P 1 P 1 P 7 (x 0, x 1 ) (y 0, y 1 ) (z 0, z 1 ) (x 0 y 0 z 0, x 0 y 0 z 1, x 0 y 1 z 0, x 0 y 1 z 1, x 1 y 0 z 0, x 1 y 0 z 1, x 1 y 1 z 0, x 1 y 1 z 1 ). 19/103 M.V. Catalisano Secant Varieties

20 Segre-Veronese Variety A Segre-Veronese Variety is the image of the embedding ν (d1,...,d t ) : P n 1... P n t P N N = Π ( d i +n i n i ) 1 via the forms of multi-degree (d 1,..., d t ) of the multi-graded homogeneous coordinate ring C[x 0,1,..., x n1,1; x 0,2,..., x n2,2;...; x 0,t,..., x nt,t]. For t = 1, X is a Veronese Variety. For d 1 = = d t = 1, X is a Segre Variety. 20/103 M.V. Catalisano Secant Varieties

21 Example ν (1,2) : P 2 P 1 P 8 (x 0, x 1, x 2 ) (y 0, y 1 ) (x 0 y 2 0, x 0y 0 y 1, x 0 y 2 1, x 1y 2 0, x 1y 0 y 1, x 1 y 2 1, x 2y 2 0, x 2y 0 y 1, x 2 y 2 1 ). 21/103 M.V. Catalisano Secant Varieties

22 Varieties of λ-reducible forms In 1954, Mammana introduced the varieties of reducible plane curves: consider the space S d of forms of degree d in 3 variables and let λ = (d 1,..., d r ) be a partition of d. ( ) 2 + d X 2,λ P N, N = 1 d is the variety parameterizing forms F which are the product of r forms F i with deg F i = d i, that is, F = F 1 F r S d. 22/103 M.V. Catalisano Secant Varieties

23 Varieties of λ-reducible forms The varieties of reducible hypersurfaces are an obvious generalization: consider the space S d of forms of degree d in n + 1 variables and let λ = (d 1,..., d r ) be a partition of d. ( ) n + d X n,λ P N, N = 1 d is the variety parameterizing forms F which are the product of r forms F i with deg F i = d i F = F 1 F r S d. 23/103 M.V. Catalisano Secant Varieties

24 METHOD FOR SEGRE-VERONESE 24/103 M.V. Catalisano Secant Varieties

25 Method for Segre-Veronese The problem of determining the dimension of σ s (X) is related (via the theory of inverse systems or, equivalently, apolarity) to determining the Hilbert Function of a 0-dimensional scheme made by the first infinitesimal neighbourhoods of s generic points (this is equivalent to what is classically known as Terracini s Lemma). 25/103 M.V. Catalisano Secant Varieties

26 Step 1 - Terracini s Lemma Let P σ s (X) be a generic point P < P 1,..., P s >, (P i X P N ). Then by Terracini s Lemma : T P (σ s (X)) =< T P1 (X),..., T Ps (X) > =< 2P 1 X P s X > hence dim σ s (X) = dim T P (σ s (X)) = dim < 2P 1 X P s X > = N dim(i 2P1 X P s X ) 1. 26/103 M.V. Catalisano Secant Varieties

27 Step 2 : Segre-Veronese map Let ν (d1,...,d t ) : (P n 1... P n t ) X P N Q i P n 1... P n t ; P i X Q i P i the scheme of double points 2P 1 X P s X X corresponds to the scheme 2Q Q s P n 1... P n t. 27/103 M.V. Catalisano Secant Varieties

28 Step 2 : Segre-Veronese map and we have hence dim(i 2P1 X P s X ) 1 = dim(i 2Q Q s ) (d1,...,d t ) dim σ s (X) = N dim(i 2P1 X P s X ) 1 = N dim(i 2Q Q s ) (d1,...,d t ) = HF (2Q Q s, (d 1,..., d t )) 1 28/103 M.V. Catalisano Secant Varieties

29 Veronese Varieties Thus, in case X is the Veronese Variety, we have dim σ s (X) = = N dim(i 2Q Q s ) d = HF (2Q Q s, d) 1 29/103 M.V. Catalisano Secant Varieties

30 Veronese Varieties The work by J. Alexander and A. Hirschowitz (1995) confirmed that apart from the quadratic Veronese varieties and a (few) well known exceptions, all the Veronese varieties X = ν d (P n ) have higher secant varieties of the expected dimension. More precisely, σ s (X) is defective only in the following cases: d = 2 and 2 s n; n = 2, d = 4, s = 5; n = 3, d = 4, s = 9; n = 4, d = 3, s = 7; n = 4, d = 4, s = 14. Hence if X is a Veronese Variety the problem of finding the dimension of σ s (X) is solved and the Waring Problem for a generic polynomial is solved also. 30/103 M.V. Catalisano Secant Varieties

31 Veronese Varieties Except the cases listed above, if F is a generic form of degree d in n + 1 variables. rk(f) = ( d+n n n + 1 ) (Alexander - Hirschowitz, 1995). 31/103 M.V. Catalisano Secant Varieties

32 Example of a defective Segre variety X = the Segre embedding of P 1 P 2 P 5 P 35 is defective for s = 5 exp dim σ 5 (X) = min{ ; N} = min{44; 35} = 35 = N that is, we expect σ 5 (X) = P N, i.e., we expect that the ideal of 5 double points in degree (1, 1, 1) is zero. But there exist F 1 of degree (1, 1, 0), and F 2 of degree (0, 0, 1) in the ideal of 5 points, hence F = F 1 F 2 is a form of degree (1, 1, 1) in the ideal of 5 double points. 32/103 M.V. Catalisano Secant Varieties

33 Step 3: the Multiprojective-Affine-Projective Method Let T = k[x 0,1,..., x n1,1; x 0,2,..., x n2,2;...; x 0,t,..., x nt,t] be the multigraded homogeneus coordinate ring of P n 1... P n t. Let S = k[t, x 1,1,..., x n1,1; x 1,2,..., x n2,2;...; x 1,t,..., x nt,t] be the coordinate ring of P n n t. 33/103 M.V. Catalisano Secant Varieties

34 Step 3: the Multiprojective-Affine-Projective Method where dim T (d1,...,d t ) = dim(i W ) (d d t ) T (d1,...,d t ) is the vector space of the forms of multidegree (d 1,..., d t ) of T ; W = (n n t )Λ 1 +(n 1 +n n t )Λ (n n t 1 )Λ t and the Λ i P n i 1 are generic linear spaces of P n n t. (Examples: ν (2,1,1) (P 1 P 5 P 5 ), ν (2,1) (P 2 P 1 )). 34/103 M.V. Catalisano Secant Varieties

35 Step 3: the Multiprojective-Affine-Projective Method Then dim(i 2Q Q s ) (d1,...,d t ) = dim(i W +2P P s ) d d t where thus W + 2P P s P n n t P 1,..., P s are generic points W = (n n t )Λ (n n t 1 )Λ t Λ i P n i 1 P n n t are generic linear spaces dim σ s (X) = N dim(i 2Q Q s ) (d1,...,d t ) = N dim(i W +2P P s ) d d t = 1 + number of conditions that 2P P s imposes to the forms of (I W ) d d t 35/103 M.V. Catalisano Secant Varieties

36 Notation P n 1... P n t (d 1,..., d t ) instead of ν (d1,...,d t )(P n 1... P n t ). 36/103 M.V. Catalisano Secant Varieties

37 Example of a defective Segre-Veronese variety P 1 P 5 P 5 (2, 1, 1) is defective for 7 s 11 N = = 107; expdim σ 7 (X) = 7x12 1 = 83; W = 2Λ 1 + 3Λ 2 + 3Λ 3 P 11 where Λ 1 P 0, Λ 2 P 4, Λ 3 P 4, i.e., W is a subscheme of P 11 formed by one double point and two triple linear spaces of dimension 4; there exists an r.n.c. C maximally intersecting the configuration Λ 2 + Λ points (, Carlini (2009)); deg C deg F = 11 4 = 44, where F (I W +2P P 7 ) 4 ; degree (C (W + 2P P 7 )) = = 46. Then we conclude that our variety is 7-defective. 37/103 M.V. Catalisano Secant Varieties

38 Example of a defective Segre-Veronese variety expdim σ 11 (X) = 107; N = 107; hence we expect that σ 11 (X) = P 107 or, equivalently, that there are not forms of degree 4 in (I W +2P P 11 ), where W is as above, that is, W is a subscheme of P 11 formed by one double point and two triple linear spaces Λ 1 and Λ 2 of dimension 4. But there is a quadric S 1 through 2Λ 1 + Λ 2 + the 12 points, and, analogously, there is a quadric S 2 through Λ 1 + 2Λ 2 + the 12 points. (In fact the quadrics of P 11 are 78, and the conditions given to the quadrics by 2Λ 1 + Λ points are = 77). The form corresponding to the quartic S 1 S 2 is in (I W +2P P 11 ) 4. 38/103 M.V. Catalisano Secant Varieties

39 Example of a non defective Segre-Veronese variety P 2 P 1 (2, 1) exp dim σ s (X) = min{4s 1; 11} hence we expect σ 3 (X) to fill P N and dim σ 2 (X) = 7. W P 3 is formed by a line, say L, and one double point, say 2P 0. Since dim(i L+2P0 +2P 1 +2P 2 ) 3 = 4 dim(i L+2P0 +2P 1 +2P 2 +2P 3 ) 3 = 0 we have dim σ 2 (X) = 11 4 = 7 dim σ 3 (X) = /103 M.V. Catalisano Secant Varieties

40 Example of a defective Segre-Veronese variety P 2 P 5 (2, 1) is defective iff s = 5 Let X = ν (2,1) (P 2 P 5 ) P 35 ; Y = ν (1,1) (P 5 P 5 ) P 35. By a direct computation, we get σ 4 (X) = σ 4 (Y)=31. Hence σ s (X) = σ s (Y) for s 4. Since dim σ 4 (Y) = 31 exp dim σ 4 (X) = 31 dim σ 5 (Y) = 34, dim σ 6 (Y) = 35 exp dim σ 5 (X) = exp dim σ 6 (X) = 35 we have that X is defective iff s = 5. 40/103 M.V. Catalisano Secant Varieties

41 Step 4: Specialization and méthode d Horace Recall that: let Z P n be a scheme corresponding to the ideal sheaf I Z and let H P n be a hyperplane. The trace of Z with respect to H is the schematic intersection Tr H (Z ) = Z H. The Residual Res H (Z ) of Z with respect to H is defined by the ideal sheaf I Z : O P n( H). Taking the global sections of the exact sequence 0 I ResH Z (d 1) I Z (d) I TrH (Z )(d) 0, we obtain the so called Castelnuovo exact sequence 0 (I ResH Z ) d 1 (I Z ) d (I TrH (Z )) d from which we get the following inequality dim(i Z ) d dim(i ResH Z ) d 1 + dim(i TrH (Z )) d. 41/103 M.V. Catalisano Secant Varieties

42 Step 4: Specialization and méthode d Horace In order to compute dim(i W +2P P s ) d d t, specialize the scheme Y = W + 2P P s by placing some of the P i on a hyperplane. Let Ỹ be the specialized scheme. In some cases, specialize Y in such a way that new linear spaces are in the base locus for the hypersurfaces defined by the forms of (IỸ ) d d t. then use Castelnuovo Lemma dim(iỹ ) d dim(i ResH Ỹ ) d 1 + dim(i TrH Ỹ ) d (where H is a hyperplane) or Horace Differential Lemma (example: σ 4 (ν (2,1) (P 2 P 1 )). 42/103 M.V. Catalisano Secant Varieties

43 Example P 1... P 1 (t times ) (1,..., 1) s even; s = 2h we have to compute dim(i X ) t where X = (t 1)Q (t 1)Q t + 2P P 2h. Example: for t = 7 and s = 16 we have to compute (Note that 2 7 = 128 = 16 8)) dim(i 6Q Q 7 +2P P 16 ) 7 43/103 M.V. Catalisano Secant Varieties

44 Specialize on a hyperplane H the points Q 2,..., Q 7, P 1,..., P 8. The 8 lines L 1 = Q 1 P 9,..., L 8 = Q 1 P 16 are in the base locus of the hypersurfaces defined by the forms of (I X ) 7, hence where (I X ) 7 = (I Y ) 7 Y = X + L L 8. Now use Castelnuovo Lemma: and induction. dim(i Y ) 7 dim(i ResH Y ) 6 + dim(i TrH Y ) 7 44/103 M.V. Catalisano Secant Varieties

45 Since the hypersurfaces defined by the forms of (I ResH Y ) 6 are cones with vertex in Q 1 we have dim(i ResH Y ) 6 = dim(i 5Q Q 7 +P 1 + +P 8 +2P P 16 ) 6. Since < Q 2,..., Q 7 > is in the base locus of the hypersurfaces defined by the forms of (I TrH Y ) 7, we get dim(i TrH Y ) 7 = dim(i 5Q Q 7 +2P P 8 +8 simple points ) 6 and now we are in P 1... P 1 (6 times ) (1,..., 1) 45/103 M.V. Catalisano Secant Varieties

46 SECANT VARIETIES OF SEGRE VARIETIES WITH 2 FACTORS 46/103 M.V. Catalisano Secant Varieties

47 Segre of two factors P m P n (1, 1) everything is well known The case for Segre varieties with two factors is very well understood since all of the theory is in terms of ranks of matrices and that is understood very well from both an algebraic and geometric standpoint. 47/103 M.V. Catalisano Secant Varieties

48 Segre of two factors ALL THE THEORY IS IN TERMS OF RANKS OF MATRICES Example: P 2 P 3 P 11. a 0 a 1 [ ] a 0 b 0 a 0 b 1 a 0 b 2 a 0 b 3 b 0 b 1 b 2 b 3 = a 1 b 0 a 1 b 1 a 1 b 2 a 1 b 3 a 2 a 2 b 0 a 2 b 1 a 2 b 2 a 2 b 3 The product of the two projective spaces (P m P n ) can be identified with the (m + 1) (n + 1) matrices of rank 1 and the general points on σ s (X) are the sums of s matrices of rank 1. (A matrix has rank s if and only if it is the sum of s matrices of rank 1.) In the example dim σ 2 (X) = 11 2 = 9 48/103 M.V. Catalisano Secant Varieties

49 By our method Example: P 2 P 3 P 11. exp dim σ s (X) = min{6s 1; 11} hence exp dim σ 2 (X) = 11 dim σ 2 (X) = 11 dim(i W +2P1 +2P 2 ) 2 W P 5 is formed by a line and a plane. By projecting from the two points P 1 and P 2 we get dim(i W +2P1 +2P 2 ) 2 = dim(i W ) 2 where W P 3 is still formed by a line and a plane, hence dim σ 2 (X) = 11 2 = 9. 49/103 M.V. Catalisano Secant Varieties

50 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES WITH 2 FACTORS 50/103 M.V. Catalisano Secant Varieties

51 Segre-Veronese of two factors P m P n (a, b) there are only partial results 51/103 M.V. Catalisano Secant Varieties

52 Segre-Veronese of two factors with b = 1 P m P n (a, 1) 52/103 M.V. Catalisano Secant Varieties

53 Segre-Veronese of two factors with b = 1 P 2 P 1 (3, 1) P 2 P 1 (a, 1) P 3 P 2 (2, 1) is defective for s = 5 London (1890); Dionisi, Fontanari (2001); Carlini, Chipalkatti(2001) is defective only for a = 3, s = 5 (δ = 1) Dionisi, Fontanari (2001) is defective for s = 5 (δ = 1) Carlini, Chipalkatti(2001) 53/103 M.V. Catalisano Secant Varieties

54 Segre-Veronese of two factors with b = 1 P 3 P 4 (2, 1) P 5 P 2 (2, 1) P 1 P n (a, 1) is defective for s = 6 (δ = 1) Carlini, Chipalkatti(2001) is defective for s = 8 (δ = 2) Carlini, Chipalkatti(2001) is never defective Chiantini, Ciliberto (2002) P m P n (n + 1, 1) is never defective -, Geramita, Gimigliano (2005) 54/103 M.V. Catalisano Secant Varieties

55 Segre-Veronese of two factors with b = 1 P m P n (a, 1) is not defective for a 3 and s s 1 or s s 2, where: (n+1)( s 1 q(n + 1) m+a a ), (q N); s 2 t(n + 1), (t N). (m+n+1) (n+1)( m+a a ) (m+n+1) Bernardi, Carlini, - (2010) 55/103 M.V. Catalisano Secant Varieties

56 Segre-Veronese of two factors with b = 1 P 2k+1 P 2 (2, 1) is defective for s = 3k + 2 Ottaviani (2008) P m P n (2, 1) is not defective for s s 1 and n m + 2; or for s s 2, where: s 1 = (n + 1) m 2 (n 2)(n+1) 2 if m is even; s 1 = (n + 1) m 2 (n 3)(n+1) 2 if m and n are odd ; s 1 = (n + 1) m 2 (n 3)(n+1)+1 2 if n is even and m is odd ; s 2 = (n + 1) m if m is even; s 2 = (n + 1) m otherwise. Abo, Brambilla (2009) 56/103 M.V. Catalisano Secant Varieties

57 Segre-Veronese of two factors with b = 1 P m P m (2, 1) is never defective Abo (2010) P m P m 1 (2, 1) is defective only for (m, m 1) = (4, 3) and s = 6 Abo (2010) 57/103 M.V. Catalisano Secant Varieties

58 Conjectures P m P n (2, 1) is never defective, except for 1) n ( ) m+2 2 m (i.e., it is unbalanced) and ( m+2 ) ( 2 m < s < min{n + 1; m+2 ) 2 }; 2) (m, n, s) = (2k + 1, 2, 3k + 2) con k 1; 3) (m, n, s) = (3, 4, 6). Abo, Brambilla (2012) 58/103 M.V. Catalisano Secant Varieties

59 Segre-Veronese of two factors with n = 1 P m P 1 (a, b) 59/103 M.V. Catalisano Secant Varieties

60 Segre-Veronese of two factors with n = 1 P 1 P 1 (a, b) is defective only for (a, b) = (2, 2d) and s = 2d + 1 -, Geramita, Gimigliano (2005) P 2 P 1 (a, b) is defective only for (a, b) = (3, 1), (a, b) = (2, 2d) Baur, Draisma (2007) 60/103 M.V. Catalisano Secant Varieties

61 Segre-Veronese of two factors with n = 1 P m P 1 (2, 2d) is defective iff d(m + 1) + 1 s d(m + 1) + m Abrescia (2008) P m P 1 (2, 2d + 1) is never defective Abrescia (2008) P m P 1 (3, b) is defective only for m = 2, b = 1 Abrescia (2008) 61/103 M.V. Catalisano Secant Varieties

62 Segre-Veronese of two factors with n = 1 P m P 1 (a, b) if b 3, it is defective only for (a, b) = (2, 2d), d(m + 1) + 1 s d(m + 1) + m Abo, Brambilla (2009) P m P 1 (a, b) is defective only for m = 2, (a, b) = (3, 1), s = 5 and for (a, b) = (2, 2d), d(m + 1) + 1 s d(m + 1) + m Ballico, Bernardi, - (2011) 62/103 M.V. Catalisano Secant Varieties

63 Segre-Veronese of two factors P 2 P 2 (2, 2) P n P 2 (2, 2) P 3 P 3 (2, 2) is defective for s = 8 -, Geramita, Gimigliano (2005) is defectivefor s = 3n + 2 -, Geramita, Gimigliano (2005) Bocci (2005) is defective for s = 15 -, Geramita, Gimigliano (2005) P 3 P 4 (2, 2) is defective for s = 19 Bocci (2005) 63/103 M.V. Catalisano Secant Varieties

64 Conjecture P m P n (a, b) is never defective, except for 1) b = 1, m 2 and it is unbalanced ; 2) (m, n) = (m, 1), (a, b) = (2, 2d) ; 3) (m, n) = (3, 4), (a, b) = (2, 1); 4) (m, n) = (m, 2), (a, b) = (2, 2) ; 5) (m, n) = (2k + 1, 2), k 1, (a, b) = (2, 1); 6) (m, n) = (2, 1), (a, b) = (3, 1); 7) (m, n) = (2, 2), (a, b) = (2, 2); 8) (m, n) = (3, 3), (a, b) = (2, 2); 9) (m, n) = (3, 4), (a, b) = (2, 2). Abo, Brambilla (2011) 64/103 M.V. Catalisano Secant Varieties

65 Conjecture P m P n (a, b) for (a, b) (3, 3), it is never defective Abo, Brambilla (2009) 65/103 M.V. Catalisano Secant Varieties

66 SECANT VARIETIES OF SEGRE and SEGRE-VERONESE WITH MANY FACTORS 66/103 M.V. Catalisano Secant Varieties

67 The unbalanced case P n 1... P n t P n (1,..., 1) P n 1... P n t P n (d 1,..., d t, 1) is defective for N t i=1 n i + 1 < s min{n; N} where N = Π t i=1 (n i + 1) 1 C. - Geramita - Gimigliano (2005) is defective for N t n i + 1 < s min{n; N}, where N = Π t ni ) +d i i=1( d i 1 C. - Geramita - Gimigliano (2008) Abo - Ottaviani - Peterson (2009) 67/103 M.V. Catalisano Secant Varieties

68 Example (unbalanced) P 1 P 2 P 7 (1, 1, 1) is 4 and 5-defective Let X denote this unbalanced Segre Variety. We have: X P 47, expdim σ 4 (X) = 43, exp dim σ 5 (X) = 47, dim(i 2P1 +2P 2 +2P 3 +2P 4 ) (1,1,1) ) (6 4)(8 4) = 8 dim(i 2P1 +2P 2 +2P 3 +2P 4 +2P 5 ) (1,1,1) ) (6 5)(8 5) = 3 hence dim σ 4 (X) 47 8 = 39, dim σ 5 (X) 47 3 = /103 M.V. Catalisano Secant Varieties

69 Segre - many copies of P n P n... P n (t times ) (1,..., 1) has the expected dimension for s = p(n + 1) for n = 1, and some t, Sloane(1982), Hill (1986), Roman (1992); for n = 1, C. - Geramita - Gimigliano (2005); for n > 1, Abo - Ottaviani - Peterson (2009). 69/103 M.V. Catalisano Secant Varieties

70 Many copies of P 1 P 1... P 1 (t times ) (1,..., 1) is never defective if t 5 t = 6: Draisma (2008); t 7: C. - Geramita - Gimigliano (2011) P 1... P 1 (t times ) (d 1,..., d r ) is never defective if t 5 Laface - Postinghel (2013) 70/103 M.V. Catalisano Secant Varieties

71 Conjecture for Segre Varieties (true for s 6) P n 1... P n t (1,..., 1) is never defective, except 1) unbalanced; 2) P 2 P 3 P 3, for s = 5 ; δ 5 = 1; 2) P 2 P n P n, with n even, for s = 3n ; δ s = 1; 3) P 1 P 1 P n P n, for s = 2n + 1 ; δ 2n+1 = 1. Abo - Ottaviani - Peterson (2009) 71/103 M.V. Catalisano Secant Varieties

72 Example P 1... P 1 (9 times ) (1,..., 1) s = 51 We have to compute dim(i X ) 9 where X = 8Q Q 9 + 2P P 51 P 9 Specialize on a hyperplane H the points Q 2,..., Q 9, P 1,..., P 25 ; specialize P 27 on < Q 1, Q 2, Q 3, P 26 >= Π 1 P 3, specialize P 29 on < Q 1, Q 4, Q 5, P 28 >= Π 2 P 3. Π 1 and Π 2 are in the base locus of the hypersurfaces defined by the forms of (I X ) 9. 72/103 M.V. Catalisano Secant Varieties

73 The 26 lines L i =< Q 1, P i > (i = 26,..., 51) are in the base locus of the hypersurfaces defined by the forms of (I X ) 9. Let Y = Π 1 + Π 2 + L L 51 + X. Now use Castelnuovo Lemma: dim(i Y ) 9 dim(i ResH Y ) 8 + dim(i TrH ) 9. 73/103 M.V. Catalisano Secant Varieties

74 METHOD FOR REDUCIBLE FORMS 74/103 M.V. Catalisano Secant Varieties

75 Method for varieties of reducible forms Recall that ( ) n + d X n,λ P N, N = 1 d is the variety parameterizing forms F which are the product of r forms F i with deg F i = d i, that is, F = F 1 F r. 75/103 M.V. Catalisano Secant Varieties

76 Varieties of λ-reducible forms Very little is known about the secant varieties of the varieties of reducible forms Arrondo -Bernardi λ = (1,..., 1) (Split hypersurfaces) Shin Secant line variety to the varieties of split plane curves Abo All the higher secant varieties of split plane curves 76/103 M.V. Catalisano Secant Varieties

77 Varieties of λ-reducible forms Conjecture (Arrondo, Bernardi) The higher secant varieties for split hypersurfaces always have the expected dimension 77/103 M.V. Catalisano Secant Varieties

78 Step 1 - Terracini s Lemma Let Q σ s (X) be a generic point Q < P 1,..., P s >, (P i X n,λ P N ). Then by Terracini s Lemma : T Q (σ s (X)) =< T P1 (X n,λ ),..., T Ps (X n,λ ) > that is, the dimension of σ s (X n,λ ) is the dimension of the linear span of T P1 (X n,λ ),..., T Ps (X n,λ ). 78/103 M.V. Catalisano Secant Varieties

79 Step 2- The tangent space to X n,λ at a general point Let P = [F] X n,λ be a general point, F = F 1 F r and let I P be the following ideal ( F I P =..., F 1, F ) F r then T P = P ((I P ) d ). 79/103 M.V. Catalisano Secant Varieties

80 Step 3- The dimension of σ s (X n,λ ) Let P 1,..., P s be general points of X n,λ, and let I = I P1 + + I Ps then dim σ s (X n,λ ) = dim (I) d 1. Note that, if s = 2 dim (I) d = dim(i P1 ) d + dim(i P2 ) d dim(i P1 I P2 ) d. 80/103 M.V. Catalisano Secant Varieties

81 Varieties of λ-reducible forms Geramita, Gimigliano, Shin All the higher secant line variety to the varieties of λ-reducible curves Geramita, Gimigliano, Harbourn, Migliore, Nagel, Shin Many higher secant line variety to the varieties of λ-reducible hypersurfaces 81/103 M.V. Catalisano Secant Varieties

82 WHAT ABOUT THE WARING RANK OF A SPECIFIC FORM? 82/103 M.V. Catalisano Secant Varieties

83 Answers The problem of finding rk(f) is solved in few cases: F has degree 2. rk(f) = rk(m) where M is the symmetric matrix associated to F. 83/103 M.V. Catalisano Secant Varieties

84 Answers F is a binary form, that is, F C[x, y]. We have the Sylvester s algorithm (Sylvester 1886; Comas Seiguer 2001; Brachat, Comon, Mourrain,Tsigaridas 2009; Bernardi, Gimigliano, Idà (2011)). Let F = { C[X, Y ] F = 0}. In this case F = (f 1, f 2 ). If deg f 1 deg f 2 we have { deg f1 if f rk(f) = 1 is square free deg f 2 otherwise 84/103 M.V. Catalisano Secant Varieties

85 Answers If some algorithms work. (Iarrobino, Kanev; Landsberg, Teitler; Buczynska, Buczynski; Brachat, Comon, Mourrain, Tsigaridas; Bernardi, Gimigliano, Idà; Oeding, Ottaviani) 85/103 M.V. Catalisano Secant Varieties

86 Answers F C[x, y, z] has degree 3, i.e. F represents a cubic curve. We have an explicit algorithm (Comon, Mourrain, Reznick; 1996). In particular we have that : the maximum rank for a ternary cubic is 5. 86/103 M.V. Catalisano Secant Varieties

87 Maximum rank A natural question arises: What is the maximum rank for a form of degree d in n variables?. 87/103 M.V. Catalisano Secant Varieties

88 Answers The maximum rank for ternary quartics is 7. The maximum rank for ternary quintics is 10. (De Paris, 2015) 88/103 M.V. Catalisano Secant Varieties

89 Answers F is a monomial. rk(x a 0 0 x am m ) = (a 1 + 1) (a m + 1) where 1 a 0... a m (Carlini, -, Geramita, 2011) (Buczynska, Buczynski, Teitler, 2012) 89/103 M.V. Catalisano Secant Varieties

90 Answers some reducible forms (b 2): F = x a 0 (x b 1 + x b 2 ) rk(f) = { 2b if a + 1 b 2(a + 1) if a + 1 b F = x0 a(x 0 b + x 1 b + x 2 b) { 2b if a + 1 b rk(f) = 2(a + 1) if a + 1 b (Carlini - Geramita, 2012) 90/103 M.V. Catalisano Secant Varieties

91 Answers F = x a 0 (x b x b m) rk(f) = (a + 1)m if a + 1 b F = x a 0 (x b 0 + x b x b m) rk(f) = (a + 1)m if a + 1 b (Carlini, -, Geramita, 2012) 91/103 M.V. Catalisano Secant Varieties

92 Answers the Vandemonde determinant V n = 1 i<j n (x i x j ) C[x 1,..., x n ] rk(v n ) = (n + 1)! (Carlini, -, Chiantini, Geramita, Woo, 2015) 92/103 M.V. Catalisano Secant Varieties

93 Answers F = x a 0 G(x 1,..., x n ) If G = (g 1,..., g n ) is a complete intersection and deg g i a + 1, then rk(f) = Π n 1 deg g i (Carlini, -, Chiantini, Geramita, Woo, 2015) 93/103 M.V. Catalisano Secant Varieties

94 METHOD TO FIND THE WARING RANK OF A SPECIFIC FORM 94/103 M.V. Catalisano Secant Varieties

95 Method Apolarity Lemma Let S = C[x 0,..., x n ], T = C[X 0,..., X n ], F S d, F = { T F = 0}. Let L 1,..., L r be pairwise linearly independent linear forms, with L i corresponding to the point P i, and X = {P 1,..., P r }, then F = a 1 L d a r L d r I X F. 95/103 M.V. Catalisano Secant Varieties

96 Method Let rk(f) = r, let X be a set of points apolar to F. Let t T be a linear form corresponding to the linear space Π, and let I Y = I X : (t), ( so Y = X \ Π). Since t is a non-zero divisor int /I Y, we have the following exact sequence 0 (T /I Y ) i 1 t (T /I Y ) i (T /(I Y + (t))) i 0, for t >> 0, we get Y = HF (T /I Y, t) =. t HF(T /(I Y + (t)), i) = l(t /(I Y + (t))). i=0 96/103 M.V. Catalisano Secant Varieties

97 Method Hence, since I X F, we get X Y = l(t /(I Y + (t))) = l(t /(I X : (t) + (t))) l(t /F : (t) + (t)), and so we have a lower bound for rk(f). 97/103 M.V. Catalisano Secant Varieties

98 Example F = x 2 y 2 z 3 C[x, y, z] Let Π = {X = 0}. Since F : (X) = (X F) = (2xy 2 z 3 ) = (X 2, Y 3, Z 4 ), we have F : (X) + (X) = (X, Y 3, Z 4 ) and so HF (T /(F : (X) + (X))) = Hence rk(f) /103 M.V. Catalisano Secant Varieties

99 Example Now since the ideal F = (x 2 y 2 z 3 ) = (X 3, Y 3, Z 4 ), (X 4 Z 4, Y 4 Z 4 ) F is the ideal of 12 distinct points. It follows that rk(f) /103 M.V. Catalisano Secant Varieties

100 Final observation Now note that F = x(y 2 + z 2 ) = xy 2 + xz 2. rk(xy 2 ) = 3 ; rk(xz 2 ) = 3 ; rk(f) = 4 < rk(xy 2 ) + rk(xz 2 ). 100/103 M.V. Catalisano Secant Varieties

101 Example But if we consider F = xy 2 + zw 2 C[x, y, z, w] since rk(xy 2 ) = 3 ; rk(zw 2 ) = 3 ; rk(f) = 6, we have rk(f) = rk(xy 2 ) + rk(zw 2 ). 101/103 M.V. Catalisano Secant Varieties

102 Strassen s Additivity Conjecture Let F and G be homogeneous polynomials in different sets of variables. rk(f + G) = rk(f) + rk(g)? 102/103 M.V. Catalisano Secant Varieties

103 Thanks for your attention! 103/103 M.V. Catalisano Secant Varieties