HIROTACHI ABO AND MARIA CHIARA BRAMBILLA

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1 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES P m P n EMBEDDED BY O(1, ) HIROTACHI ABO AND MARIA CHIARA BRAMBILLA Abstract. Let X m,n be the Segre-Veronese variety P m P n embee by the morphism given by O(1, ). In this paper, we provie two functions s(m, n) s(m, n) such that the s th secant variety of X m,n has the expecte imension if s s(m, n) or s(m, n) s. We also present a conjecturally complete list of efective secant varieties of such Segre-Veronese varieties. 1. Introuction Let X P N be an irreucible non-singular variety of imension. Then the s th secant variety of X, enote σ s (X), is efine to be the Zariski closure of the union of the linear spans of all s-tuples of points of X. The stuy of secant varieties has a long history. The interest in this subject goes back to the Italian school at the turn of the 0 th century. This topic has receive renewe interest over the past several ecaes, mainly ue to its increasing importance in an ever wiening collection of isciplines incluing algebraic complexity theory [Bürgisser et al. 1997, Lansberg 006, Lansberg 008], algebraic statistics [Garcia et al. 005, Eriksson et al. 005, Aoki et al. 007], an combinatorics [Sturmfels an Sullivant 006, Sullivant 008]. The major questions surrouning secant varieties center aroun fining invariants of those objects such as imension. A simple imension count suggests that the expecte imension of σ s (X) is min {N, s( + 1) 1}. We say that X has a efective s th secant variety if σ s (X) oes not have the expecte imension. In particular, X is sai to be efective if X has a efective s th secant variety for some s. For instance, the Veronese surface X in P 5 is efective, because the imension of σ (X) is four while its expecte imension is five. A well-known classification of the efective Veronese varieties was complete in a series of papers by Alexaner an Hirschowitz [Alexaner an Hirschowitz 1995, Brambilla an Ottaviani 008]. There are corresponing conjecturally complete lists of efective Segre varieties [Abo et al. 006] an efective Grassmann varieties [Baur et al. 007]. Secant varieties of Segre-Veronese varieties are however less well-unerstoo. In recent years, a lot of efforts have been mae to evelop techniques to stuy secant varieties of such varieties (see for example [Catalisano et al. 005, Carlini an Chipalkatti 003, Carlini an Catalisano 007, Ottaviani 006, Catalisano et al. 008, Ballico 006, Abrescia 008]). But even the classification of efective two-factor Segre-Veronese varieties is still far from complete. In orer to classify efective Segre-Veronese varieties, a crucial step is to prove the existence of a large family of non-efective such varieties. A powerful tool to establish non-efectiveness of large classes of Segre-Veronese varieties is the inuctive approach base on specialization techniques, which consist in placing a certain number of points on a chosen ivisor. For a given n = (n 1,..., n k ) N k, we write P n for P n 1 P n k. Let Xn a be the Segre-Veronese variety P n embee by the morphism given by O(a) with a = (a 1,..., a k ) N k. As we shall see in Section, the problem of etermining the imension of σ s (Xn) a is equivalent to the problem of etermining the value of the Hilbert function h P n(z, ) of a collection Z of s general ouble points in P n at a, i.e., h P n(z, a) = im H 0 (P n, O(a)) im H 0 (P n, I Z (a)). 000 Mathematics Subject Classification. 14M99, 14Q99, 15A69, 15A7. Key wors an phrases. Secant varieties, Segre-Veronese varieties, efectiveness. 1

2 HIROTACHI ABO AND MARIA CHIARA BRAMBILLA Suppose that a k. Denote by n an a the k-tuples (n 1, n,..., n k 1) an (a 1, a,..., a k 1) respectively. Given a P n P n, we have an exact sequence 0 I ez (a ) I Z (a) I Z P n,pn (a) 0, where Z is the resiual scheme of Z with respect to P n. This exact sequence gives rise to the so-calle Castelnuovo inequality Thus, we can conclue that h P n(z, a) h P n( Z, a ) + h P n (Z Pn, a). - if h P n( Z, a ) an h P n (Z Pn, a ) are the expecte values an - if the egrees of Z an Z P n are both less than or both greater than im H 0 (P n, O(a )) an im H 0 (P n, O(a)) respectively, then h P n(z, a) is also the expecte value. By semicontinuity, the Hilbert function of a general collection of s ouble points in P n has the expecte value at a. This enables one to check whether or not σ s (Xn) a has the expecte imension by inuction on n an a. To apply this inuctive approach, we nee some initial cases regaring either imensions or egrees. The class of secant varieties of two-factor Segre-Veronese varieties embee by the morphism given by O(1, ) can be viewe as one of such initial cases. In fact, the above-mentione specialization technique involves secant varieties of two-factor Segre varieties, most of which are known to be efective, an thus we cannot apply this technique to fin im σ s (Xn) a for n = (m, n) an a = (1, ). To siestep this problem, we therefore nee an a hoc approach. This paper is evote to stuying secant varieties of Segre-Veronese varieties P m P n embee by the morphism given by O(1, ). Let ( (m + 1) n+ ) q(m, n) =. m + n + 1 Our main goal is to prove the following theorem: Theorem 1.1. Let n = (m, n) an let a = (1, ). If n is sufficiently large, then σ s (X a n) has the expecte imension for s = q(m, n). A straightforwar consequence of this theorem is the following: Corollary 1.. Let n = (m, n) an let a = (1, ). If n is sufficiently large, then σ s (X a n) has the expecte imension for all s q(m, n). In orer to prove Theorem 1.1, we show that if m n +, then σ s(m,n) (Xn) a has the expecte imension, where (m + 1) n/ (m )(m+1) if n is even; s(m, n) = (m + 1) n/ (m 3)(m+1) if m an n are o; (m + 1) n/ (m 3)(m+1)+1 if m is even an if n is o. Theorem 1.1 then follows immeiately, because s(m, n) = q(m, n) for a sufficiently large n (an explicit boun for n can be foun just before Corollary 3.14). To prove that σ s(m,n) (X a n) has the expecte imension, we will use ouble inuction on m an n. More precisely, we will show the following two claims: (i) Let n = (n+1, n). Then the secant variety σ s(n+1,n) (X a n) has the expecte imension. Note that the case n = (n +, n) is trivial since s(n +, n) = 0. (ii) Let n = (m, n ) an n = (m, n). If σ s(m,n ) (X a n ) has the expecte imension, then σ s(m,n) (X a n) has also the expecte imension.

3 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES P m P n EMBEDDED BY O(1, ) 3 Claim (i) can be prove by an inuctive approach that specializes a certain number of points on a subvariety of P m P n of the form P m P n (see Section for more etails). Note that a similar approach was successfully applie to stuy secant varieties of Segre varieties (see for example [Abo et al. 006]). The proof of (ii) relies on a ifferent specialization technique which allows to place a certain number of points on a two-coimensional subvariety of P m P n of the form P m P n (see Section 3 for more etails). This approach can be regare as a moification of the approach introuce in [Brambilla an Ottaviani 008] that simplifies the proof of the Alexaner-Hirschowitz theorem for cubic Veronese varieties. We also woul like to mention that the same approach was extene to secant varieties of Grassmannians of planes in [Abo et al. 008]. In Section 4, we will moify the above techniques to prove the following theorem: Theorem 1.3. Let n = (m, n) an let a = (1, ) an let { (m + 1) n/ + 1 if n is even; s(m, n) = (m + 1) n/ + 3 otherwise. Then σ s (X a n) has the expecte imension for any s s(m, n). Theorems 1.1 an 1.3 complete the classification of efective Segre-Veronese varieties X 1, m,n for m = 1,. To be more precise, the following is an immeiate consequence of these theorems: Corollary 1.4. Let n = (m, n) an let a = (1, ). (i) If m = 1, then σ s (X a n) has the expecte imension for any s. (ii) If m =, then σ s (X a n) has the expecte imension unless (n, s) = (k + 1, 3k + ) with k 1. Note that (i) is well-known, see for example [Carlini an Chipalkatti 003]. We also mention that Theorem 1.3 of [Baur an Draisma 007] gives a complete classification of the case m = 1, n = for any egree a = ( 1, ), where 1, 1. On the other han, to our best knowlege, (ii) was previously unknown. The efectiveness of the (3k + ) n secant variety of X 1,,k+1 has alreay been establishe (see [Carlini an Chipalkatti 003, Ottaviani 006] for the proofs). Thus Corollary 1.4 (ii) completes the classification of efective secant varieties of X 1,,n. In Section 5, we will give a conjecturally complete list of efective secant varieties of Xm,n. 1, Evience for the conjecture was provie by results in [Catalisano et al. 005, Carlini an Chipalkatti 003, Ottaviani 006]. Further evience in support of the conjecture was obtaine via the computational experiments we carrie out. Thus the first part of this section will be evote to explaining these experiments, which were one with the computer algebra system Macaulay evelope by Dan Grayson an Mike Stillman [Grayson an Stillman]. The proofs of Lemmas 3.10 an 4.5 are also base on computations in Macaulay. All the Macaulay scripts neee to make these computations are available at Splitting Theorem Let V be an (m+1)-imensional vector space over C an let W be an (n+1)-imensional vector space over C. For simplicity, we write P m,n for P m P n = P(V ) P(W ). In this section, we inicate by X m,n for the Segre-Veronese variety P m,n embee by the morphism ν 1, given by O(1, ) for simplicity. Let T p (X m,n ) be the affine cone over the tangent space T p (X m,n ) to X m,n at a point p X m,n. For each p X m,n, there are two vectors u V \ {0} an v W \ {0}, such that p = [u v ] P(V S (W )). In this way, p can be ientifie with ([u], [v]) P m,n through ν 1,. Thus p is also enote by ([u], [v]). Let p = [u v ] X m,n. Then T p (X m,n ) = V v + u v 1 W. We enote by Y p (X m,n ) (or just by Y p ) the (m + 1)-imensional subspace V v of V S (W ).

4 4 HIROTACHI ABO AND MARIA CHIARA BRAMBILLA Definition.1. Let p 1,..., p s, q 1,..., q t be general points of X m,n an let U m,n (s, t) be the subspace of V S (W ) spanne by s T p i (X m,n ) an t Y q i (X m,n ). Then U m,n (s, t) is expecte to have imension { ( )} n + min s(m + n + 1) + t(m + 1), (m + 1). We say that S(m, n; 1, ; s; t) is true if U m,n (s, t) has the expecte imension. For simplicity, we enote S(m, n; 1, ; s; 0) by T (m, n; 1, ; s). Note that U m,n (s, 0) is the affine cone of σ s (X m,n ). Remark.. Let q 1,..., q t be general points of X m,n an let σ s (X m,n ) be the s th secant variety of X m,n. By Terracini s lemma [Terracini 1911], the span of the tangent spaces to X m,n at s generic points is equal to the tangent space to σ s (X m,n ) at the generic z point in the linear subspace spanne by the s points. Thus the vector space U m,n (s, t) can be thought of as the affine cone over the tangent space to the join J(P(Y q1 ),..., P(Y qt ), σ s (X m,n )) of P(Y q1 ),..., P(Y qt ) an σ s (X m,n ) at a general point in the linear subspace spanne by q 1,..., q t an z. Therefore, S(m, n; 1, ; s; t) is true if an only if J(P(Y q1 ),..., P(Y qt ), σ s (X m,n )) has the expecte imension. In particular, σ s (X m,n ) has the expecte imension if an only if S(m, n; 1, ; s; 0) is true. Remark.3. Let N = (m + 1) ( ) n+. Then H 0 (P m,n, O(1, )) can be ientifie with the set of hyperplanes in P N 1. Since the conition that a hyperplane H P N contains T p (X m,n ) is equivalent to the conition that H intersects X m,n in the first infinitesimal neighborhoo of p, the elements of H 0 (P m,n, I p (1, )) can be viewe as hyperplanes containing T p (X m,n ). Let q X m,n. A similar argument shows that the elements of H 0 (P m,n, I q P(Yq) (1, )) can be ientifie with hyperplanes containing Y q, where q P(Yq) is a zero-imensional subscheme of X m,n of length m + 1. Let p 1,..., p s, q 1,..., q t X m,n an let Z = {p 1,..., p s, q 1 P(Y q1 ),..., q t P(Yqt )}. Recall that Terracini s lemma says that the linear subspace spanne by T p1 (X m,n ),..., T ps (X m,n ) is the tangent space to σ s (X m,n ) at a general point in the linear subspace spanne by p 1,..., p s. This implies that im J(P(Y q1 ),..., P(Y qt ), σ s (X m,n )) equals the value of the Hilbert function h P m,n(z, ) of Z at (1, ), i.e., h P m,n(z, (1, )) = im H 0 (P m,n, O(1, )) im H 0 (P m,n, I Z (1, )). In particular, if an only if S(m, n; 1, ; s; t) is true. h P m,n(z, (1, )) = min {s(m + n + 1) + t(m + 1), N}. Definition.4. A six-tuple (m, n; 1, ; s; t) is calle subabunant (resp. superabunant) ( ) n + s(m + n + 1) + t(m + 1) (m + 1) (resp. ). We say that (m, n; 1, ; s; t) is equiabunant if it is both subabunant an superabunant. brevity, we will write the five-tuple (m, n; 1, ; s) instea of the six-tuple (m, n; 1, ; s; 0). Assume that S(m, n; 1, ; s; t) is true. Note that when (m, n; 1, ; s; t) is superabunant, U m,n (s, t) coincies with the whole space V S (W ), whereas for subabunant (m, n; 1, ; s; t), U m,n (s, t) can be a proper subspace of the whole space. Remark.5. Given two vectors (s, t) an (s, t ), we say that (s, t) (s, t ) if s s an t t. Suppose that S(m, n; 1, ; s; t) is true an that (m, n; 1, ; s; t) is subabunant (resp. superabunant). Then S(m, n; 1, ; s ; t ) is true for any choice of s an t with (s, t) (s, t ) (resp. with (s, t) (s, t )). Remark.6. Suppose that m = 0. We make the following simple remarks: For

5 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES P m P n EMBEDDED BY O(1, ) 5 (i) Let q X 0,m. Then P(Y q (X 0,n )) is just q itself. Thus if q 1,..., q t are general points of X 0,n an if (0, n; 1, ; s; t) is subabunant, then S(0, n; 1, ; s; t) is true if an only if T (0, n; 1, ; s) is true. (ii) By the Alexaner-Hirschowitz theorem, [Alexaner an Hirschowitz 1995], we know that T (0, n; 1, ; n + 1) is true. Then if (0, n; 1, ; s) is superabunant an if s n + 1, then T (0, n; 1, ; s) is true. Theorem.7. Let m = m + m + 1 an let s = s + s. If (m, n; 1, ; s ; s + t) an (m, n; 1, ; s ; s + t) are subabunant (resp. superabunant, resp. equiabunant) an if S(m, n; 1, ; s ; s + t) an S(m, n; 1, ; s ; s + t) are true, then (m, n; 1, ; s; t) is subabunant (resp. superabunant, resp. equiabunant) an S(m, n; 1, ; s; t) is true. Proof. Here we only prove the theorem in the case where (m, n; 1, ; s ; s +t) an (m, n; 1, ; s ; s + t) are both subabunant, because the remaining cases can be prove in a similar manner. Let V an V be subspaces of V of imensions m + 1 an m + 1 respectively. Suppose that V is the irect sum of V an V. Let p = [u v ] X m,n. If u V, then we have T p (X m,n ) = V v + u v 1 W = (V v + u v 1 W ) (V v ) = T p (X m,n) Y p (X m,n) for some p X m,n (p must be of the form [u v ] with u V ). Similarly, one can prove that if u V, then T p (X m,n ) = Y p (X m,n) T p (X m,n) for some p X m,n. Let q = [u v ] X m,n. Then there exist q X m,n an q X m,n such that Y q (X m,n ) = V v = (V v ) (V v ) = Y q (X m,n) Y q (X m,n). Thus one can conclue that U m,n (s, t) U m,n(s, s + t) U m,n(s, s + t). By assumption, im U m,n(s, s + t) = s (m + n + 1) + (s + t)(m + 1) an im U m,n(s, s + t) = s (m + n + 1) + (s + t)(m + 1). Thus im U m,n (s, t) = im U m,n(s, s + t) + im U m,n(s, s + t) = s(m + n + 1) + t(m + 1) (m + 1) ( ) n+ + (m + 1) ( ) ( n+ = (m + 1) n+ ), an hence (m, n; 1, ; s, t) is subabunant an S(m, n; 1, ; s; t) is true We will iscuss three examples to illustrate how to use Theorem.7 below. These examples will be use in later sections. Example.8. In this example, we apply Theorem.7 to prove that T (, ; 1, ; s) is true for every s 3. Note that (, ; 1, ; s) is subabunant for s 3. Thus it suffices to show that T (, ; 1, ; 3) is true. Taking m = 1, m = 0 an s =, s = 1, one can reuce T (, ; 1, ; 3) to S(1, ; 1, ; ; 1) an S(0, ; 1, ; 1; ). Inee (1, ; 1, ; ; 1) an (0, ; 1, ; 1; ) are both subabunant. The statement S(1, ; 1, ; ; 1) can be reuce again to twice S(0, ; 1, ; 1; ), taking m = m = 0 an s = s = 1. This means that T (, ; 1, ; 3) is reuce to triple S(0, ; 1, ; 1; ). Clearly S(0, ; 1, ; 1; 0) is true, an so is S(0, ; 1, ; 1; ) by Remark.6 (i). Hence we complete the proof. Example.9. We prove that T (m, 1; 1, ; 3) is true for any m. The proof is by inuction. It has been alreay prove that T (1, 1; 1, ; 3) is true (see [Catalisano et al. 005]). Now suppose that T (m 1, 1; 1, ; 3) is true for some m. Note that (m, 1; 1, ; 3) is superabunant. Since (m 1, 1; 1, ; 3; 0) an (0, 1; 1, ; 0; 3) are also superabunant, we can reuce T (m, 1; 1, ; 3) to T (m 1, 1; 1, ; 3) an S(0, 1; 1, ; 0; 3). Clearly, S(0, 1; 1, ; 0; 3) is true, by Remark.6 (i). Since T (m 1, 1; 1, ; 3) is true by inuction hypothesis, T (m, 1; 1, ; 3) is also true.

6 6 HIROTACHI ABO AND MARIA CHIARA BRAMBILLA Example.10. Here we prove that T (n + 1, n; 1, ; s) is true for any s n an any n 1. Note that (n + 1, n; 1, ; s) is subabunant for such an s. Thus it is sufficient to prove that T (n + 1, n; 1, ; s) is true if s = n First suppose that n is even, i.e., n = k for some integer k 1. Then s = k + 1. Since (k, k; 1, ; k; 1) an (0, k; 1, ; 1; k) are both subabunant, then T (k + 1, k; 1, ; k + 1) can be reuce to S(k, k; 1, ; k; 1) an S(0, k; 1, ; 1; k). Analogously, S(k, k; 1, ; k; 1) can be reuce to S(k 1, k; 1, ; k 1; ) an S(0, k; 1, ; 1; k). That means T (k+1, k; 1, ; k+1) is now reuce to S(k 1, k; 1, ; k 1; ) an twice S(0, k; 1, ; 1; k) (we will enote it by S(0, k; 1, ; 1; k)). We can repeat the same process (k ) times to reuce T (k+1, k; 1, ; k+1) to S(k, k; 1, ; 0; k+1) an (k + 1) S(0, k; 1, ; 1; k). Inee we have only to check that (k + 1 h, k; 1, ; k + 1 h; h) is subabunant for any 1 h k + 1, which is true. Now the statement S(k, k; 1, ; 0; k + 1) can be reuce to S(k 1, k; 1, ; 0; k + 1) an S(0, k; 1, ; 0; k + 1), since both (k, k; 1, ; 0; k + 1) an (k 1, k; 1, ; 0; k + 1) are subabunant. Analogously S(k 1, k; 1, ; 0; k + 1) can be reuce to S(k, k; 1, ; 0; k + 1) an S(0, k; 1, ; 0; k + 1). Repeating the same process k times, we can reuce S(k, k; 1, ; 0; k + 1) to (k + 1) S(0, k; 1, ; 0; k + 1). Recall that (0, k; 1, ; 1; k) an (0, k; 1, ; 0; k + 1) are subabunant. Thus S(0, k; 1, ; 1; k) an S(0, k; 1, ; 0; k + 1) are true, because T (0, k; 1, ; 1) an T (0, k; 1, ; 0) are true an by Remark.6 (i). This implies that T (k + 1, k; 1, ; k + 1) is true. In the same way, we can also prove that T (n + 1, n; 1, ; s) is true when n is o. Inee, T (k +, k + 1; 1, ; k + ) can be reuce to (k + ) S(0, k + 1; 1, ; 1; k + 1) an (k + 1) S(0, k + 1; 1, ; 0; k + ). Since S(0, k + 1; 1, ; 1; k + 1) an S(0, k + 1; 1, ; 0; k + ) are true, so is T (k +, k + 1; 1, ; k + ). As immeiate consequences of Theorem.7, we can prove the following two propositions: Proposition.11. T (m, n; 1, ; s) is true if s m + 1 an m ( ) n+1 or if s (m + 1)(n + 1). Proof. We first prove that if m ( ) n+1, then T (m, n; 1, ; s) is true for any s m + 1. Since (m, n; 1, ; s) is subabunant for any s m + 1, it is enough to prove that T (m, n; 1, ; m + 1) is true. Applying Theorem.7 (m + 1) times, we can reuce to (m + 1) S(0, n; 1, ; 1; m). Inee (0, n; 1, ; 1; m) is subabunant, since from the assumption m ( ) n+1 it follows ( ) n + (n + 1) + m. It follows also that (m h, n; 1, ; m + 1 h; h) is subabunant for any 1 h m 1. Since S(0, n; 1, ; 1; 0) is true, so is S(0, n; 1, ; 1; m). This implies that T (m, n; 1, ; m + 1) is true. To show that T (m, n; 1, ; s) is true for any s (m + 1)(n + 1), it is enough to prove that T (m, n; 1, ; (m + 1)(n + 1)) is true, since (m, n; 1, ; (m + 1)(n + 1)) is superabunant. In the same way as in the previous case the statement can be reuce to (m + 1) S(0, n; 1, ; m + 1; (m + 1)n). Since (0, n; 1, ; m + 1) is superabunant an T (0, n; 1, ; m + 1) is true, it follows that (0, n; 1, ; m + 1; (m + 1)n) is superabunant an S(0, n; 1, ; m + 1; (m + 1)n) is true. Thus T (m, n; 1, ; (m + 1)(n + 1)) is true. Remark.1. In Sections 3 an 4, we will use ifferent techniques to improve the bouns given in Proposition.11. ( n+ ) ( n+ ) Proposition.13. Suppose that m 1 an 3. Let l = an let h =. m + n + 1 n + 1 Then (i) T (m, n; 1, ; s) is true for any s l(m + 1). (ii) If (n, ) (, 4), (3, 4), (4, 3), (4, 4) an if s h(m + 1), then T (m, n; 1, ; s) is true. (iii) If (n, ) = (, 4), (3, 4), (4, 3) or (4, 4), then T (m, n; 1, ; s) is true for any s (h+1)(m+1).

7 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES P m P n EMBEDDED BY O(1, ) 7 Proof. (i) Suppose that s = l(m + 1). Since l(n + 1) + lm = l(m + n + 1) ( n+ ) (m + n + 1) = m + n + 1 ( n + then (0, n; 1, ; l; lm) is subabunant (this implies that (h, n; 1, ; l + hl; l(m h)) is also subabunant for all 1 h m). Then T (m, n; 1, ; l(m + 1)) can be reuce to (m + 1) S(0, n; 1, ; l; lm). ) Furthermore, since l < ( n+ n + 1, then S(0, n; 1, ; l; 0) is true by the Alexaner-Hirschowitz theorem. Thus S(0, n; 1, ; l; lm) is true by Remark.6 (i). This implies, by Theorem.7, that T (m, n; 1, ; l(m + 1)) is true. (ii) Let s = h(m + 1). Then (m, n; 1, ; s) is clearly superabunant. The statement T (m, n; 1, ; s) can be reuce to (m + 1) S(0, n; 1, ; h; hm). Suppose that n 3, 4. Then the Alexaner- Hirschowitz theorem says that T (0, n; 1, ; h) is true, an so is S(0, n; 1, ; h; hm). Hence by Theorem.7 it follows that T (m, n; 1, ; h(m + 1)) is true. (iii) Suppose that (n, ) = (, 4), (3, 4), (4, 3) or (4, 4). Then T (0, n; 1, ; h + 1) is true by the Alexaner-Hirschowitz theorem, an thus S(0, n; 1, ; h; (h + 1)m) is also true. Therefore the same argument as in (ii) proves that T (m, n; 1, ; (h + 1)(m + 1)) is true. 3. Segre-Veronese varieties P m P n embee by O(1, ): Subabunant Case Let V be an (m + 1)-imensional vector space over C with basis {e 0,..., e m } an let W be an (n + 1)-imensional vector space over C with basis {f 0,..., f n }. As in the previous section, X m,n enotes Xm,n. 1, Let U L be a two-coimensional subspace of W an let L = P(V ) P(U L ). Note that if p is a point of ν 1, (L), then the affine cone T p (X m,n ) over the tangent space to X m,n at p moulo V S (U L ) has imension (m + n + 1) (m + n + 1) =. Definition 3.1. Let k = n/ an let s(m, n) = (m + 1)k (m )(m+1) if n is even; (m + 1)k (m 3)(m+1) if m an n are o; (m + 1)k (m 3)(m+1)+1 if m is even an if n is o. Note that s(m, m ) = 0. We will sometimes rop the parameters m, n, when they are clear from the context. The goal of this section is to prove that if m n +, then T (m, n; 1, ; s) is true for any s s(m, n). Since (m, n; 1, ; s) is subabunant, it is sufficient to prove that T (m, n; 1, ; s(m, n)) is true. The key point is to restrict to subspaces of coimension an to use two-step inuction on n. It is obvious that T (m, m ; 1, ; 0) is true. It also follows from Example.10 that T (m, m 1; 1, ; s(m, m 1)) is true. Thus it remains only to show that if T (m, n ; 1, ; s(m, n )) is true, then so is T (m, n; 1, ; s(m, n)). To o this, we nee to introuce the auxiliary statements R(m, n) an Q(m, n) (see Definitions 3. an 3.6) an use ouble inuction on m an n to prove such auxiliary statements. Definition 3.. Let k an s = s(m, n) be as given in Definition 3.1. Note that s(m, n ) = s (m+1). Let p 1,..., p s (m+1) be general points of L, let q 1,..., q m+1 be general points of P m,n \L an let V m,n be the vector space V S (U L ), s (m+1) T pi (X m,n ), m+1 T q i (X m,n ). Note that the following inequality hols: ( ) n im V m,n (m + 1) + [s (m + 1)] + (m + 1)(m + n + 1) { ( (m + 1) n+ ) = if n is even, or if m an n are o; (m + 1) ( ) n+ 1 if m is even an if n is o. ),

8 8 HIROTACHI ABO AND MARIA CHIARA BRAMBILLA We say that R(m, n) is true if the equality hols. Remark.3 implies that R(m, n) is true if an only if { im H 0 (P m,n 0 if n is even or if m an n are o;, I Z L (1, )) = 1 if m is even an if n is o, where Z = {p 1,..., p s (m+1), q 1,..., q m+1 }. Proposition 3.3. Let k an s = s(m, n) be as given in Definition 3.1. If R(m, n) is true an if T (m, n ; 1, ; s (m + 1)) is true, then T (m, n; 1, ; s) is true. Proof. Let p 1,..., p s P m,n an let Z = {p 1,..., p s}. Then it is easy to check that m 3 m if n is even or if m an n are o; im H 0 (P m,n, I Z (1, )) k m3 if m is even an n is o. Suppose that p 1,..., p s (m+1) L an that p s m,..., p s P m,n \ L. Let Z = {p 1,..., p s}. Let Z = Z L = {p 1,..., p s (m+1) }. Then we have the following short exact sequence: Taking cohomology, we have Thus we must have 0 I Z L (1, ) I Z (1, ) I Z,L(1, ) 0. 0 H 0 (P m,n, I Z L (1, )) H 0 (P m,n, I Z (1, )) H 0 (L, I Z (1, )). im H 0 (P m,n, I Z (1, )) im H 0 (P m,n, I Z L (1, )) + im H 0 (L, I Z (1, )). Since R(m, n) an T (m, n ; 1, ; s (m + 1)) are true, we have { im H 0 (P m,n im H, I Z (1, )) 0 (L, I Z (1, )) + 1 if m is even an if n is o; im H 0 (L, I Z (1, )) otherwise, from which the proposition follows. To prove that T (m, n; 1, ; s(m, n)) is true, it is therefore enough to show that R(m, n) is true if m n. The proof is again by two-step inuction on n. To be more precise, we first prove R(m, m) an R(m, m + 1) are true. Then we show that if R(m, n ) is true, then R(m, n) is also true. Proposition 3.4. R(m, m) is true for any m 1. Proof. Without loss of generality, we may assume that U L = f,..., f m+1. Let p 0..., p m P m,m \ L. For each i {0,..., m}, we have p i = [u i vi ] where u i V an v i W \ U L. Recall that T pi (X m,m ) = V vi + u i v i W. To prove the proposition, we will fin explicit vectors u i s an v i s such that m V S (W ) T pi (X m,m ) (mo V S (U L )). Let u i = e i for each i {0,..., m} an let { fi for i = 0, 1, v i = if 0 + f 1 + f i for i m. i=0 Then we have e0 f0,..., e m f0, e 0 f 0 f 1,..., e 0 f 0 f m if i = 0; e0 f T pi (X m,m ) = 1,..., e m f1, e 1 f 0 f 1,..., e 1 f 1 f m if i = 1; e0 (if 0 + f 1 + f i ),..., e m (if 0 + f 1 + f i ), e i (if 0 + f 1 + f i )f 0,..., e i (if 0 + f 1 + f i )f m if i.

9 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES P m P n EMBEDDED BY O(1, ) 9 Now we prove that every monomial in { e i f j f k 0 i, j 1, j k m } lies in V S (U L ), m i=0 T p i (X m,m ). For each i {,..., m}, we have e 0 (if 0 + f 1 + f i ) e 0 (i f 0 + f 1 + f i + if 0 f 1 + if 0 f i + f 1 f i ) e 0 f 1 f i (mo V S (U L ), T p1 (X m,m ), T p (X m,m ) ). Inee, e 0 f0, e 0 f 0 f 1 an e 0 f 0 f i are in T p1 (X m,m ), e 0 f1 is in T p (X m,m ), e 0 fi V S (U L ). Similarly, one can prove that e 1 (if 0 + f 1 + f i ) e 1 if 0 f i (mo V S (U L ), T p1 (X m,m ), T p (X m,m ) ) for each i {,..., m}. So we have prove that e i f j f k m i=0 T p i (X m,m ) if i, j {0, 1} an k {0,..., m}. Note that, for each i {,..., m}, e i (if 0 + f 1 + f i )f 0 ie i f 0 f 1 + e i f 0 f i ; e i (if 0 + f 1 + f i ) ie i f 0 f 1 + ie i f 0 f i + e i f 1 f i ; e i (if 0 + f 1 + f i )f 1 ie i f 0 f 1 + e i f 1 f i moulo V S (U L ), m i=0 T p i (X m,m ). Thus e i (if 0 + f 1 + f i ) e i (if 0 + f 1 + f i )f 0 ( /i)e i (if 0 + f 1 + f i )f 1 is congruent to (/i)e i f 1 f i moulo V S (U L ), m i=0 T p i (X m,m ). Thus e i f 1 f i, an hence e i f 0 f 1 an e i f 0 f i, is in V S (U L ), m i=0 T p i (X m,m ). For every integer j such that i j an j, we have e i (if 0 + f 1 + f i )f j ie i f 0 f j + e i f 1 f j ; e i (jf 0 + f 1 + f j ) je i f 0 f j + e i f 1 f j moulo V S (U L ), m i=0 T p i (X m,m ). Hence e i (jf 0 + f 1 + f j ) (j/i)e i (if 0 + f 1 + f i )f j ( j/i)e i f 1 f j. This implies that e i f 1 f j, an hence e i f 0 f j, is containe in V S (U L ), m i=0 T p i (X m,m ), which completes the proof. Proposition 3.5. R(m, m + 1) is true for any m 1. Proof. We only prove the statement for m even, since the other case can be prove in the same way. If m is even, then s(m, m+1) = 3m/+1. Let p 1,..., p m/ L an let q 1,..., q m+1 P m,m+1 \L. Choose a subvariety P m,m = P(V ) P(W ) P m,m+1 in such a way that the intersection of P m,m with L is P m,m. We enote it by H. Specialize q 1,..., q m+1 on H \ L. Suppose that p 1,..., p m/ H. Let Z = {p 1,..., p m/, q 1,..., q m+1 }. Then we have an exact sequence 0 I Z L H (1, ) I Z L (1, ) I (Z L) H,H (1, ) 0. By Proposition 3.4, R(m, m) is true. Thus im H 0 (I (Z L) H,H (1, )) = 0. So we have im H 0 (P m,m+1, I Z L H (1, )) = im H 0 (P m,m+1, I Z L (1, )). Thus we nee to prove that im H 0 (P m,m+1, I Z L H (1, )) = 1. Let Z be the resiual of Z L by H. Then H 0 (P m,m+1, I Z L H (1, )) H 0 (P m,m+1, I ez (1, 1)). Note that Z consists of L, m + 1 simple points q 1,..., q m+1 an m/ ouble points p 1,..., p m/. We enote by X m,m+1 the Segre variety X1,1 m,m+1 obtaine embeing Pm,m+1 by the morphism given by O(1, 1). The conition that im H 0 (P m,m+1, I ez (1, 1)) = 1, i.e., h P m,m+1( Z, (1, 1)) = is in

10 10 HIROTACHI ABO AND MARIA CHIARA BRAMBILLA (m + 1)(m + ) 1, is equivalent to the conition that the following subspace of V W has imension (m + 1)(m + ) 1: m/ m+1 V U L, T pi (X m,m+1), u i v i, where q i = [u i v i ]. Without loss of generality, we may assume that U L = f 0,..., f m 1 an that W = f 1,..., f m+1. Since p i L for each i {1,..., m/}, there are u i V an v i U L such that p i = [u i v i ]. Recall that T pi (X m,m+1 ) = V v i + u i W. So we have which implies that Thus T pi (X m,m+1) u i f m, f m+1 (mo V U L ), V U L, T pi (X m,m+1) = (V f 0 ) V U L, m/ T pi (X m,m+1), = (V f 0 ) V (U L W ), V (U L W ), m+1 u i v i m/ m/ u i f m, f m+1. u i f m, f m+1, m+1 u i v i. Note that T 1 = { e i f 0 0 i m } an T = { e i f j 0 i m, 1 j m 1 } are bases for V f 0 an V (U L W ) respectively. Let u i = e i 1 for every i {1,..., m/}. Then T 3 = { e i f j 0 i m/ 1, m j m + 1}. Let T 4 be the set of vectors of the stanar basis for V W not inclue in the set T 1 T T 3. Then T 4 consists of m + istinct non-zero vectors. Choose m + 1 istinct elements of T 4 as u i v i s. Then 4 T i spans a vector space of imension (m + 1)(m + ) 1. Thus we complete the proof. Let U M be another two-coimensional subspaces of W an let M be the subvariety of P m,n of the forms P(V ) P(U M ). If L an M are general, then we have [( ) ( ) ( )] n + n n im H 0 (P m,n, I L M (1, )) = (m + 1) + = 4(m + 1). This is equivalent to the conition that the subspace of V W spanne by V U L an V U M has coimension 4(m + 1). Definition 3.6. Let p 1,..., p m+1 be general points of L an let q 1,..., q m+1 be general points of M. We enote by W m,n the subspace of V S (W ) spanne by V S (U L ), V S (U M ), m+1 T p i (X m,n ) an m+1 T q i (X m,n ). Then im W m,n is expecte to be (m + 1) ( ) n+. We say that Q(m, n) is true if W m,n has the expecte imension. Remark 3.7. Keeping the same notation as in the previous efinition, we enote by Z the zero-imensional subscheme {p 1,..., p m+1, q 1,..., q m+1 }. Then Q(m, n) is true if an only if im H 0 (P m,n, I Z L M (1, )) = 0. Proposition 3.8. If Q(m, n) an R(m, n ) are true, then R(m, n) is also true. Proof. Let s = s(m, n), let p 1,..., p s (m+1) L an let q 1,..., q m+1 P m,n \ L. Suppose that p 1,..., p s (m+1) L M, p s m 1..., p s (m+1) L \ (L M) an q 1,..., q m+1 M. Let Z = Z M. Then we have an exact sequence 0 I Z L M (1, ) I Z L (1, ) I Z (L M),M(1, ) 0.

11 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES P m P n EMBEDDED BY O(1, ) 11 Taking cohomology gives rise to the following exact sequence: 0 H 0 (P m,n, I Z L M (1, )) H 0 (P m,n, I Z L (1, )) H 0 (M, I Z (L M),M(1, )). By the assumption that Q(m, n) is true, we have im H 0 (P m,n, I Z L M (1, )) = 0. following inequality hols: im H 0 (P m,n, I Z L (1, )) im H 0 (M, I Z (L M),M(1, )). Hence if R(m, n ) is true, then so is R(m, n). Lemma 3.9. If Q(m, n) an Q(1, n) are true, then Q(m, n) is also true. Thus the Proof. Let V be a (m 1)-imensional subspace of V an let V be a two-imensional subspace of V. Suppose that V can be written as the irect sum of V an V. Let U = V S (U L ), V S (U M ), T p (X m,n ). Suppose that p = ([u], [v]) P m,n = P(V ) P(U L ). Then V v V S (W ). Thus T p (X m,n ) T p (X m,n ) (mo V S (U L )). Similarly, one can prove that T q (X m,n ) T q (X 1,n ) (mo V S (U L )) if q = ([u], [v]) P(V ) P(W ). This means that if p 1,..., p m+1 P(V ) P(W ) an if q 1,..., q m+1 P(V ) P(W ), then m+1 m+1 V S (U L ), T pi (X m,n ), T qi (X m,n ) = V S (U L ), m+1 T pi (X m,n ) V S (U L ), m+1 T qi (X 1,n ) In other wors, W m,n W m,n W 1,n. Thus if Q(m, n) an Q(1, n) are true, so is Q(m, n). Lemma Let n 3. Then Q(1, n) an Q(, n) are true. Proof. Here we only prove that Q(1, n) is true for any n 3, because the proof of the remaining case follows the same path. Without loss of generality, we may assume that U L = f 0,..., f n an U M = f,..., f n. Let U K = f 0, f 1, f n 1, f n an let K = P(V ) P(U K ). Note that S (W ) = S (U L ), S (U M ), S (U K ), so V S (W ) = V S (U L ), V S (U M ), V S (U K ). In another wor, H 0 (P 1,n, I L M K (1, )) = 0. Specializing p 1 an p on K L an q 1 an q on K M yiels the following short exact sequence, where Z = {p 1, p, q 1, q }: 0 I Z L M K (1, ) I Z L M (1, ) I (Z L M) K,K (1, ) 0. Since H 0 (P 1,n, I L M K (1, )) vanishes, so oes H 0 (P 1,n, I Z L M K (1, )). Thus, in orer to show that Q(1, n) is true, it is enough to prove that Q(1, 3) is true. Let p 1 an p be general points of L an let q 1 an q be general points of M. To prove that Q(1, 3) is true, we irectly show that (3.1) W 1,3 = V S (U L ), V S (U M ), T p1 (X 1,3 ), T p (X 1,3 ), T q1 (X 1,3 ), T q (X 1,3 ). Recall that T p (X 1,3 ) for p = [u v ] is isomorphic to V v + u vw. Thus we can check equality (3.1) as follows. Let S = C[e 0, e 1, f 0,..., f 3 ]. Choose ranomly u 1,..., u 4 V, v 1, v U L an v 3, v 4 U M. For each i {1,..., 4}, let T i be the ieal of S generate by u i vi, e 1 vi, u i v i f 0,..., u i v i f 3. Let I L an I M be the ieals of S generate by V S (U L ) an V S (U M ) respectively an let I = 4 T i + I L + I M. The minimal set of generators for I can be compute in Macaulay an we checke that the members of the minimal generating set form a basis for V S (W )..

12 1 HIROTACHI ABO AND MARIA CHIARA BRAMBILLA Proposition Let n 3. Then Q(m, n) is true for any m. Proof. The proof is by two-step inuction on m. Since we have alreay prove this proposition for m = 1 an, we may assume that m 3. By inuction hypothesis, Q(m, n) is true. Since Q(1, n) is true by Lemma 3.10, it immeiately follows from Lemma 3.9 that Q(m, n) is true. As we alreay mentione, the following is an immeiate consequence of Proposition 3.11: Corollary 3.1. Let m n. Then R(m, n) is true. Proof. The proof is by inuction on n. By Proposition 3.4, R(m, m) is true. The statement R(m, m + 1) is also true by Proposition 3.5. Assume that R(m, n ) is true for some n m. We may also assume that n 3. From Proposition 3.8 an Proposition 3.11 it follows, therefore, that R(m, n) is true. Thus we have complete the proof. Theorem Let k an s = s(m, n) be as given in Definition 3.1 an suppose that m n +. Then T (m, n; 1, ; s) is true for any s s. Proof. Since (m, n; 1, ; s) is subabunant, it is enough to prove that T (m, n; 1, ; s) is true. The proof is by inuction on n. If n = m, then s(m, m ) = 0. Thus T (m, m ; 1, ; 0) is clearly true. If n = m 1, then s(m, m 1) = (m 1)/ + 1. By Example.10, T (m, m 1; 1, ; s) is true for any s m/ + 1. Now suppose that the statement is true for some m n. By Proposition 3.3, T (m, n; 1, ; s) reuces to T (m, n ; 1, ; s (m + 1)) an R(m, n). By Corollary 3.1, R(m, n) is true for any m n. It follows therefore that T (m, n; 1, ; s) is true, which completes the proof. Define a function r(m, n) as follows: m 3 m r(m, n) = (m )(m + 1) if m is even an if n is o; otherwise. Corollary Suppose that n > r(m, n). Then T (m, n; 1, ; s) is true for any ( (m + 1) n+ ) s. m + n + 1 Proof. Since (m, n; 1, ; s) is subabunant, it suffices to show that T (m, n; 1, ; s) is true for s = (m+1)( n+ ). Note that m+n+1 (m + 1)k (m )(m+1) + s = (m + 1)k (m 3)(m+1) + (m + 1)k (m 3)(m+1)+1 + m 3 m (m+n+1) m 3 m (m+n+1) n+m 3 + (m+n+1) if n is even; if m an n are o; otherwise. It is straightforwar to show that if n > r(m, n), then s = s(m, n). Thus it follows immeiately from Theorem 3.13 that T (m, n; 1, ; s) is true. Remark If m = 1, then r(1, n) < 0. Since s(1, n) = n + 1, it follows that T (1, n; 1, ; n + 1) is true. Since (1, n; 1, ; n + 1) is equiabunant, T (1, n; 1, ; s) is therefore true for any s. 4. Segre-Veronese varieties P m P n embee by O(1, ): Superabunant Case In this section, we keep the same notation as in Section 3. Let k = n/ an let { (m + 1)k + 1 if n is even; s(m, n) = (m + 1)k + 3 otherwise.

13 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES P m P n EMBEDDED BY O(1, ) 13 It is straightforwar to show that (m, n; 1, ; s(m, n)) is superabunant. The main goal of this section is to prove that T (m, n; 1, ; s(m, n)) is true, which implies that T (m, n; 1, ; s) is true for any s s(m, n). Definition 4.1. Let s = s(m, n), let p 1,..., p s (m+1) be general points of L, let q 1,..., q m+1 be general points of P m,n \ L an let V m,n be the vector space V S (U L ), s (m+1) T pi (X m,n ), m+1 T q i (X m,n ). Then the following inequality hols: ( ) n + im V m,n (m + 1). We say that R(m, n) is true if the equality hols. Remark 4.. In the same way as in the proofs of Propositions 3.3 an 3.8, one can prove the following: (i) If R(m, n) an T (m, n ; 1, ; s(m, n )) are true, then T (m, n; 1, ; s(m, n)) is true. (ii) If Q(m, n) an R(m, n ) are true, then R(m, n) is true. In particular, if R(m, n ) is true, then R(m, n) is true, because Q(m, n) is true for n 3 by Proposition Definition 4.3. Suppose that (m, n) (1, 1). A 4-tuple (m, n; 1, ) is sai to be balance if ( ) n + m. Otherwise, we say that (m, n; 1, ) is unbalance. Remark 4.4. The notion of unbalance was first introuce for Segre varieties (see for example [Catalisano et al. 00, Abo et al. 006]). Then it was extene to Segre-Veronese varieties in [Catalisano et al. 008]. In the same paper it is also prove that if (m, n; 1, ) is unbalance, then T (m, n; 1, ; s) fails if an only if ( ) { ( )} n + n + (4.1) n < s < min m + 1,. In particular, T (m, ; 1, ; m + 1) is true if m 5 an T (m, 3; 1, ; m + 1) is true if m 8. Here we woul like to briefly explain why if s satisfies the above inequalities, then σ s (X m,n ) is efective. Let p 1,..., p s be generic points in X m,n. By assumption, we have s < n + 1. Thus there is a proper subvariety of P m,n of type P s 1,n that contains p 1,..., p s. Thus we have ( ) n + im σ s (X m,n ) s(im P m,n im P s 1,n ) + s [( ) ] n + = s + m + 1 s. It is straightforwar to show that if s fulfills inequalities (4.1), then [( ) ] { ( )} n + n + s + m + 1 s < min s(m + n + 1), (m + 1). Thus [ σ s (X m,n ) is efective. This also says that, for such an s, the expecte imension of σ s (X m,n ) (n+ ) ] is s + m + 1 s. Lemma 4.5. (i) If m 3, then R(m, n) is true for any n ; (ii) R(, n) is true for any n 3. Proof. We first prove (i) for m 8. By Proposition 3.11 an Remark 4. (ii), it suffices to show that R(m, ) an R(m, 3) are true for any m 8. Suppose that n {, 3}. If m 8, then (m, n; 1, ) is unbalance. Furthermore, (m, n; 1, ; m + 1) is superabunant. Thus R(m, n) can be

14 14 HIROTACHI ABO AND MARIA CHIARA BRAMBILLA reuce to T (m, n; 1, ; m + 1). By Remark 4.4, T (m, n; 1, ; m + 1) is true for n {, 3}. Thus R(m, n) is also true for m 8 an n {, 3}. The remaining cases of (i) can be checke irectly as follows: Let S = C[e 0,..., e m, f 0,..., f n ] an let s = s(m, n). Choose ranomly u 1,..., u s V, v 1,..., v s (m+1) U L an v s m,..., v s W. For each i {1,..., s}, let T i be the ieal of S generate by e 0 v i,..., e m v i, u i v i f 0,..., u i v i f n an let I L be the ieal generate by V S (U L ). Let I = s T i + I L. Computing the minimal generating set of I, we can check in Macaulay that the vector space spanne by homogeneous elements of I of the multi-egree (1, ) coincies with V S (W ). Claim (ii) can be also checke in the same way. Theorem 4.6. T (m, n; 1, ; s) is true for any s s(m, n). Proof. In Example.9, we showe that T (m, 1; 1, ; 3) is true for any m. One can irectly check that T (, ; 1, ; 4) is true. So, since R(, n) is true for any n 3 by Proposition 4.5, it follows from Remark 4. (i) that T (, n; 1, ; s) is true for any n 1. Suppose now that m 3. If n = 0, then s(m, 0) = 1, an obviously T (m, 0; 1, ; 1) is true. If n = 1, T (m, 1; 1, ; 3) is true. Moreover by Proposition 4.5, we know that R(m, n) is true for any n. Hence, from Remark 4. (i) it follows that T (m, n; 1, ; s(m, n)) is true for any n an any m 3. This conclues the proof. 5. Conjecture Let X m,n be the Segre-Veronese variety P m,n embee by the morphism given by O(1, ). The main purpose of this section is to give a conjecturally complete list of efective secant varieties of X m,n. Let V be an m-imensional vector space over C with basis {e 0,..., e m } an let W be an n- imensional vector space over C with basis {f 0,..., f n }. As mentione at the beginning of Section, for a given point p = [u v ] X m,n, the affine cone T p (X m,n ) over the tangent space to X m,n at p is isomorphic to V v + u vw. Let A(p) be the (m + 1) (m + 1) ( ) n+ matrix whose i th row correspons to e i v an let B(p) be the (n + 1) (m + 1) ( ) n+ matrix whose i th row correspons to u vf i. Then T p (X m,n ) is represente by the (m + n + ) (m + 1) ( ) n+ matrix C(p) obtaine by stacking A(p) an B(p): C(p) = ( A(p) B(p) ). For ranomly chosen points p 1,..., p s X m,n, let T s (X m,n ) = s T p i (X m,n ). Then T s (X m,n ) is represente by the s(m + n + ) (m + 1) ( ) n+ matrix C(p1,..., p s ) efine by C(p 1,..., p s ) = ( C(p 1 ) C(p ) C(p s ) ). Thus Remark. an semicontinuity imply that if { ( )} n + rank C(p 1,..., p s ) = min s(m + n + 1), (m + 1), then σ s (X m,n ) has the expecte imension. We programe this in Macaulay an compute the imension of σ s (X m,n ) for m, n 10 to etect potential efective secant varieties of X m,n. This experiment shows that X m,n is nonefective except for (m, n; 1, ) unbalance; (m, n) = (, n), where n is o an n 10; (m, n) = (4, 3). Remark 5.1. The efective cases we foun in the experiments are all well-known. In Remark 4.4, we gave an explanation of why if (m, n; 1, ) is unbalance, then X m,n is efective. Here we will iscuss the remaining known efective cases.

15 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES P m P n EMBEDDED BY O(1, ) 15 (i) It is classically known that σ 5 (X,3 ) is efective (see [Carlini an Chipalkatti 003] an [Carlini an Catalisano 007] for moern proofs). Carlini an Chipalkatti prove in their work on Waring s problem for several algebraic forms [Carlini an Chipalkatti 003] that T (, 5; 1, ; 8) is false. In [Ottaviani 006], Ottaviani then prove, as a generalization of the Strassen theorem [Strassen 1983] on three-factor Segre varieties, that T (, n; 1, ; s) fails if (n, s) = (k + 1, 3k + ) for any k 1. Here we sketch his proof for the efectiveness of X,k+1. Recall that X,k+1 is the image of the Segre-Veronese embeing ν 1, : P(V ) P(W ) P(V S W ), where V an W have imensions 3 an k + respectively. For every tensor φ V S W, let S φ : V W V W = V W be the contraction operator inuce by φ. If P, Q an R are the three symmetric slices of φ, then S φ can be written as a skew-symmetric matrix of orer 3(k + ) of the form S φ = P 0 R 0 P Q Q R 0 The rank of S φ is 3(k + ) for a general tensor φ V S W. On the other han, since the contraction operator corresponing to a ecomposable tensor has rank, we have rank S φ s, if φ is the sum of s ecomposable tensors. Since the ecomposable tensors correspon to the points of the Segre-Veronese variety, we can euce that if s = 3k +, then σ s (X,k+1 ) oes not fill P 3(k+3 ) 1. (ii) The efectiveness of σ 6 (X 4,3 ) can be prove by the existence of a certain rational normal curve in X 4,3 passing through generic six points of X 4,3. Let π 1 : P 4,3 P 4 an π : P 4,3 P 3 be the canonical projections. Given generic points p 1,..., p 6 P 4,3, there is a unique twiste cubic ν 3 : P 1 C 3 P 3 that passes through π (p 1 ),..., π (p 6 ). Let q i = ν3 1 (π (p i )) for each i {1,..., 6}. Since any orere subset of six points in general position in P 4 is projectively equivalent to the orere set {π 1 (p 1 ),..., π 1 (p 6 )}, there is a rational quartic curve ν 4 : P 1 C 4 P 4 such that ν 4 (q i ) = π 1 (p i ) for all i {1,..., 6}. Let ν = (ν 4, ν 3 ) an let C = ν(p 1 ). Then C passes through p 1,..., p 6. The image of C uner the morphism given by O(1, ) is a rational normal curve of egree 10(= ) in P 10. Thus we have im σ 6 (X 4,3 ) (7 1) = 46 < 6( ) 1 = 47, an so σ 6 (X 4,3 ) is efective. See [Carlini an Chipalkatti 003] for an alternative proof. The experiments with our program an Remark 5.1 suggest the following conjecture: Conjecture 5.. Let X m,n be the Segre-Veronese variety P m,n embee by the morphism given by O(1, ). Then σ s (X m,n ) is efective if an only if (m, n, s) falls into one of the following cases: (a) (m, n; 1, ) is unbalance an ( ) { ( n+ n < s < min m + 1, n+ )} ; (b) (m, n, s) = (, k + 1, 3k + ) with k 1; (c) (m, n, s) = (4, 3, 6). It is known that the conjecture is true for m = 1 (see [Carlini an Chipalkatti 003]). Here we prove that the conjecture is true for m = as a consequence of Theorems 3.13 an 4.6. Theorem 5.3. T (, n; 1, ; s) is true for any s except (n, s) = (k + 1, 3k + ) with k 1. Proof. Assume first that n = k is even. Then we have s = s = 3k + 1. Hence, from Theorems 3.13 an 4.6, it follows that T (, k; 1, ; s) is true for any s. Suppose now that n = k+1 is o. Then we have s = 3k+1 an s = 3k+3. Thus T (, n; 1, ; s) is true for any s 3k + 1, by Theorem 3.13, an for any s 3k + 3, by Theorem 4.6..

16 16 HIROTACHI ABO AND MARIA CHIARA BRAMBILLA If n = 1, then s = 1 an s = 3. So it remains only to prove that also T (, 1; 1, ; ) is true. But this has been alreay prove in Example.10. So we complete the proof. In [Ottaviani 006] it is also claime that σ 3k+ (X,k+1 ) is a hypersurface if k 1 an that this can be prove by moifying Strassen s argument in [Strassen 1983]. Then it follows that the equation of σ 3k+ (X,k+1 ) is given by the pfaffian of S φ, where S φ is the skew-symmetric matrix introuce in Remark 5.1 (i). We conclue this paper by giving an alternative proof of the fact that σ 3k+ (X,k+1 ) is a hypersurface for k 1. Definition 5.4. Suppose that n is o. Let s = 3 n/ +, let p 1,..., p s 3 be general points of L, let q 1, q, q 3 be general points of P,n \ L an let V,n be the vector space V S (U L ), s 3 T p i (X,n ), 3 T q i (X,n ). Then the following inequality hols: ( ) n + im V,n 3. We say that R(, n) is true if the equality hols. Lemma 5.5. Let n 3 be an o integer. Then R(, n) is true. Proof. The proof is very similar to that of Proposition 3.5. One can easily prove that if Q(, n) is true an if R(, n ) is true, then R(, n) is true. Since we have alreay prove that Q(, n) is true, it suffices to show that R(, 3) is true. Let p 1, p L an let q 1, q, q 3 P,3. Choose a subvariety H of P,3 of the form P, = P(V ) P(W ) such that P, intersects L in P,0. Suppose that p 1, p H. Specializing q 1, q an q 3 in H \ L, we obtain an exact sequence: 0 I Z L H (1, ) I Z L (1, ) I (Z L) H,H (1, ) 0, where Z = {p 1, p, q 1, q, q 3 }. Since we have alreay prove that R(, ) is true, we can conclue that im H 0 (I (Z L) H,H (1, )) = 0. Thus it is enough to prove that H 0 (I Z L H (1, )) = 0 or H 0 (I ez (1, 1)) = 0, where Z is the resiual of Z L by H. Note that Z consists of two ouble points p 1, p, three simple points q 1, q, q 3 in H an L. Let X,3 be the Segre-Veronese variety P,3 embee by O(1, 1). We want to prove that L, T p i (X,3 ) an 3 T q i (X,3 ) span V W. Note that if p = [u v], then T p (X,3 ) = V v + u W. Now assume the following: U L = f 0, f 1 an W = f 1, f, f 3 ; p 1 = e 0 f 0, p = e 1 f 1 V U L ; q 1 = e f, q = e f 3 V W. For any non-zero q 3 V W, one can show that V W = L, T pi (X,3), Thus we complete the proof. 3 T qi (X,3). Proposition 5.6. If (n, s) = (k + 1, 3k + ) for k 1, then im σ s (X,n ) = 3 ( n+ ). Proof. The proof is by inuction on k. It is well-known that σ 5 (X,3 ) is a hypersurface. Thus we may assume that k. Let p 1,..., p s P,n. Then there is a subvariety L of P,n of the form P,n such that p 1, p, p 3 L. Let us suppose that p 4,..., p s P.n \ L. Then we have an exact sequence 0 I Z L (1, ) I Z (1, ) I Z L,L (1, ) 0. Taking cohomology, we get im H 0 (I Z (1, )) im H 0 (I Z L (1, )) + im H 0 (I Z L,L (1, )).

17 SECANT VARIETIES OF SEGRE-VERONESE VARIETIES P m P n EMBEDDED BY O(1, ) 17 By Lemma 5.5, im H 0 (I Z L (1, )) = 0. Thus, by inuction hypothesis, im H 0 (I Z (1, )) im H 0 (I Z L,L (1, )) 1. As alreay claime, it is known that T (, n; 1, ; s) oes not hol, i.e. im H 0 (I Z (1, )) 1. It follows that σ s (X,n ) is a hypersurface in the ambient space P 3(n+ ) 1. Acknowlegements. We thank Tony Geramita an Chris Peterson for organizing the Special Session on Secant Varieties an Relate Topics at the Joint Mathematics Meetings in San Diego, January 008. Our collaboration starte in the stimulating atmosphere of that meeting. We also thank Giorgio Ottaviani for useful suggestions an Enrico Carlini for his communications regaring this paper an a careful reaing. The first author woul like to thank FY 008 See Grant Program of the University of Iaho Research Office. The secon author (partially supporte by Italian MIUR) woul like to thank the University of Iaho for the warm hospitality an for financial support. References [Abo et al. 008] H. Abo, G. Ottaviani an C. Peterson, Defectiveness of Grassmannians of planes, in preparation. [Abo et al. 006] H. Abo, G. Ottaviani an C. Peterson, Inuction for secant varieties of Segre varieties, arxiv: math.ag/ , to appear in Trans. AMS. [Abrescia 008] S. Abrescia, About efectivity of certain Segre-Veronese varieties, to appear in Cana. J. Math. [Alexaner an Hirschowitz 1995] J. Alexaner an A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995), no., 01. [Aoki et al. 007] S. Aoki, T. Hibi, H. Ohsugi an A. Takemura, Markov basis an Groebner basis of Segre-Veronese configuration for testing inepenence in group-wise selections, arxiv: [Ballico 006] E. Ballico, On the non-efectivity an non weak-efectivity of Segre-Veronese embeings of proucts of projective spaces Port. Math. 63 (006), no. 1, [Baur et al. 007] K. Baur, J. Draisma, W. e Graaf, Secant imensions of minimal orbits: computations an conjectures, Experimental Mathematics 16():39 50, 007. [Baur an Draisma 007] K. Baur an J. Draisma, Secant imensions of low-imensional homogeneous varieties, arxiv: [Brambilla an Ottaviani 008] M. C. Brambilla an G. Ottaviani, On the Alexaner-Hirschowitz theorem, J. Pure an Applie Algebra 1 (008), no. 5, [Bürgisser et al. 1997] P. Bürgisser, M. Clausen, M. A. Shokrollahi, Algebraic Complexity theory, vol. 315, Grunl. Math. Wiss., Springer, [Carlini an Catalisano 007] E. Carlini an M. V. Catalisano, Existence results for rational normal curves, J. Lon. Math. Soc. () 76 (007), no. 1, [Carlini an Chipalkatti 003] E. Carlini an J. Chipalkatti, On Waring s problem for several algebraic forms, Comment. Math. Helv. 78 (003), no. 3, [Catalisano et al. 00] M.V. Catalisano, A.V. Geramita an A. Gimigliano, Ranks of tensors, secant varieties of Segre varieties an fat points, Linear Algebra Appl. 355 (00), Erratum, Linear Algebra Appl. 367 (003), [Catalisano et al. 005] M. V. Catalisano, A. V. Geramita an A. Gimigliano, Higher secant varieties of Segre- Veronese varieties, Projective varieties with unexpecte properties, Walter e Gruyter GmbH & Co. KG, Berlin, 005, [Catalisano et al. 007] M. V. Catalisano, A.V. Geramita an A. Gimigliano, Segre-Veronese embeings of P 1 P 1 P 1 an their secant varities, Collect. Math. 58 (007), no. 1, 1 4. [Catalisano et al. 008] M. V. Catalisano, A.V. Geramita an A. Gimigliano, On the ieals of secant varieties to certain rational varieties, J. Algebra 319 (008), no. 5, [Eriksson et al. 005] N. Eriksson, K. Ranesta, B. Sturmfels an S. Sullivant, Phylogenetic algebraic geometry. In Projective varieties with unexpecte properties, 37 55, Walter e Gruyter, Berlin, 005. [Garcia et al. 005] L. D. Garcia, M. Stillman an B. Sturmfels, Algebraic geometry of Bayesian networks, J. Symbolic Comput. 39 (005), no. 3-4, [Grayson an Stillman] D. Grayson an M. Stillman, Macaulay, a software system for research in algebraic geometry, Available at [Lansberg 006] J. M. Lansberg, The borer rank of the multiplication of two by two matrices is seven, J. Amer. Math. Soc. 19 (006), [Lansberg 008] J. M. Lansberg, Geometry an the complexity of matrix multiplication, Bull. Amer. Math. Soc. (N.S.) 45 (008),