THE 2-SECANT VARIETIES OF THE VERONESE EMBEDDING

THE 2-SECANT VARIETIES OF THE VERONESE EMBEDDING ALEX MASSARENTI Cortona - July 4, 2012 1 Abstract. We prove that a general polynomial F k[x 0,..., x n] amits a ecomposition as sum of h = 2 powers of linear forms if an only if its secon partial erivatives lie on a line. In this way we work out set-theoretical equations for the variety of secant lines Sec 2 (V n of the Veronese variety V n. In [Ka] V. Kanev, aopting a ierent approach, prove that the same equations cut out ieal-theoretically Sec 2 (V n. Contents Introuction 1 1. Preliminaries on secant varieties 2 2. Veronese embeing an homogeneous polynomials 3 2.1. Catalecticant Varieties 4 2.2. Secant varieties of rational normal curves 5 3. The rst secant variety of V n 8 References 10 Introuction A variation on the Waring problem (coming from a question in number theory state by E. Waring in 1770, see [Wa], which states that every integer is a sum of at most 9 positive cubes aske which is the minimum positive integer h such that the generic polynomial of egree on P n amits a ecomposition as a sum of h -powers of linear forms. In 1995 J. Alexaner an A. Hirshowitz solve completely this problem over an algebraically close base el k of characteristic zero, see [AH]. They prove that the minimum integer h is the expecte one h = 1 n+1 ( n+, except in the following cases: n h n 2 2 h n 2 4 5 3 4 9 4 3 7 4 4 14 Polynomials often appear in issues of applie mathematics, As instance in signal theory [CM], algebraic complexity theory [BCS], coing an information theory [Ro]. In particular issues relate to ecompositions of homogeneous polynomials in sums of powers are of particular interest in signal theory an clearly in pure mathematics. Inee egree homogeneous polynomials can be seen as points if the projective space P N = Proj(k[x 0,..., x n ], while -powers of linear forms are 1 These are notes for a seminar I gave on July 4, 2012 in Cortona uring the summer school "Tensors: Geometric Complexity Theory an Waring problems" hel by J. M. Lansberg an Massimiliano Mella. 1

2 ALEX MASSARENTI parametrize by the Veronese variety V n PN. Therefore the geometric counterpart of this type of problems is the stuy of secant varieties to Veronese varieties. There is a line of research that stuies varieties parametrizing ecomposition of the form F = L 1 +... + L h of a general homogeneous polynomial F k[x 0,..., x n ]. These varieties, calle varieties of sums of powers, V SP for short are the main object of a series of papers [DK], [IK], [RS], [MM] for a birational approach, an [Do] for a survey on the theme. However, for applie sciences, is more interesting to etermine: - whether a polynomial amits a ecomposition into a number of linear forms, - an eventually to calculate explicitly the ecomposition. We focus the attention on the case Sec h (V n PN an aopt the philosophy ictate by the following trivial but crucial statement: "If F = h i=1 λ il i then its partial erivatives of orer l lie in the linear space L l 1,..., L l h for any l = 1,..., 1." In the case n = 2 we prove that in orer to establish if a homogeneous polynomials F k[x 0, x 1 ] amits a ecomposition as sum of h powers it is enough to verify that im(h h 1, where H is the linear space spanne by the partial erivatives of orer h of F. Furthermore, if im(h = h 1 we get a metho to work out the linear forms relate to F, 2.13. Finally trying to exten the metho in higher imension we compute the imension of the linear space of polynomials whose ( 1-erivatives lie in a general linear subspace H (P N, this space is also calle the ( 1-prolongation of H. Consequently we n the formula for the imension of Sec h (V2 n, an the secant efect of V2 n. Furthermore we obtain a criterion to etermine whether a polynomial amits a ecomposition in the cases = 2 an = 3, h = 2. Finally, in theorem 3.1, we will prove that a general polynomial F k[x 0,..., x n ] amits a ecomposition as sum of h = 2 powers of linear forms if an only if its secon partial erivatives lie on a line. In [Ka] V. Kanev, aopting a ierent approach, prove that the same conitions cut out ieal-theoretically Sec 2 (V n. 1. Preliminaries on secant varieties Let X P N be an irreucible an reuce non egenerate variety, the reuce closure of the graph of Γ h (X X... X G(h 1, N, α : X... X G(h 1, N, taking h general points to their linear span x 1,..., x h. Observe that Γ h (X is irreucible an reuce of imension hn. Let π 2 : Γ h (X G(h 1, N be the natural projection. Denote by S h (X := π 2 (Γ h (X G(h 1, N. Again S h (X is irreucible an reuce of imension hn. Finally let I h = {(x, Λ x Λ} P N G(h 1, N, with natural projections π h an ψ h onto the factors. Furthermore observe that ψ h : I h G(h 1, N is a P h 1 -bunle on G(h 1, N. Denition 1.1. Let X P N be an irreucible an reuce, non egenerate variety. The abstract h-secant variety is the irreucible an reuce variety Sec h (X := (ψ h 1 (S h (X I h.

THE 2-SECANT VARIETIES OF THE VERONESE EMBEDDING 3 While the h-secant variety is Sec h (X := π h (Sec h (X P N. It is immeiate that Sec h (X is a (hn + h 1-imensional variety with a P h 1 -bunle structure on S h (X. One says that X is h-efective if im Sec h (X < min{im Sec h (X, N}. 2. Veronese embeing an homogeneous polynomials Let ν : P n P N be the -Veronese embeing, an let V n = ν(p n be its image. Let [F ] P N = Proj(k[x 0,..., x n ] be a egree homogeneous polynomial. Fixe a positive integer h such that Sec h (V n PN we want to etermine whether [F ] Sec h (V n. We begin with the following simple observation: Remark 2.1. If F = h i=1 λ il i then its partial erivatives of orer l lie in the linear space L l 1,..., L l h for any l = 1,..., 1. The partial erivatives of orer l are ( n+l l homogeneous polynomials of egree l, so the previous observation is meaningful when h < ( ( n+l l an h < l+n n. The latter conition ensures that L l 1,..., L l h is a proper subspace of the projective space PN l parametrizing homogeneous polynomials of egree l. Consier the partial erivatives Fl l 0,...,l n := l F an the x l 0 0,..., x ln n incience variety I l,h = {(F, H Fl l 0,...,l n H, l 0 +... + l n = l} P N G(h 1, N l π 1 P N G(h 1, N l Let S h V l n G(h 1, N l be the abstract h-secant variety of V l n. Note that when h < ( n+l l the map π 1 is generically injective. Let X l,h = π 1 (I l,h P N be its image. By remark 2.1 we have Sec h (V n X l,h. We want to n cases when the equality hols in orer to get a simple criterion to establish whether [F ] Sec h (V n. Remark 2.2. The equality hols trivially when = 2. Let F k[x 0,..., x n ] 2 be a polynomial an let M F be the matrix of the quaratic symmetric form associate to F. Then F Sec h (V2 n if an only if rank(m F h. On the other han the rows of M F are exactly the partial erivatives of F. The Waring rank. Let h be the smallest integer such that Sec h (V n = PN. computation we expect: ( 1 h = n+ n+1. π 2 By a imensions This is almost always true, J. Alexaner an A. Hirschowitz in [AH] prove that the following are the only exceptional cases: n h 2 arbitrary n + 1 3 4 8 4 2 6 4 3 10 4 4 15

4 ALEX MASSARENTI 2.1. Catalecticant Varieties. Let us look closer at the variety X l,h. This variety parametrizes polynomials F k[x 0,..., x n ] whose partial erivatives of orer l span a (h 1-plane. Let M l,h be the ( ( n+l l n+ l l matrix whose lines are the l-th erivatives of F = i α 0+...+i n= i 0,...,i n x i0 0...xin n. Then X l,h is the eterminantal variety ene in P N by rank(m l,h h, where the α i0,...,i n are the homogeneous coorinates on P N. Let P M be the projective space parametrizing ( ( n+l l n+ l l matrices, an let M h P M be the variety ( of matrices of rank less or equal than h. Then M h is (n+l ( (n+ l a irreucible variety of imension M l h l h. Clearly the variety X l,h is a special linear section of M h. Lemma 2.3. The varieties X l,h an X l,h are isomorphic. Proof. The matrix M l,h whose lines are the ( l-th partial erivatives of F is the ( ( n+ l l n+l l matrix given by M l,h = M t l,h, where M t l,h is the transpose matrix of M l,h. Then the assertion follows. Example 2.4. Consier a polynomial of egree three in three variables F = a 0 x 3 + a 1 x 2 y + a 2 x 2 z + a 3 xy 2 + a 4 xyz + a 5 xz 2 + a 6 y 3 + a 7 y 2 z + a 8 yz 2 + a 9 z 3. The variety X 1,2 is ene by rank F x F y = rank 3a 0 2a 1 2a 2 a 3 a 4 a 5 a 1 2a 3 a 4 3a 6 2a 7 a 8 2. F z a 2 a 4 2a 5 a 7 2a 8 3a 9 Consier the projective space P 17 of 3 6 matrix with homogeneous coorinates The eterminantal variety M 2 ene by rank X 0,0,..., X 0,5, X 1,0,..., X 1,5, X 2,0,..., X 2,5. X 0,0 X 0,1 X 0,2 X 0,3 X 0,4 X 0,5 X 1,0 X 1,1 X 1,2 X 1,3 X 1,4 X 1,5 X 2,0 X 2,1 X 2,2 X 2,3 X 2,4 X 2,5 is irreucible of imension 17 4 = 13. The linear space 2X 1,0 X 0,1 = 0, 2X 2,0 X 0,2 = 0, 2X 0,3 X 1,1 = 0, X H = 0,4 X 1,2 = 0, 2X 0,5 X 2,2 = 0, 2X 2,3 X 1,4 = 0, 2X 2,4 X 1,5 = 0, X 0,4 X 2,1 = 0. 2 cuts out on M 2 the variety X 1,2, which is irreucible of imension 5 = im(sec 2 (V 2 3. Remark 2.5. Consiering a polynomial F k[x, y, z] 4 an proceeing as in example 2.4 one get im(x 1,2 = 6, so Sec 2 (V4 2 X 1,2. N k, where N k = ( k+n Proposition 2.6. Let = 2k be an even integer such that ( n+k k 1. The variety X k,n k is an irreucible hypersurface of egree ( n+k k in P N. n

THE 2-SECANT VARIETIES OF THE VERONESE EMBEDDING 5 Proof. The map π 2 : I k,n k G(N k 1, N k = P N k is ominant, so Ik,N k an X k,n k are irreucible. The assertion follows observing that X k,n k is ene by the vanishing of the eterminant of a ( ( n+k k n+k k matrix. Let us look at some consequences of the previous proposition. Example 2.7. Consier a polynomial F = a 0 x 4 + a 1 x 3 y + a 2 x 3 z + a 3 x 2 y 2 + a 4 x 2 yz + a 5 x 2 z 2 + a 6 xy 3 + a 7 xy 2 z + a 8 xyz 2 +a 9 xz 3 + a 10 y 4 + a 11 y 3 z + a 12 y 2 z 2 + a 13 yz 3 + a 14 z 4. The map π 2 : I 2,4 G(3, 5 is ominant, so X 2,4 is irreucible. Let Z 0, Z 1, Z 2, Z 3, Z 4, Z 5 be homogeneous coorinates on P 5 corresponing to x 2, xy, xz, y 2, yz, z 2 respectively. To compute the imension of the general ber of π 2 we can take the 3 plane H = {Z 0 = Z 3 = 0} which intersect V2 2 in a subscheme of imension zero. Computing the secon partial erivatives of F it turns out that π2 1 (H = {a 0 = a 1 = a 2 = a 3 = a 4 = a 5 = a 6 = a 7 = a 10 = a 11 = a 12 = 0}. So im(π2 1 (H = 14 11 = 3 an im(x 2,4 = 3 + 8 = 11. Since im(sec 4 V4 2 = 11 we get Sec 4 V 2 4 = X 2,4. Consier now π 2 : I 2,5 P 5. This map is ominant, so X 2,5 is irreucible. We have im(π 1 2 (H = 14 6 = 8, where H = {Z 0 = 0}. So im(x 2,5 = 13 an Sec 5 V 2 4 = X 2,5 is an hypersurface of egree 6 in P 14. Consier now the case = 4, n = 3, h = 9 an the secon partial erivatives. The map π 2 : I 2,9 P 9 is ominant an X 2,9 is irreucible. The general ber of π 2 has imension 24. Then im(x 2,9 = 24 + 9 = 33 an Sec 9 V4 3 = X 2,9 is an hypersurface of egree 10 in P 34. Finally in the case = 4, n = 4, h = 14 as before one can verify that X 2,14 is irreucible of imension 68, so is an hypersurface of egree 15 in P 69. Sec 14 V 4 4 = X 2,14 Example 2.8. Consier now a polynomial F k[x, y, z] 6 an the partial erivative of orer 3. For h = 8, 9 the map π 2 is ominant, so X 3,8 an X 3,9 are irreucible. First let us take h = 8. Proceeing as before we get im(π2 1 (H = 27 19 = 8 an im(x 3,8 = 24. So Sec 8 V6 2 X 3,8 is a ivisor. In the case h = 9 we have im(π2 1 (H = 27 10 = 17 an im(x 3,9 = 17 + 9 = 26. So is an hypersurface of egree 10 in P 27. Sec 9 V 2 6 = X 3,9 2.2. Secant varieties of rational normal curves. We begin with the simplest case n = 1. We enote by C P the egree rational normal curve, in this case Sec h (C P if an only if h 2. Lemma 2.9. Let F = i+j= α i,jx i 0x j 1 k[x 0, x 1 ] be a homogeneous polynomial, an let c = c(α i,j be the coecient of x h 0 in the partial erivative where C is a constant. h F x, with h 1. Then c = C α m s,s, 0 xs 1 Proof. Since the only monomial of F proucing c is x s 0 x s 1 the assertion follows.

6 ALEX MASSARENTI Theorem 2.10. For any h 2 we have Sec h(c = X h,h. Consequently if the partial erivatives of orer h of a homogeneous polynomial F k[x 0, x 1 ] lie in a hyperplane of P h then [F ] lies in Sec h (C. Proof. The partial erivatives of orer h of F are h + 1 homogeneous polynomials of egree h. If F = h i=1 λ il i the partial erivatives lie in L h 1,..., L h h which is a hyperplane h-secant to C h, but eg(c h = h an the latter conition is irrelevant. Let H be a general hyperplane in P h, forcing the partial erivatives of a egree polynomial G = i+j= α i,jx i 0x j 1 k[x 0, x 1 ] to lie in H gives h + 1 linear equations in the coecients of G. Without loss of generality we can suppose H to be the ene by the vanishing of the rst homogeneous coorinate on P h, then by 2.9 the ber of π 2 is the linear subspace of P N ene by The equations of π2 1 (H are inepenent so an the imension of X h,h is π 1 2 (H = {α s,s = 0, s = 0,..., h}. im(π2 1 (H = ( h + 1 = h 1, im(x h,h = im(i h,h = h 1 + h = 2h 1. Finally im(sec h (C = h + h 1 = 2h 1 yiels Sec h (C = X h,h. Remark 2.11. The partial erivatives of orer h of a homogeneous polynomial F k[x 0, x 1 ] epen on + 1 parameters. We consier the matrix M,h whose lines are the partial erivatives. From 2.10 we get equations for Sec h (C imposing rank(m,h h, that is the classical eterminantal escription of Sec h (C. Proposition 2.12. If [F ] Sec h (C is general then its ecomposition in powers of linear forms is unique. Proof. Let H P h be the hyperplane spanne by the partial erivatives of orer h of F. Since eg(c h = h an F is general we have H C h = {L h 1,..., L h h }. Then {L 1,..., L h } is the unique h-polyheron of F. Theorem 2.10 an proposition 2.12 immeiately suggest an algorithm. Construction 2.13. Given F k[x 0, x 1 ] to establish if F amits a ecomposition in h 2 linear forms, an eventually to n it we procee as explaine in the following iagram. {Compute im(h } im(h =h im(h =h 1 {F amits a h polyheron} {F oes not amit a h polyheron} {Compute H C h } Then H C h = {L h 1,..., L h h } an F = h i=1 λ il i. Example 2.14. Consier the case = 4, h = 2 an write F = i 0+i 1=4 α i,jx i 0x j 1. 2 F x 0 x 1 2 F x 2 0, 2 F we get x 2 1 Sec 2 (C 4 = {54α 2 3,1α 0,4 18α 3,1 α 2,2 α 1,3 144α 4,0 α 2,2 α 0,4 + 4α 3 2,2 + 54α 4,0 α 2 1,3 = 0}. Now consier the polynomial F = 9(x 4 0 + x 3 0x 1 + x 2 0x 1 + x 0 x 3 1 + 4x 4 1. Forcing

THE 2-SECANT VARIETIES OF THE VERONESE EMBEDDING 7 The secon partial erivatives of F lie on the line H = {X 0 3X 1 + 3X 2 = 0} Proj(k[x 0, x 1 ] 2. Now we have to compute the intersection H C 2, where C 2 = {X 2 1 4X 0 X 2 = 0} is the conic parametrizing squares of linear forms, we have H C 2 = {[15 + 6 6 : 6 + 2 6 : 1], [15 6 6 : 6 2 6 : 1]}. Finally we compute the linear forms giving the ecomposition L 1 = 5.44948x 0 + x 1 an L 2 = 0.55051x 0 + x 1. Now we consier the variety X 1,h. First we compute the imension of the general ber of π 2 : I 1,h G(h 1, n. Theorem 2.15. The ber of π 2 : I 1,h G(h 1, n on a general (h 1-plane H G(h 1, n is a linear subspace of P N of imension ( + h 1 im(π2 1 (H = 1. Furthermore the imension of X 1 is given by ( + h 1 im(x 1,h = h(n h + 1 + 1. Proof. We can suppose H = {X 0 =... = X n h = 0}, where {X 0,..., X n } are homogeneous coorinates on P n. We write a general polynomial [F ] P N in the form F = α i0,...,i n x i0 0...xin n. i 0+...+i n= The ber π2 1 (H is the linear subspace of PN ene by the vanishing of the coecients of x 0,..., x n h in the erivatives of F. Many of these equations are reunant, the iculty is in counting the exact number of inepenent equations. We prove that this number is ( +n 1 1 + ( +n 1 ( +h 1 by inuction on n h. If n h = 0 then H is an hyperplane an the conition on the ( erivatives are all inepenent, so the number of conitions is exactly the number of erivatives 1+n ( 1. Furthermore our formula for n h = 0 gives +n 1 ( 1 + +n 1 ( +n 1 ( = +n 1 1, an the case n h = 0 is verie. Consier now the general case, let H = {X 0 =... = X n h 1 = 0}, let C n h 1 the number of inepenent conitions obtaine forcing the partial erivatives to lie in H. Aing the conition {X n h = 0} gives new equations coming from the coecients of the form α 0,...,0,in h,i n h+1,...,i n, with i n h 0. These correspons to monomials of egree in the variables x n h,..., x n that contain the variable x n h. Now the monomials of egree not containing x n h are the monomials of egree in x n h+1,..., x n. So in the nal step we are aing ( ( + h + h 1 conitions. Then the number if inepenent equations is C n h = C n h 1 + ( ( +h +h 1, by inuction hypothesis ( ( ( + n 1 + n 1 + n (n h 1 1 C n h 1 = +. 1 So C n h = ( ( +n 1 1 + +n 1 ( +n (n h 1 1 ( + +h ( +h 1 ( = +n 1 ( 1 + +n 1 ( +h 1. Finally we have im(x 1,h = im(g(h 1, n + im(π2 1 (H = h(n h + 1 + ( +h 1 1. Proposition 2.16. If h n. The variety X 1,h is irreucible.

8 ALEX MASSARENTI Proof. By Lemma 2.3 it is equivalent to prove that X 1,h is irreucible. Consier the map π 2 : I 1,h G(h 1, n. By Theorem 2.15 the general ber of π 2 is a linear subspace of P N of imension im(π2 1 (H = ( +h 1 1 an π2 is surjective on G(h 1, n, so X 1,h is irreucible. In the cases = 2 an = 3, h = 2 we have that im(x 1,h = im(sec h (V n, since X 1,h is irreucible we get Sec h (V n = X 1,h. So if the rst partial erivatives of a polynomial F span a linear space of imension h 1 then F can be ecompose into a sum of h powers of linear forms. Remark 2.17. Consier the case = 2. By Alexaner-Hirshowitz theorem, see [AH], Sec h (V2 n P N if an only if h n. By theorem 2.15 an remark 2.2 we recover the eective imension of Sec h (V2 n, im(sec h (V2 n = 2nh h2 + 3h 2, 2 an consequently the formula for the h-secant efect of V n δ h (V n 2 = 2, h(h 1. 2 Up to now we have a complete escription for polynomials of arbitrary egree in two variables an for polynomials of egree two in any number of variables. So we concentrate on the case n 2 an 3. Theorem 2.18. Let n 2, 3, h n be positive integers. Then Sec h (V n is a subvariety of X 1,h of coimension ( + h 1 coim Sech (V n (X 1,h = h 2. Proof. Since n 2, 3, an h n, by Alexaner-Hirshowitz theorem the eective imension of Sec h (V n is the expecte one im(sec h (V n = min{hn + (h 1, N }. Furthermore n 2, 3, h n implies hn + (h 1 < N. So im(sec h (V n = hn + (h 1. Finally coim Sech (V n (X 1,h = h(n h + 1 + ( ( +h 1 1 hn (h 1 = +h 1 h 2. Corollary 2.19. If = 3 then Sec 2 (V3 n = X 2,2 for any n 2. Consequently if the secon partial erivatives of a homogeneous polynomial F k[x 0,..., x n ] 3 lie in a line of P n then [F ] lies in Sec 2 (V3 n. Proof. For h = 2, = 3 we have ( +h 1 h 2 = 0. We conclue by theorem 2.18. 3. The first secant variety of V n We focus on the case h = 2 without any assumptions on an n. We will use the equality n ( ( 1 + k + n =, 1 k=0 which can be easily prove by inuction on n. Theorem 3.1. If h = 2 for the rst secant variety of V n for any n an 3. Sec 2 (V n = X 2, 2 we have

THE 2-SECANT VARIETIES OF THE VERONESE EMBEDDING 9 Proof. Consier the iagram I 2, 2 = {(F, H Fl l 0,...,l n H, l 0 +... + l n = 2} P N G(1, N 2 π 1 π 2 P N G(1, N 2 clearly S 2 V2 n Im(π 2. Let F k[x 0,..., x n ] be a polynomial whose partial erivatives of orer 2 lie on a line H P N2. The erivatives of orer 3 of F are cubic polynomials whose rst partial erivatives are collinear. By 2.19 X 2,1 = X 2,2 = Sec 2 V3 n, so if we enote by G a partial erivative of orer 3 of F we get a ecomposition G = L 3 1 + L 3 2. Then G x0,..., G xn (which are partial erivatives of orer 2 of F lie on the line L 2 1, L 2 2, an so the line containing the partial erivative of orer 2 of F is exactly the secant line to V2 n given by L 2 1, L 2 2. This means that S 2 V n 2 = Im(π 2. Since the ber of π 2 are linear spaces we conclue that I 2, 2 an X 2, 2 are irreucible. We compute now the imension of the ber of π 2. We x on P N2 homogeneous coorinates Z 0,..., Z N2 corresponing to the monomials in lexicographic orer x 2 0, x 0 x 1,..., x 2 n, an consier the line H = {Z 0 = Z 1 =... = Z N2 2 = 0}. First consier monomials containing x 0. Forcing the erivatives to lie in {Z 0 = 0} we get ( 2+n n conitions (the monomials containing x 2 0, whose number is equal to the number of egree 2 monomials in x 0,..., x n. Imposing {Z 1 = 0} we get ( 2+n 1 n 1 conitions (the monomials containing x 0 x 1, whose number is equal to the number of egree 2 monomials in x 1,..., x n. Proceeing in this way forcing {Z n = 0} we get ( 2+n n n n = 1 conition (the monomials containing x 0 x n, whose number is equal to the number of egree 2 monomials in x n. Up to now we have n ( ( 2 + k 1 + n = k 1 k=0 conitions. Consier now the monomials containing x 1. Forcing {Z n+1 = 0} we get ( 2+n 1 n 1 conitions (the monomials containing x 2 1, whose number is equal to the number of egree 2 monomials in x 1,..., x n. Imposing {Z n+2 = 0} we get ( 2+n 2 n 2 conitions (the monomials containing x1 x 2, whose number is equal to the number of egree 2 monomials in x 2,..., x n. Proceeing in this way we get n 1 ( ( 2 + k 1 + n 1 = k 1 conitions. At the step x n 2 we have k=2 k=0 2 ( ( 2 + k 1 + 2 = k 1 k=0 more conitions. At the step x n 1 we have only to force {Z N2 2 = 0}, an we get ( 1 1 = 1 conitions. Summing up the ber π2 1 (H is a linear subspace of PN ene by n ( 1 + k n ( ( 1 + k + n + 1 = 1 + 1 = 2 1 1 k=0

10 ALEX MASSARENTI equations. So the ber has imension ( + n im(π2 1 (H = N ( + n + 2 = 1 ( + n Finally we look at the map π 2 : I 2, 2 S 2 V n 2, since π 2 is ominant we have im(x 2, 2 = im(i 2, 2 = 2n + 1. Since im(sec 2 V n = 2n + 1 the assertion follows. + 2 = 1. The case n = 2, h = 4. In the same spirit of Theorem 3.1 we obtain the following result. Theorem 3.2. If n = 2, h = 4 for the variety of 4-secant 3-planes of V 2 for any 2. Sec 4 (V 2 = X 4, 2 we have Proof. The case = 4 is example 2.7. Consier now the case = 5. The map π 2 : I 4,3 G(3, 5 is ominant, so X 4,3 an hence X 4,2 are irreucible. Let F k[x, y, z] 5 be a polynomial, looking at the proof of theorem 3.1 we get that forcing the partial erivatives of orer 3 of F to lie in {Z 0 = Z 3 = 0} gives ( 5 2 + 2 2 + ( 5 2 + 2 2 {monomials containing x 2 y 2 } = 20 3 = 17 conitions. Since im(x 4,2 = im(x 4,3 = 20 17 + im(g(3, 5 = 11 we conclue Sec 4 (V 2 5 = X 4,2. Consier the case = 6 an the partial erivative of orer 3. If the 3-th erivatives of F lie in a 3-plane then the rst partial erivative of F are egree 5 polynomials whose secon partial erivatives lie in a 3-plane. By the same trick of Theorem 3.1 we prove that the 3-plane containing the 3-th partial erivative has to be 4 secant to V3 2. So X 4,3 is irreucible, an as usual by counting imension we get the equality Sec 4 (V6 2 = X 4,3. Now we treat the general case by inuction on. Let F k[x, y, z] be a polynomial whose 2 -th erivative lies in a 3-plane. Then the rst partial erivative of F are polynomials of egree 1 whose 1 2 -th erivatives lie in a 3-plane. So F x, F y, F z can be ecompose as sums of four powers of linear forms. As before we conclue that the map π 2 : I 4, 2 G(3, N 2 is ominant, so X 4, 2 is irreucible. By combinatorial calculations similar to previous we compute im(x 4, = im(sec 4(V 2 2. Remark 3.3. In a completely analogous way one can show that Sec 5 (V 2 is ene by size 6 minors of the matrix of partial erivatives of orer 2 for = 4 an 6. References [AH] J. Alexaner, A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom. 4, 2, (1995, 201-222. [BCS] P. Bürgisser, M. Clausen, M.A. Shokrollahi, Algebraic Complexity Theory, volume 315 of Grun. er Math. Wiss. Springer, Berlin, 1997. [CM] P. Comon, B. Mourrain, Decomposition of quantics in sums of power of linear forms, Signal Processing, 53(2:93-107, 1996. Special issue on High-Orer Statistics. [Do] I. Dolgachev, Dual homogeneous forms an varieties of power sums, Milan Journal of Mathematics, 99. [DK] I. Dolgachev, V. Kanev, Polar covariants of plane cubics an quartics, Av. in Math. 98 (1993, 216301. [IK] A. Iarrobino, V. Kanev, Power Sums, Gorenstein Algebras an Determinantal Loci, Lecture notes in Mathematics, 1721, Springer, 1999.

THE 2-SECANT VARIETIES OF THE VERONESE EMBEDDING 11 [Ka] V. Kanev, Choral varieties of Veronese varieties an catalecticant matrices, J. Math. Sci. (New York 94 (1999, no. 1, 11141125, Algebraic geometry, 9. MR MR1703911 (2001b:14078 [MM] A. Massarenti, M. Mella, Birational aspects of the geometry of Varieties of Sum of Powers, arxiv:1010.1707v2 [math.ag]. [RS] K. Ranesta, F.O. Schreier, Varieties of Sums of Powers, J. Reine Angew. Math, 525, 2000. [Ro] S. Roman, Coing an Information Theory, Springer, New York, 1992. [Wa] E. Waring, Meitationes Algebricae, Cambrige: J. Archlate from the Latin by D.Weeks, American Mathematical Sociey, Provience, RI, 1991. Alex Massarenti, SISSA, via Bonomea 265, 34136 Trieste, Italy E-mail aress: alex.massarenti@sissa.it