Characterization of Veronese varieties via projection in Grassmannians Enrique Arrondo and Raffaella Paoletti A Giacomo Paoletti, amico e padre Abstract. We characterize for any d the d-uple Veronese embedding of Pn as the only variety that, under certain general conditions, can be projected from the Grassmannian of (d − 1)-planes in Pnd+d−1 to the Grassmannian of (d − 1)-planes in Pn+2d−3 in such a way that two (d− 1)-planes meet at most in one point. We also study the relation of this problem with the Steiner bundles over Pn. 2000 Mathematics Subject Classification: 14N05, 14N20 Introduction In [2], the first author characterized, under certain conditions, the embedding of Pn in G(1, 2n+1) via OPn(1)⊕2 as the only n-dimensional subvariety of G(1, 2n+ 1) that is isomorphically projectable into a subvariety of G(1, n + 1) (under a projection induced by a linear projection from P2n+1 to Pn+1). In this paper we deal with the analogous problem for the embedding of Pn in G(d− 1, nd+ d− 1) via OPn(1)⊕d. One first problem is that the naive generalization to arbitrary d of the case d = 2 does not work (see Remark 2.2). As we will see in Section 2, the right notion for a subvariety of a Grassmann variety of (d − 1)-planes to be special under projection is to be projectable into a subvariety of G(d − 1, n + 2d − 3) in such a way that any two (d− 1)-planes meet at most in one point. With this notion in mind, in Section 3 we prove our main projectability result (Theorem 3.1) by essentially repeating most of the steps in the proof of [2]. The only step we cannot reproduce is the result that a projectable variety has positive defect, so that we will need to add this as a hypothesis. Finally, in Section 4 we observe that the projectability of Pn in Grassmannians is closely related to Steiner bundles. However, not all of them appear. We will see for instance that, if n ≥ 3, we never obtain the most particular case, namely the Schwarzenberger bundles. 2 Enrique Arrondo and Raffaella Paoletti Both authors acknowledge the financial support of the Italy-Spain project “Acción Integrada” HI00-128. The first author has also been supported by the Spanish Ministry of Science and Technology project BFM2000-0621. The second author has been partially supported by Italian MURST and GNSAGA-INdAM. Both authors belong to EAGER (European contract PRN-CT-2000-00099). 1. Preliminaries and projectability in Grassmannians of lines We will work over an algebraically closed field k of characteristic zero. We will denote by G(r,m) the Grassmann variety of r-linear spaces in Pm. A linear pro- jection G(r,m) 99K G(r,m′) will mean the natural (rational) map induced by the we changed -----> to 99K in the whole document corresponding linear projection Pm 99K Pm′ . The main example we are going to consider is the following. Example 1.1. Consider the natural embedding of Pn in G(d − 1, nd + d − 1) defined by OPn(1)⊕d and call V its image. In coordinates, it can be described by associating to each (x0 : . . . : xn) ∈ Pn the (d − 1)-plane spanned by the rows of the matrix  x0 . . . xn 0 . . . 0 . . . 0 . . . 0 0 . . . 0 x0 . . . xn . . . 0 . . . 0 ... . . . ... 0 . . . 0 0 . . . 0 . . . x0 . . . xn  . We consider now the linear projection Pnd+d−1 99K Pn+d−1 defined by (z10 : z11 : . . . : z1n : z20 : . . . : zd,n−1 : zdn) 7→ (z10 : z11 + z20 : . . . : z1,d−1 + . . .+ zd0 : . . . : z1n + . . . . . .+ zd,n+1−d : . . . : zd−1,n + zd,n−1 : zdn). This projection induces a projection from G(d−1, nd+d−1) to G(d−1, n+d−1) under which the image of the above Pn consists of the (d− 1)-planes spanned by the rows of the matrix x0 x1 . . . xn 0 . . . 0 0 x0 x1 . . . xn . . . 0 ... . . . . . . . . . ... 0 . . . 0 x0 x1 . . . xn  . (1) This is still an embedding of Pn in G(d− 1, n+ d− 1), since the maximal minors of the above matrix (which give the image of Pn after the Plücker embedding of Characterization of Veronese varieties via projection in Grassmannians 3 G(d − 1, n + d − 1) in P(n+d d )−1) define the d-uple Veronese embedding of Pn in P(n+d d )−1. By this reason we will call this subvariety V ⊂ G(d− 1, nd+ d− 1) (or any of its isomorphic projections) the n-dimensional Veronese variety. Following [2] we will use the following definitions: Definition 1.2. We will say that a subvariety X ⊂ G(d − 1, N) of dimension n is nondegenerate if the union of all the (d − 1)-planes parametrized by X is not contained in a hyperplane of PN . We will also say that X is uncompressed if the union in PN of all of the (d − 1)-planes parametrized by X has the expected dimension n + d − 1. Otherwise, if the dimension is smaller, we will say that X is compressed. The main result of [2] is the following: Theorem 1.3. The only nondegenerate uncompressed n-dimensional subvariety of G(1, 2n+ 1) that can be isomorphically projected to G(1, n+ 1) is the Veronese variety (i.e. the one of Example 1.1 for d = 2). The main tool to prove that result is the use of the generalization to the Grass- mannians context of the notions of secant variety and secant defect. The precise definitions are as follows: Definition 1.4. The k-secant variety of a variety X ⊂ G(1, N) is the variety SkX ⊂ G(rk, N) consisting of the closure of the set of linear spans of k+1 general lines of X (observe that, although one should expect rk to be 2k+1, its value could be actually smaller). The k-secant defect of X is the dimension δk of the set of lines of X contained in a general space of SkX. It follows that SkX has dimension (k + 1)(n− δk). The two main ingredients in the proof of Theorem 1.3 are the fact that a projectable variety has positive first defect δ1 (actually in this case one can prove that δ1 = 1), and the fact that more precisely the set of lines contained in a general 3-plane of S1X is a conic (specifically, the set of lines meeting two skew lines). It is thus easy to prove that a projectable variety in G(1, 2n+1) has defects δk = k for k = 1, . . . , n and that the divisors of X obtained for k = n− 1 yield an isomorphism of X to Pn whose inverse is the double Veronese embedding. 2. Projectability in arbitrary Grassmannians The obvious natural generalizations of the notions of k-secant varieties and k- secant defects in arbitrary Grassmannians are the following: 4 Enrique Arrondo and Raffaella Paoletti Definition 2.1. The k-secant variety of X ⊂ G(d− 1, N) is the closure SkX in G(rk, N) of the set of linear spans of k + 1 general (d − 1)-planes of X. The k- secant defect of X is the dimension δk of the set of (d− 1)-planes of X contained in a general space of SkX. It follows that SkX has dimension (k+ 1)(n− δk). In the case k = 1, we will just speak about secant variety SX and secant defect δ. In general, one should expect r1 = 2d− 1. Example 2.2. The Veronese variety V defined in Example 1.1 has first secant variety of dimension 2n−2, i.e. less than expected. Indeed, let (z10 : z11 : . . . : z1n : z20 : . . . : zdn) be homogeneous coordinates on Pnd+d−1 and for each k ∈ {1, . . . , d} consider the n-dimensional linear subspaces Πk defined by the equations: zij = 0 for j = 1, . . . , d and i 6= k. Then each point of V can be represented as the span of d corresponding points P1, . . . , Pd s.t. Pk ∈ Πk. Let ϕ : V×V 99K G(2d−1, nd+d−1) defined by ϕ ((Λ1,Λ2)) = 〈Λ1,Λ2〉. Then SV = Im ϕ. Let π ∈ Im ϕ, π = 〈Λ1,Λ2〉 with Λ1 = 〈P1, . . . , Pd〉 and Λ2 = 〈Q1, . . . , Qd〉. Then π = 〈P1Q1, . . . , PdQd〉 where PkQk is the line through Pk and Qk. Thus ϕ−1(π) = { (〈P̃1, . . . , P̃d〉, 〈Q̃1, . . . , Q̃d〉) / P̃1 6= Q̃1 ∈ P1Q1 } has dimension 2 and dimSV = 2n − 2. In the same way, it is easy to see that V has secant defects δk = k for k = 1, . . . , n. Remark 2.3. Let us see that the Veronese varieties cannot be characterized by means of the projectability to G(d−1, n+d−1) that we have seen in Example 1.1. In fact, let X ⊂ G(d−1, N) be an irreducible nondegenerate n-dimensional variety and let us analyze when it is possible to project it isomorphically to G(d− 1, n+ d − 1). A projection will be induced by a linear projection πA : PN → Pn+d−1 with center a linear space A of dimension N −n− d. The fact that πA induces an isomorphism between X and its image in G(d− 1, n+ d− 1) is equivalent to the following property: for any Λ1,Λ2 ∈ X (maybe infinitely close), dim(〈Λ1,Λ2〉 ∩A) < dim(〈Λ1,Λ2〉)− d. Actually this condition says that dimπA(〈Λ1,Λ2〉) ≥ d, so that πA(Λ1), πA(Λ2) represent distinct points in G(d − 1, n + d − 1). In particular, if Λ1,Λ2 are skew the above condition states that dim(〈Λ1,Λ2〉 ∩A) < d− 1. Consider the following incidence variety: I = {(A,Π) | dim(A ∩Π) ≥ d− 1} ⊂ G(N − n− d,N)× SX and let p, q be the corresponding projections. The elements of p(I) represent “bad centers” of projections. A dimensional count on the fibers of q shows that dim(I) = dim(SX)+dimG(N−n−d,N)−nd. Since dim(SX) ≤ 2n (and in fact one expects to have an equality), it follows that dim(I) ≤ n(2 − d) + dimG(N − n − d,N). Therefore, only if d = 2 one can expect p to be surjective, which means that all the possible centers of projection should be bad. On the contrary, if d ≥ 3, Characterization of Veronese varieties via projection in Grassmannians 5 one should be able to always find a good center of projection. Hence only for d = 2 the Veronese varieties would be special by means of its projectability to G(d− 1, n+ d− 1) as in Example 1.1. This is in fact the content of Theorem 1.3. The right notion of projectability will be the following: Definition 2.4. Let X ⊂ G(d − 1, N) a smooth irreducible variety and let k be an integer such that 0 ≤ k ≤ d − 1. We will say that X is k-projectable to G(d− 1,M) if there exists a projection from G(d− 1, N) to G(d− 1,M) such that any two (d−1)-planes of the image of X (maybe infinitely close) do not meet along a linear space of dimension greater than or equal to k. Remark 2.5. 1) If X is k-projectable, then any two (d − 1)-planes of X itself do not meet along a linear space of dimension greater than or equal to k. 2) If k = d − 1, then X is (d − 1)-projectable to G(d − 1,M) if and only if X is isomorphically projectable to G(d− 1,M). 3) If k = 0, then X is 0-projectable to G(d−1,M) if and only if any two (d−1)- planes of the image of X are skew. In other words, the union of the (d− 1)-planes of X is a smooth scroll and it is isomorphic to its image in PM . 4) If X is k-projectable to G(d− 1,M), it is also (k + 1)-projectable to G(d− 1,M−1). In particular, ifX is 1-projectable to G(d−1,M) it is also isomorphically projectable to G(d− 1,M − d+ 2). 5) If X is uncompressed, then it is certainly not (d − 1)-projectable to G(d − 1, n+ d− 2). Therefore, X is not 0-projectable to G(d− 1, n+ 2d− 3). Example 2.6. We have seen (Example 1.1) that the Veronese variety is isomor- phically projectable toG(d−1, n+d−1), but this was not a sufficiently special prop- erty to characterize it (Remark 2.3). By the above fourth remark, 1-projectability to G(d− 1, n+ 2d− 3) is a stronger condition. Let us see that this is now a very restrictive property satisfied by the Veronese variety. Let us repeat first the dimension count of Remark 2.3 considering now 1- projectability to G(d − 1, n + 2d − 3) instead of (d − 1)-projectability to G(d − 1, n+ d− 1) (obviously the two notions coincide if d = 2). So let X ⊂ G(d− 1, N) be an irreducible nondegenerate n-dimensional variety. A projection to G(d − 1, n + 2d − 3) will be induced by a linear projection πA : PN → Pn+2d−3 with center a linear space A of dimension N − n − 2d + 2. For any Λ1,Λ2 ∈ X (maybe infinitely close), the fact that πA(Λ1) and πA(Λ2) meet at most in one point is equivalent to dim(〈Λ1,Λ2〉 ∩ A) < 1. Consider now the following incidence variety: I = {(A,Π) | dim(A ∩Π) ≥ 1} ⊂ G(N − n− 2d+ 2, N)× SX and let p, q be the corresponding projections. The elements of p(I) represent “bad centers” for 1-projectability. A new dimensional count on the fibers of q shows that this time dim(I) = dim(SX) + dimG(N − n− d,N)− 2n. Therefore, 6 Enrique Arrondo and Raffaella Paoletti dim(I) ≤ dimG(N − n − d,N) and in general one expects to have an equality. This shows that 1-projectability to G(d − 1, n + 2d − 3) is the right property to study (observe also that the fifth remark above shows that we can never have 0-projectability unless X is compressed). On the other hand, the Veronese varieties V are 1-projectable to G(d− 1, n+ 2d − 3): this follows immediately from the above dimensional count and the fact that the dimension of SV is smaller than expected (Example 2.2). We thus strengthen the conjecture in [2] to the following: Conjecture 2.7. The only smooth irreducible nondegenerate n-dimensional va- riety of G(d − 1, nd + d − 1) that is 1-projectable to G(d − 1, n + 2d − 3) is the Veronese variety. 3. Characterization of Veronese varieties in arbitrary Grassmannians. In this section we will prove the following evidence of Conjecture 2.7: Theorem 3.1. Let X ⊂ G(d− 1, nd+ d− 1) a smooth irreducible nondegenerate n-dimensional variety such that any two (possibly infinitely close) (d−1)-planes of X do not meet. If X has positive defect and is 1-projectable to G(d−1, n+2d−3), then X is the Veronese variety. The case n = 1 of Conjecture 2.7 is very easy to prove. Before proving it, we state without proof an easy technical lemma that we will need. Lemma 3.2. The set Ω ⊂ G(d − 1, 2d − 1) of (d − 1)-planes of P2d−1 meeting a fixed (d − 1)-plane Λ is (after the Plücker embedding) a hyperplane section of G(d−1, 2d−1) having a point of multiplicity d at the point represented by Λ. The intersection of G(d − 1, 2d − 1) with the tangent cone of Ω at this singular point consists of the set of (d − 1)-planes meeting Λ in a space of dimension at least d− 2. Proposition 3.3. The only smooth irreducible nondegenerate curve X in G(d − 1, 2d − 1) that is 1-projectable to G(d − 1, 2d − 2) is the embedding of P1 via the vector bundle OP1(1)⊕d. Proof. Let m be the degree of X after the Plücker embedding. For any Λ ∈ X, consider the set of (d−1)-planes of X meeting Λ. From Lemma 3.2 there are other m− d of them besides Λ. Hence there are m− d hyperplanes of P2d−1 containing two (d − 1)-planes of X (one of them being Λ). Of course everything is counted with multiplicity, and in particular it could happen that some or all of the above Characterization of Veronese varieties via projection in Grassmannians 7 (d − 1)-planes are infinitely close to Λ. But using in this case the second part of Lemma 3.2 we also get hyperplanes containing two infinitely close (d − 1)-planes of X (and in fact we get not just hyperplanes but even linear spaces of dimension d). As a consequence, if m 6= d we would get, varying Λ, an infinite family of hyperplanes containing two (d − 1)-planes of X. Therefore, through any point p ∈ P2d−1 we find a hyperplane containing two (maybe infinitely close) (d − 1)- planes ofX. This implies that the projection ofX from any point would produce in the image two (d−1)-planes meeting along a line, which contradicts our hypothesis on X. We thus get that X has degree d. We then have that the union of the (d − 1)-planes of X is a d-dimensional scroll in P2d−1 of degree d, and hence it is a rational normal scroll. Since two (d− 1)-planes of the scroll do not meet, the splitting type is necessarily (1, . . . , 1), i.e. X is the embedding of P1 via the vector bundle OP1(1)⊕d. ut The proof of Theorem 3.1 is based, following the steps in [2], on two main lemmas that we will state and prove first. Lemma 3.4. Let X ⊂ G(d− 1, N) a smooth irreducible nondegenerate variety of dimension n with positive defect and such that any two (maybe infinitely close) (d−1)-planes are skew. Then, for general Λ1,Λ2 ∈ X, the set YΠ of (d−1)-planes of X contained in Π := 〈Λ1,Λ2〉 is the one-dimensional family (or one of the two families if d = 2) of the Pd−1’s of the Segre variety P1 × Pd−1 ⊂ Π. In particular, the secant defect δ of X is equal to 1. Proof. Since the defect is positive, YΠ has dimension at least one. In the Grass- mann variety of (d−1)-planes of Π, the set of all those meeting Λ1 is a hyperplane section (under the Plücker embedding). Therefore, if YΠ had dimension bigger than or equal to two, then it would meet that hyperplane section in at least a curve. This implies that there would be infinitely many (d− 1)-planes of X meet- ing Λ1. This contradicts the assumption that any two (d − 1)-planes of X are skew. Hence YΠ is a curve, and the result follows now from [3], Theorem 5.1 or from (the proof of) Proposition 3.3. ut Lemma 3.5. Let X ⊂ G(d− 1, N) a smooth irreducible nondegenerate variety of dimension n with positive defect and such that any two (maybe infinitely close) (d− 1)-planes are skew. Then, for any integer k > 1, δk ≥ min {δk−1 + 1, n}. Proof. Since clearly δk ≥ δk−1, there is nothing to prove if δk−1 = n. We hence assume δk−1 < n. Take k + 1 general (d − 1)-planes Λ0, . . . ,Λk of X and write Π′ = 〈Λ0, . . . ,Λk−1〉 and Π = 〈Λ0, . . . ,Λk〉 = 〈Π′,Λk〉. By the generality of these (d − 1)-planes, YΠ′ has dimension δk−1 and Λk is not contained in Π′. Consider the incidence variety I = {(Λ′,Λ) | Λ ⊂ 〈Λ′,Λk〉} ⊂ YΠ′ × YΠ 8 Enrique Arrondo and Raffaella Paoletti and let p, q be the corresponding projections. Consider a general element Λ′ ∈ YΠ′ ; then p−1(Λ′) = G(d−1, 〈Λ′,Λk〉)∩X = Y〈Λ′,Λk〉 and this has dimension δ1 ≥ 1 by hypothesis. Therefore, I has dimension δk−1(X)+ δ1, so it is enough to show that q is generically finite over its image. We thus take a general element Λ ∈ Im q. Then q−1(Λ) = {Λ′ ∈ YΠ′ | Λ′ ⊂ 〈Λ,Λk〉} = {Λ′ ∈ X | Λ′ ⊂ Π′, Λ′ ⊂ 〈Λ,Λk〉} = X ∩ G(d − 1,Π′ ∩ 〈Λ,Λk〉). Since 〈Λ,Λk〉 is not contained in Π′ (because Λk is not) then Π′ ∩ 〈Λ,Λk〉 has dimension at most 2d − 2. Therefore any two (d − 1)-planes inside Π′ ∩ 〈Λ,Λk〉 should meet. Since two (d − 1)-planes of X cannot meet, it follows that X ∩ G(d − 1,Π′ ∩ 〈Λ,Λk〉) consists of at most one element. This completes the proof. ut Proof of Theorem 3.1: By Lemma 3.4, we have δ1 = 1 and iterating Lemma 3.5 we get δk ≥ min{k, n}. This implies in particular that δm = n for some m ≤ n. Therefore all the (d − 1)-planes of X are contained in the span of m + 1 general elements of X, which has dimension at most md+d−1. Since X is nondegenerate in G(d−1, nd+d−1), it follows that m = n and the span of n+1 general elements of X is the whole Pnd+d−1, i.e. the elements are in general position. Moreover, we get that, for k = 1, . . . , n, it holds δk = k and k + 1 general elements of X span a linear space of dimension kd+ d− 1. In the particular case k = n− 1 we have that n general elements of X span a linear space Π of codimension d in Pnd+d−1 and YΠ is a hypersurface of X. This hypersurface is contained in the hyperplane section HΠ of G(d − 1, nd + d − 1) consisting of the (d−1)-planes meeting Π. By Lemma 3.2, this hyperplane section is singular with multiplicity d along the set of (d−1)-planes contained in Π. Hence, as a divisor on X, the hyperplane section HΠ can be written as aYΠ + EΠ, with a ≥ d. If now Π′ is the span of two general elements ofX, we know from Lemma 3.4 that YΠ′ is the set of (d−1)-planes of the Segre embedding of P1×Pd−1 in Π′. This has degree (after the Plücker embedding) equal to d, i.e. the intersection product of HΠ and YΠ′ is d. This means, using the above identity, that d = a+ EΠ · YΠ′ . Therefore a = d and EΠ · YΠ′ = 0. Since YΠ′ “moves”, we obtain as in [2] that EΠ = 0. Summing up, we got that d · YΠ is linearly equivalent to the hyperplane section of X. Now the same reasoning as in [2] shows that X ∼= Pn and that X is the Veronese variety. ut 4. Relation with Steiner bundles In this section we will study the relation of the projection of Veronese varieties in Grassmannians with the so-called Steiner bundles on Pn. Remark 4.1. Let us analyze more closely Example 1.1. The matrix 1 represents a set of (d − 1)-planes in Pn+d−1 parametrized by Pn or, dually, a set of (n− 1)- planes in Pn+d−1. It is well-known that this dual representation corresponds to the set of n-secant spaces to the rational normal curve. Indeed we can take the Characterization of Veronese varieties via projection in Grassmannians 9 standard rational normal curve in Pn+d−1 (u0 : u1 : . . . : un+d−1) = (λn+d−1 : λn+d−2µ : . . . : µn+d−1). Then a set of (maybe infinitely close) n points on the curve is given by the zeros of a homogeneous polynomial x0λ n + x1λ n−1µ+ . . .+ xnµ n in the variables λ, µ. On the other hand, the system of hyperplanes of Pn+d−1 containing the (n − 1)- plane spanned by this set of points is generated by the independent hyperplanes of Pn+d−1: x0u0 + x1u1 + . . .+ xnun = 0 x0u1 + x1u2 + . . .+ xnun+1 = 0 ... x0ud−1 + x1ud + . . .+ xnun+d−1 = 0 whose coefficients yield the matrix 1 in Example 1.1. Of course this is not the most general situation, in the sense that only after a very particular projection we will get such a dual representation. In the language of vector bundles, the embedding of Pn in G(d − 1, nd + d − 1) is equivalent to the evaluation epimorphism O⊕(n+1)d Pn → OPn(1)d (with kernel ΩPn(1)⊕d) while the projection to G(d−1, n+d−1) is equivalent to an epimorphism O⊕n+d Pn → OPn(1)d. The dual of its kernel is called a Steiner bundle. And the bundle corresponding to the n-secant spaces to a rational normal curve (called Schwarzenberger bundle) is the most particular case. We refer to [1] for a thorough study of these bundles. Definition 4.2. We will call Schwarzenberger-Veronese variety a Veronese vari- ety in G(d− 1, n+ d− 1) that corresponds to a Schwarzenberger bundle. A first fact showing how special Schwarzenberger-Veronese varieties are is the following: Proposition 4.3. If n ≥ 3, the Schwarzenberger-Veronese variety with d ≥ 3 is never obtained as a projection of a Veronese variety V ′ ⊂ G(d−1, n+2d−3) such that any two (d− 1)-planes of V ′ meet at most in one point. Proof. Following the notations introduced in Example 1.1, we want to show that if n ≥ 3 it is not possible to factorize the projection ϕ of the Veronese variety V ' Pn ↪→ G(d− 1, nd+ d− 1) to G(d− 1, n+ d− 1) by means of a projection ψ to V ′ ⊂ G(d− 1, n+ 2d− 3) such that any two (d− 1)-planes of V ′ meet at most in one point. Assume for contradiction that it were possible. Then this projection would be induced by a linear map Pnd+d−1 99K Pn+2d−3 defined by (z10 : z11 : . . . : z1n : z20 : . . . : zd,n−1 : zdn) 7→ (z10 : z11 + z20 : . . . : z1,d−1 + . . .+ zd0 : . . . : z1n + . . . . . .+ zd,n+1−d : . . . : zdn : L1 : . . . : Ld−2) 10 Enrique Arrondo and Raffaella Paoletti where L1, . . . , Ld−2 are linear forms in z10, . . . , zdn. Then the image under ψ of a point (x0 : . . . : xn) ∈ Pn ' V would be represented by the (d− 1)-plane spanned by the rows of a matrix of the following type:  x0 x1 . . . xn 0 . . . 0 | l1,1 . . . l1,d−2 0 x0 x1 . . . xn . . . 0 | l2,1 . . . l2,d−2 ... . . . . . . . . . ... | . . . 0 . . . 0 x0 x1 . . . xn | ld,1 . . . ld,d−2  (2) where l1,1, . . . , ld,d−2 are linear forms in x0, . . . , xn. We will find a contradiction by showing that there exist two of these (d− 1)-planes meeting in more than one point. The span in Pn+2d−3 of the (d−1)-planes corresponding to the (clearly distinct) points (t0 : . . . : tn−1 : 0), (0 : t0 : . . . : tn−1) would be the linear space generated by the rows of the matrix t0 t1 . . . tn−1 0 0 . . . 0 | l′1,1 . . . l′1,d−2 0 t0 t1 . . . tn−1 0 . . . 0 | l′2,1 . . . l′2,d−2 ... . . . . . . . . . . . . ... | . . . 0 . . . 0 t0 t1 . . . tn−1 0 | l′d,1 . . . l′d,d−2 0 t0 t1 . . . tn−1 0 . . . 0 | l′′1,1 . . . l′′1,d−2 0 0 t0 t1 . . . tn−1 . . . 0 | l′′2,1 . . . l′′2,d−2 ... ... . . . . . . . . . ... | . . . 0 0 . . . 0 t0 t1 . . . tn−1 | l′′d,1 . . . l′′d,d−2  where l′i,j = li,j(t0, . . . , tn−1, 0) and l′′i,j = lij(0, t0, . . . , tn−1). Performing the obvious elementary transformations on the rows, the matrix can be reduced to the matrix: t0 . . . tn−1 0 0 . . . 0 l′1,1 . . . l′1,d−2 0 t0 . . . tn−1 0 . . . 0 l′2,1 . . . l′2,d−2 ... . . . . . . . . . ... . . . 0 . . . 0 t0 . . . tn−1 0 l′d,1 . . . l′d,d−2 0 0 . . . 0 t0 . . . tn−1 l′′d,1 . . . l′′d,d−2 0 . . . . . . 0 l′′1,1 − l′2,1 . . . l′′1,d−2 − l′2,d−2 ... ... ... ... 0 . . . . . . 0 l′′d−1,1 − l′d,1 . . . l′′d−1,d−2 − l′d,d−2  . Obviously, the upper-left block of the matrix has rank d + 1. The lower-right block can be regarded as a (d − 1) × (d − 2) matrix of linear forms in the Pn−1 of coordinates t0, . . . , tn−1. Thus this block has rank ≤ d − 3 on a subvariety of Pn−1 of codimension at most 2. Since n ≥ 3, it follows that there exist t0, . . . , tn−1 such that the rank of matrix is at most 2d− 2. In other words, the (d− 1)-planes Characterization of Veronese varieties via projection in Grassmannians 11 corresponding to the points (t0 : . . . : tn−1 : 0), (0 : t0 : . . . : tn−1) meet in more than one point. This yields the wanted contradiction. ut Let us consider now the case n = 2. Denote by (a : b : c) a system of Plücker coordinates on the space P2∗ of lines in P2: if l is the line passing through the points (x0 : x1 : x2) and (y0 : y1 : y2), then a = x0y1 − x1y0, b = x0y2 − x2y0, c = x1y2 − x2y1. (3) Remark 4.4. Observe that the line trough the points (t0 : t1 : 0) and (0 : t0 : t1) has coordinates (t20 : t0t1 : t21) which satisfy the equation b2 − ac = 0; viceversa, each line whose coordinates (a : b : c) satisfy the same equation is a line generated by two points of type (t0 : t1 : 0), (0 : t0 : t1). Proposition 4.5. If n = 2, the Schwarzenberger-Veronese variety with d ≥ 3 can be obtained as a projection of a Veronese variety V ′ ⊂ G(d− 1, 2d− 1) such that any two (d− 1)-planes of V ′ meet at most in one point. Proof. According to the proof of Proposition 4.3, we need to find linear forms l1,1, . . . , ld,d−2 in the matrix (4.1) such that for any two (a priori possibly infinitely close) points (x0 : x1 : x2), (y0 : y1 : y2) in P2 the matrix 0 BBBBBBBBBBBBBBBBBB@ x0 x1 x2 0 0 . . . 0 l′1,1 . . . l′1,d−2 0 x0 x1 x2 0 . . . 0 l′2,1 . . . l′2,d−2 ... . . . . . . . . . ... . . . 0 . . . 0 x0 x1 x2 0 l′d−1,1 . . . l′d−1,d−2 0 . . . 0 0 x0 x1 x2 l′d,1 . . . l′d,d−2 y0 y1 y2 0 0 . . . 0 l′′1,1 . . . l′′1,d−2 0 y0 y1 y2 0 . . . 0 l′′2,1 . . . l′′2,d−2 ... . . . . . . . . . ... ... ... 0 . . . 0 y0 y1 y2 0 l′′d−1,1 . . . l′′d−1,d−2 0 . . . 0 0 y0 y1 y2 l′′d,1 . . . l′′d,d−2 1 CCCCCCCCCCCCCCCCCCA (4) where l′i,j = li,j(x0, x1, x2) and l′′i,j = lij(y0, y1, y2) has rank at least 2d− 1. A long series of matrix manipulations shows that the choice  l1,1 . . . l1,d−2 l2,1 . . . l2,d−2 . . . ld,1 . . . ld,d−2  =  t2 0 . . . 0 t0 + 2t1 + t2 t2 0 −t0 t0 + 2t1 + t2 ... 0 −t0 . . . 0 0 0 t2 ... ... . . . t0 + 2t1 + t2 0 0 −t0  12 Enrique Arrondo and Raffaella Paoletti satisfies the required property. It is helpful for the computations to remark that the rank of the matrix 4 does not change if we substitute the points P = (x0 : x1 : x2) and Q = (y0 : y1 : y2) with any two distinct points of the line 〈P,Q〉. In particular, we should not care about the possibility that P and Q become infinitely close. ut References [1] Ancona V., Ottaviani G., Unstable hyperplanes for Steiner bundles and mul- tidimensional matrices, Adv. Geom. 1 (2001), 165-192. [2] Arrondo E., Projections of Grassmannians of lines and characterization of Veronese varieties, J. Algebraic Geom. 8 (1999), 85-101. [3] Lanteri A., Turrini C., Some formulas concerning nonsingular algebraic vari- eties embedded in some ambient variety, Atti. Accad. Sci. Torino 116 (1982), 463-474. [4] Schwarzenberger R.L.E., Vector bundles on the projective plane, Proc. London Math. Soc. 11 (1961), 623-640. Enrique Arrondo Departamento de Algebra, Facultad de Ciencias Matemáticas Universidad Complutense de Madrid 28040 Madrid, Spain Email: enrique arrondo@mat.ucm.es Raffaella Paoletti Dipartimento di Matematica e Applicazioni per l’Architettura Piazza Ghiberti 23 50123 Firenze, Italy Email: raffy@math.unifi.it