On the ranks of the third secant variety of Segre-Veronese embeddings

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by  Edoardo Ballico

ON THE RANKS OF THE THIRD SECANT VARIETY OF
SEGRE-VERONESE EMBEDDINGS
EDOARDOBALLICOANDALESSANDRABERNARDI
7 Abstract. We give an upper bound for the rank of the border rank 3 partially symmetric
1 tensors. In the special case of border rank 3 tensors of k factors without any symmetry(Segre
0 case)wecanshowthatallranksamong3andk−1arise.
2

n Introduction
a
J In this paper we deal with the problem of finding a bound for the minimum integer r needed

4 to write a given tensor T as a linear combination of r indecomposable tensors. Such a minimum
2 number is now known under the name of rank of T (Definition 1.4). In order to be as general as

possible we will consider the tensor T to be partially symmetric, i.e.
]
G T ∈Sd1V1⊗···⊗SdkVk (1)
A
where the d ’s are positive integers and V ’s are finte dimensional vector spaces defined over an
i i
.
h algebraically closed field K. The decomposition that will give us the rank of such a tensor will be
t of the following type:
a
m r
T = λ v⊗d1 ⊗···⊗v⊗dk (2)
[ X i 1,i k,i
i=1

1 where λi ∈K and vj,i ∈Vj, i=1,...,r and j =1,...,k.
v Another very interesting and useful notion of “ rank ” is the minimum r such that a tensor T
5
canbe writtenas a limit ofa sequenceofrankr tensors. This lastinteger is calledthe border rank
4
of T (Definition 1.5) and clearly it can be strictly smaller than the rank of T (Remark 1.6). It
8
6 has become a common technique to fix a class of tensors of given border rank and then study all
0 the possible ranks arising in that family (cf. [6, 3, 9, 13]). The rank of tensors of border rank 2 is
. well known (cf. [6] for symmetric tensors, [2] for tensors without any symmetry, [4] for partially
1
0 symmetric tensors). The first not completely classified case is the one of border rank 3 tensors.
7 In [6, Theorem 37] the rank of any symmetric order d tensor of border rank 3 has been computed
1 and it is shown that the maximum rank reached is 2d−1. In the present paper, Theorem 1.7, we
:
v prove that the rank of partially symmetric tensors T as in (1) of border rank 3 can be at most
i
X k
r(T)≤−1+ 2d .
r X i
a i=1
In[9,Theorem1.8]J.Buczyn´skiandJ.M.Landsbergdescribedthecasesinwhichtheinequalityin
Theorem 1.7 is an equality: when k =3 and d =d =d =1 they show that there is an element
1 2 3
ofrank 5. All ranks for border rank 3 partially symmetric tensors aredescribed in [8]when k =3,
d = d = d = 1 and n = 1 for at least one integer i. Therefore our Theorem 1.7 is the natural
1 2 3 i
extensionofthe twoextremecases(tensorswithoutanysymmetrywhered =1foralli=1,...,k
i
and totally symmetric case where k =1).
In the special case of tensors without any symmetry, i.e. T ∈ V ⊗···⊗V , we will be able to
1 k
show, in Theorem 1.8, that any rank between 3 and k−1 arises among border rank 3 tensors. In
theproofofthistheoremwewilldescribethe structureofoursolutions: theyareallobtainedfrom
(P1)k taking as a border scheme a degree 3 connected curvilinear scheme (Proposition 3.1 gives
the case of rank k−1 when k ≥4 and the other cases follows taking a smaller number of factors).
In[5]wedefinedthe notionofcurvilinearrankforsymmetrictensorstobe the minimumlength
ofacurvilinearschemewhosespancontainsagivensymmetrictensor. Wecanextendssomeofthe
ideas in [5] and some of those used in our proof of Theorem 1.7 to the case of partially symmetric
1
2 E.BALLICOANDA.BERNARDI
tensorsandprovethat,ifapartiallysymmetrictensoriscontainedinthespanofaspecialdegreec
k
curvilinearschemewithαcomponents,therankofthistensorisboundedby2α+c −1+ d
(cid:16) Pi=1 i(cid:17)
(cf. Theorem 1.10).
1. Notation, Definitions and Statements
In this section we introduce the basic geometric tools that we will use all along the paper.
Notation 1.1. We indicate with
ν :Pn1 ×···×Pnk →PM,
M = (cid:16)Qki=1(ni+1)(cid:17)−1 the Segre embedding of the multi-projective space Pn1 ×···×Pnk, i.e.
the embedding of Pn1 ×···×Pnk by the complete linear system |OPn1×···×Pnk(1,...,1)|.
For each i∈{1,...,k} let
πi :Pn1 ×···×Pnk →Pni
denote the projection onto the i-th factor.
Let
τ :Pn1 ×···×Pnk →Pn1 ×···×Pˆni ×···×Pnk
i
denote the projection onto all the factors different from Pni.
Let ε ∈ Nk be the k-ple of integers ε = (0,...,1,...,0) with 1 only in the i-th position. We
i i
say that a curve C ⊂ Pn1 ×···×Pnk has multi-degree (a1,...,ak) if for all i = 1,...,k the line
bundle O (ε ) has degree a .
C i i
We say that a morphism h : P1 → Pn1 ×···×Pnk has multi-degree (a1,...,ak) if, for all
i=1,...,k:
h∗(OPn1×···×Pnk(εi))∼=OP1(ai).
Let
ν :Pn1 ×···×Pnk →PN,
d1,...,dk
N = k di+ni −1 denote the Segre-Veronese embedding of Pn1 ×···×Pnk of multi-degree
(cid:16)Qi=1(cid:0) ni (cid:1)(cid:17)
(d ,...,d ) and define
1 k
X :=ν (Pn1 ×···×Pnk)
d1,...,dk
to be the Segre-Veronese variety.
The name “ Segre-Veronese” is classically due to the fact that when the d ’s are all equal to 1,
i
thenthevarietyX iscalled“Segrevariety”;whilewhenk =1thenX isknowntobea“Veronese
variety ”.
Remark 1.2. An element of X is the projective class of an indecomposable partially symmetric
tensor T ∈ Sd1V1 ⊗···⊗SdkVk where P(Vi) = Pni. More precisely p ∈ X if there exists T ∈
Sd1V1⊗···⊗SdkVk such that p=[T]=[v1⊗,id1 ⊗···⊗vk⊗,dik] with [vj,i]∈Pni.
Definition 1.3. The s-th secant variety σ (X) of X is the Zariski closure of the union of all
s
s-secant Ps−1 to X. The tangential variety τ(X) is the Zariski closure of the union of all tangent
lines to X.
Observe that
X =σ (X)⊂τ(X)⊂σ (X)⊂···⊂σ (X)⊂σ (X)⊂···⊂PN. (3)
1 2 s−1 s
Definition 1.4. The X-rank r (p) of an element p ∈ PN is the minimum integer s such that
X
there exist a Ps−1 ⊂PN which is s-secant to X and containing p.
We indicate with S(p) the set of sets of points of Pn1 ×···×Pnk “ evincing ” the X-rank of
p∈PN, i.e.
S(p):={{x1,...,xs}⊂Pn1 ×···×Pnk|rX(p)=s and p∈hνd1,...,dk(x1),...,νd1,...,dk(xs)i}.
THIRD SECANT VARIETY OF SEGRE-VERONESE 3
Definition 1.5. The X-border rank br (p) of an element p ∈PN is the minimum integer s such
X
that p∈σ (X).
s
Remark 1.6. For any p∈PN =P(Sd1V1⊗···⊗SdkVk) we obviously have that brX(p)≤rX(P).
In fact p ∈ PN of rank r is such that there exist a tensor T ∈ Sd1V1 ⊗···⊗SdkVk that can be
minimally written as in (2); while an element p ∈ PN has border rank s if and only there exist a
sequence of rank r tensors Ti ∈Sd1V1⊗···⊗SdkVk such that limi→∞Ti =T and p=[T].
The first result that we prove in Section 2 is an upper bound for the rank of points in σ (X).
3
Theorem 1.7. The rank of an element p∈σ (X) is r (p)≤−1+ k 2d .
3 X Pi=1 i
In the case d =1 for all i=1,...,k, i.e. if X is the Segre variety, we fill in all low ranks with
i
points of σ (X)\σ (X). In Section 3 we prove the following result.
3 2
Theorem 1.8. Assume k ≥ 3 and let X ⊂ PM be the Segre variety of k factors. For each
x∈{3,...,k−1} there is p∈σ (X)\σ (X) with r (p)=x.
3 2 X
As remarked in the Introduction, we can extends some of the ideas of [5] on the notion of
curvilinearrank to some ideas used in our proof ofTheorem1.7 to the case of partially symmetric
tensors.
Definition 1.9. A scheme Z ⊂ Pn1 ×···×Pnk is curvilinear if it is a finite union of disjoint
schemes of the form OCi,Pi/mepii for smooth points pi ∈ Pn1 ×···×Pnk on reduced curves Ci ⊂
Pn1×···×Pnk,orequivalentlythatthetangentspaceateachconnectedcomponentofZ supported
at the P ’s has Zariski dimension ≤1. We define the curvilinear rank Cr(p) of a point p∈PN as:
i
Cr(p):=min{deg(Z)|ν (Z)⊂X, Z curvilinear, p∈hν (Z)i}.
d1,...,dk d1,...,dk
In Section 4 we prove the following result.
Theorem 1.10. If there exists a degree c curvilinear scheme Z ⊂ Pn1 ×···×Pnk such that p ∈
hν (Z)i and Z has α connected components, each of them mapped by ν into a linearly
d1,...,dk d1,...,dk
independent zero-dimensional sub-scheme of PN, then r (p)≤2α+c −1+ k d .
X (cid:16) Pi=1 i(cid:17)
2. Proof of Theorem 1.7
Remark 2.1. Fix a degree 3 connected curvilinear scheme E ⊂ P2 not contained in a line and
a point u ∈ P1. The scheme E is contained in a smooth conic. Hence there is an embedding
f :P1 →P2 with f(P1)=C and f(3u)=E.
Remark 2.2. For any couple of points u,o ∈P1, there is an isomorphism f :P1 →P1 such that
i
f(u)=o . For any such f we have f(3u)=3o .
i i
Remark 2.3. Fix two points u,o ∈P1. There is a morphism f :P1 →P1 such that f(u)=o , f
i i
is ramified at u and deg(f) = 2, i.e. f∗(OP1(1)) ∼= OP1(2). Since deg(f) = 2, f has only order 1
ramification at u. Thus f(3u)=2o (as schemes).
i
We recall the following lemma proved in [4, Lemma 3.3].
Lemma 2.4 (Autarky). Let p∈hXi with X being the Segre-Veronese variety of Pn1 ×···×Pnk.
If there exist Pmi, i=1,...,k, with mi ≤ni, such that p∈hνd1,...,dk(Pm1 ×···×Pmk)i, then the
X-rankofpisthesameastheY-rankofpwhereY istheSegre-Veroneseνd1,...,dk(Pm1×···×Pmk).
Corollary 2.5. Let Γ ⊂ Pn1 ×···×Pnk be the minimal 0-dimensional scheme such that p ∈
hν (Γ)i, then the X-rank of p is equal to its Y-rank where Y is the Segre-Veroneseembedding
d1,...,dk
of Pm1×···×Pmk where each mi =dimhπi(Γ)i≤deg(πi(Γ))−1 (πi as in Notation 1.1); moreover
if there exists an index i such that deg(π (Γ)) = 1, then we can take Y to be the Segre-Veronese
i
embedding of Pm1 ×···×Pˆmi ×···×Pmk.
Proof. Consider the projections πi :Pn1 ×···×Pnk onto the i-th factor Pni as in Notation 1.1. It
may happen that deg(π (Γ)) can be any value from 1 to deg(Γ).
i
By the just recalled Autarky Lemma, we may assume that each πi(Γ) spans the whole Pni.
Therefore if there is an index i∈{1,...,k} such that deg(π (Γ))=1 we can take p∈Pn1 ×···×
i
Pˆni×···×Pnk. Moreoverthe autarchicfact that we canassume Pni to be hπi(Γ)i implies that we
can replace each Pni with Pdimhπi(Γ)i and clearly dimhπi(Γ)i≤deg(πi(Γ)). (cid:3)
4 E.BALLICOANDA.BERNARDI
Proof of Theorem 1.7: Because of the filtration of secants varieties (3), for a given element p ∈
σ (X), it may happen that either p∈X, or p∈σ (X)\X or p∈σ (X)\σ (X). We distinguish
3 2 3 2
among these cases.
(1) If p∈X, then r (p)=1.
X
(2) If p ∈ σ (X) \ X then either p lies on a honest be-secant line to X (and in this case
2
obviouslyr (p)=2)orpbelongstocertaintangentlinetoX. Inthislattercase,consider
X
theminimumnumberhoffactorscontainingsuchatangentline,i.e. h≤kistheminimum
integersuchthatp∈hνd1,...,dh(Pn1×···×Pnh)i(maybereorderingfactors). In[4,Theorem
h
3.1] we proved that, if this is the case, then r (p)= d .
X Pi=1 i
(3) Fromnowonweassumethatp∈σ (X)\σ (X). By[9,Theorem1.2]thereareashortlist
3 2
of zero-dimensional schemes Γ⊂Pn1 ×···×Pnk such that p∈hνd1,...,dk(Γ)i, therefore, in
ordertoproveTheorem1.7,itis sufficientto boundthe rankofthe points inhν (Γ)i
d1,...,dk
for each Γ in their list.
Since p ∈ σ (X)\σ (X), The possibilities for Γ are only the following: either Γ is a
3 2
smooth degree 3 zero-dimensional scheme (case (3a) below), or it is the union of a degree
2 scheme supported at one point and a simple point (case (3b)), or it is a curvilinear
degree 3 scheme (case (3c)) or, finally, a very particular degree 4 scheme with 2 connected
components of degree 2 (case (3d)).
(3a) IfΓisasetof3distinctpoints,thenobviouslyr (p)=3([9,Case(i),Theorem1.2]).
X
(3b) If Γ is a disjoint union of a simple point a and a degree 2 connected scheme ([9,
Case (ii), Theorem 1.2]), then there is a point q on a tangent line to X such that
k
p ∈ h{ν (a),q}i. Hence r (p) ≤ 1+r (q) ≤ 1+ d (for the rank on the
d1,...,dk X X Pi=1 i
k
tangentialvarietyofX see [2]). Since d >0for all i’s andk≥2,then 1+ d ≤
i Pi=1 i
−1+ k 2d .
Pi=1 i
(3c) Assume deg(Γ)=3 and that Γ is connected ([9, Case (iii), Theorem 1.2]) supported
at a point {o}:=Γ . Since the case k =1 is true by [6, Theorem 37], we can prove
red
the theorem by using induction on k, with the case k =1 as the starting case.
Sincedeg(Γ)=3,byCorollary2.5,wecanassumethatpbelongstoaSegre-Veronese
variety of k factors all of them being either P1’s or P2’s, i.e., after having reordered
the factors,
p∈ν (P1×···×P1×P2×···×P2).
d1,...,dk
The P1’s correspond to the cases in which either deg(π (Γ))=3 and dimhπ (Γ)i=1
i i
(i.e. πi(Γ) is contained in a line of the original Pni), or deg(πi(Γ)) = 2 (notice that
in this case π is not an embedding). The P2’s correspond to the cases in which
i|Γ
dimhπ (Γ)i = 2, = deg(π (Γ)) = 3. Finally we can exclude all the cases in which
i i
deg(π (Γ)) = 1 because, again by Corollary 2.5, we would have that p belongs to a
i
Segre-Veronse variety of less factors and then this won’t give the highest bound for
the rank of p.
Now fix a point u∈P1. By Remarks 2.1, 2.2 and 2.3 there is
f :P1 →Pni with f (3u)=π (Γ). (4)
i i i
Consider the map
f =(f1,...,fk):P1 →Pn1 ×···×Pnk.
We have f(u)={o} and π (f(3u))=f (3u)=π (Γ). Since π (f(3u))=π (Γ) for all
i i i i i
i’s, the universal property of products gives f(3u)=Γ. The map f has multi-degree
(a ,...,a ) where a := 1 if n = 1 and deg(π (Γ)) = 3, and a = 2 in all other
1 k i i i i
cases. Notice that f is an embedding if deg(π (Γ))6=2. Since deg(π (Γ))=2 if and
i i i
only if π−1(o ) contains the line spanned by the degree 2 sub-scheme of Γ, we have
i i
deg(π (Γ))=2 for at most one index i. Since k ≥2, f is an embedding. Set
i
C :=ν f P1 and Z :=ν (Γ).
d1,...,dk(cid:0) (cid:0) (cid:1)(cid:1) d1,...,dk
ThecurveC issmoothandrationalofdegreeδ := k a d . Notethatδ ≤ k 2d .
Pi=1 i i Pi=1 i
Hence to prove Theorem 1.7 in this case it is sufficient to prove that r (p) ≤ δ−1
C
because clearly r (p)≤r (p) since C ⊂X.
C X
THIRD SECANT VARIETY OF SEGRE-VERONESE 5
By assumption p ∈ hZi. Since p ∈/ σ (X), hZi is not a line of PN. Hence hZi is a
2
plane. Since C is a degree δ smooth rational curve, we have dimhCi ≤ δ. By [13,
Proposition 5.1] we have r (p) ≤ dimhCi. Hence it is sufficient to prove the case
C
δ =dimhCi, i.e. we may assume that C is a rational normal curve in its linear span.
If δ ≥ 4, since Z is connected and of degree 3, by Sylvester’s theorem (cf. [10]) we
have p has C-border rank 3 and r (p)=δ−1, concluding the proof in this case.
C
Ifδ ≤3,wehaveσ (C)=hCiandhencep∈σ (X),contradictingp∈σ (X)\σ (X).
2 2 3 2
(3d) Assume that Γ has degree 4 ([9, Case (iv), Theorem 1.2]). J. Buczyn´ski and J.M.
Landsberg show that p belongs to the span of two tangent lines to X whose set
theoretic intersections with X span a line which is contained in X. This means
that Γ = v⊔w with v,w degree 2 reduced zero-dimensional schemes with support
contained in a line L ⊂ Pn1 ×···×Pnk and moreover that the multi-degree of L is
ε for some i = 1,...,k (cfr. Notation 1.1). This case occurs only when d = 1, i.e.
i i
when ν (L)=L˜ is a line.
d1,...,dk
Observe that v˜:=ν (v) and w˜ :=ν (w) are two tangent vectors to X. In
d1,...,dk d1,...,dk
[2, Theorem 1] we prove that the X-rank of a point p ∈T (X) for certain point o=
o
(o ,...,o )∈X, is the minimum number η (p) for which there exist E ⊆{1,...,k}
1 k X
such that ♯(E)=η (p) and T (X)⊆h∪ Y i where Y the n -dimensionallinear
X o i∈E o,i o,i i
subspaceobtainedbyfixingallcoordinatesj ∈{1,...,k}\{i}equaltoo ∈Pn. LetI
j i
andJ bethesetsplayingtheroleofE forhv˜iandhw˜irespectivelyandsetI′ =I\{i}
(meaning that I′ =I if i∈/ I and I′ =I\{i} otherwise) and J′ =J \{i} . Now take
α:= d + d +d
X j X j i
j∈I′ j∈J′
k
andnote thatα≤−1+ 2d ,therefore ifwe provethatr (p)≤α weare done.
Let Dj ⊂ Pn1 ×···×PPnk,h=j1∈ Ih′, be the line of multi-degreeXεJ containing πj(v),
and let T , j ∈ J′, be the line of X of multi-degree ε containing π (w). The curve
j j j
L∪ D containsv andthe curveL∪ T containsw. Hencethecurve
(cid:16)Sj∈I′ j(cid:17) (cid:16)Sj∈J′ j(cid:17)
   
T :=L∪ D ∪ T
[ j [ j
j∈I′  j∈J′ 
is a reduced and connected curve containing Γ. Since p∈hν (Γ)i, we have that
d1,...,dk
if we call T˜ :=ν (T) then p∈hT˜i andr (p)≤r (p). The curve T˜ is a smooth
d1,...,dk X T˜
and connected curve whose irreducible components are smooth rational curves and
with deg(T˜) = α. Hence dimhT˜i ≤ α. Since T˜ is reduced and connected, as in [13,
Proposition4.1]wegetr (p)≤α. Summingupr (p)≤r (p)≤α≤−1+ k 2d .
T˜ X T˜ Ph=1 h
(cid:3)
3. Proof of Theorem 1.8
Autarky Lemma (provedin [4, Lemma 3.3] and recalled here in Lemma 2.4) is true also for the
border rank ([8, Proposition 2.1]). This allows to formulate the analog of Corollary 2.5 for border
rank. Therefore, in order to prove Theorem 1.8 we can limit ourselves to the study of the case
n =1 for all i’s. This is the reasonwhy in this sectionwe will alwayswork with the Segre variety
i
of P1’s. Let
ν :(P1)k →Pr, r =2k−1
be the Segre embedding of k copies of P1’s and X :=ν((P1)k); and let
ν′ :(P1)k−1 →Pr′, r =2k−1
be the the Segre variety of k−1 copies of P1’s and X′ :=ν′((P1)k−1).
Proposition 3.1. Assume k ≥ 3. Let Γ ⊂ (P1)k be a degree 3 connected curvilinear scheme
such that deg(π (Γ)) = 3 for all i’s, and let β be the only degree 2 sub-scheme of Γ. For all
i
p∈hν(Γ)i\hν(β)i we have that
(a) if k =3, then 2≤r (p)≤3 and r (p)=2 if p is general in hν(Γ)i;
X X
6 E.BALLICOANDA.BERNARDI
(b) if k ≥4, then r (p)=k−1.
X
Proof. Since Γ⊂(P1)k is connected, it has support on only one point; all along this proof we set
o:=Supp(Γ)∈(P1)k. (5)
First of all recall that in step (3a) of the proof of Theorem 1.7 we obtained an embedding
f = (f ,...,f ) with f : P1 → P1 an isomorphism (see (4)); moreover we can fix a point u ∈ P1
1 k i
such that f(u)=o and Γ=f(3u). We proved that
C :=ν(f(P1))
is a degree k rational normal curve in its linear span. Obviously
r (p)≤r (p).
X C
Ifk ≥4Sylvester’stheoremimpliesr (p)=k−1. Nowassumek =3. Sinceadegree3rational
C
plane curve has a unique singular point, for any q ∈hCi there is a unique line L⊂hCi=P3 with
deg(L∩C) = 2. Thus r (p) = 2 (resp. r (p) = 3) if and only if p ∈/ τ(C) (resp. p ∈ τ(C),
C C
cfr. Definition 1.3). Since τ(C) is a degree 4 surface,by Riemann-Hurwitz, we see that both cases
occur and that r (p)=2 (and hence r (p)≤2 if p is general in hν(Γ)i).
C X
Claim 1. Let the point o ∈ (P1)k be, as in (5), the support of Γ. Fix any F ∈ |O(P1)k(εk)| such
that o∈/ F. Then hν(Γ)i∩hν(F)i=∅.
Proof of Claim 1. It is sufficient to show that h0(I (1,...,1)) = h0(I (1,...,1)) − 3, i.e.
F∪Γ F
h0(IΓ(1,...,1,0)) = h0(O(P1)k(1,...,1,0))−3. This is true because f1 and f2 (recalled at the
beginning of the proof this Proposition3.1 and introduced in (4)) are isomorphisms. (cid:3)
(a) Assume k = 3. Since r (p) ≤ r (p) ≤ 3 and r (p) = 2 for a general p in hν(Γ)i, we only
X C C
need to prove that r (p)>1. The case r (p)=1 corresponds to a completely decomposable
X X
tensor: p = ν(q) for some q ∈ (P1)3. Clearly r (ν(o)) = 1 but o ∈ hβi then, since we took
X
p∈hν(Γ)i\hν(β)i, wehavep6=ν(o)andinparticularq 6=o. InthiscasewecanaddqtoΓand
get that h1(I (1,1,1))>0 by [1, Lemma 1]. Since deg(f (Γ))=3, for all i’s, every point of
q∪Γ i
hβi\{o}hasrank2. Sinceq :=(q ,q ,q )6=o:=(o ,o ,o )wehaveq 6=o forsomei,sayfor
1 2 3 1 2 3 i i
i=3. Take F ∈|O(P1)3(ε3)| such that q ∈F and o∈/ F. Hence F ∩(Γ∪{q})={q}. We have
h1(F,Iq,F(1,1,1)) = 0, because O(P1)3(1,1,1) is spanned. Claim 1 gives h1(IΓ(1,1,0)) = 0.
The residual exact sequence of F in (P1)3 gives h1(I )(1,1,1)=0, a contradiction.
Γ∪q
(b) Fromnowonweassumek ≥4andthatProposition3.1istrueforasmallernumberoffactors.
Since X ⊃ C, we have r (p) ≤ k−1 (in fact, as we already recalled above, r (p) = k−1
X C
by Sylvester’s theorem). We need to prove that we actually have an equality, so we assume
r (p)≤k−2 and we will get a contradiction.
X
Take a set of points S ∈ S(p) of (P1)k evincing the X-rank of p (see Definition 1.4) and
considerv =(v ,...,v )∈S ⊂(P1)k to be apointappearinginadecompositionofp. We can
1 k
always assume that, if o= (o ,...,o ), then v 6= o : such a v ∈S ⊂ S(p) exists because, by
1 k k k
Autarky (here recalled in Lemma 2.4), no element of S(p) is contained in (P1)k−1×{o }.
k
Consider the pre-image
D :=π−1(v ).
k k
Clearly by construction o ∈/ D hence for any q ∈ (P1)k \D we have h1(I (1,...,1)) =
q∪D
h1(Iq(1,...,1,0)) = 0, because O(P1)k(1,...,1,0) is globally generated. This implies that
hν(D)i intersects X only in ν(D).
Now consider
ℓ:Pr\hν(D)i
the linear projection from hν(D)i. Since p ∈/ hν(D)i (Claim 1), ℓ is defined at p. Moreover
the map ℓ induces a rational map ν((P1)k \D) → ν((P1)k), which extends to the projection
τ :(P1)k →(P1)k−1 defined in Notation 1.1. We have
k
ℓ◦ν =ν′.
Since o ∈/ D, ℓ(hΓi) = hν′(Γ′)i, where Γ′ = τ (Γ). Hence p′ := ℓ(p) ∈ hν′(Γ′)i. By [2] every
k
element of hν′(β)i\ν′(o′), with o′ :=τ (o), has X′-rank k−1. Since deg(π (Γ))=3 for all i,
k i
we have deg(π (β)) = 2 for i = 1,...,k−1. This implies that the minimal sub-scheme α of
i
THIRD SECANT VARIETY OF SEGRE-VERONESE 7
Γ′ such that p′ ∈ hν′(α)i is such that α 6= β where β is the degree 2 sub-scheme of Γ′. Now
let S′ ⊂ (P1)k−1 be the projection by τ of the set of points of S ⊂ S(p) that are not in D,
k
i.e. S′ := τ (S \S ∩D). Since ♯(S′) ≤ k −2 and p′ ∈ hν′(Γ′)i, the inductive assumption
k
gives α 6= Γ′ (it works even when k = 4). Hence α = {o′}. Thus p ∈ hν({o}∪D)i. Hence
dim(hν(Γ∪D)i)≤dim(hν(D)i)+2, contradicting Claim 1.
(cid:3)
Proof of Theorem 1.8: If x = 3, then we may take as p a general point of σ (X). Now assume
3
x≥4 andhence k ≥5. Apply Proposition1.8 to (P1)x+1 andthen use Autarky (Lemma 2.4). (cid:3)
Question 3.2. Assume k ≥3 and n ≥ 2 for all i. Let Γ ⊂(P1)k be a connected and curvilinear
i
zero-dimensional scheme such that deg(Γ) = 3 and for all i π (Γ) is not contained in a line. Let
i
X ⊂ Pr with r = −1+ k (n +1), be the Segre embedding of (P1)k. Is r (p) = 2k−1 for a
Qi=1 i X
general p∈hν(Γ)i ?
By Autarky, Question 3.2 is true if and only if it is true when n = 2 for all i. Note that any
i
two different Γ’s are in the same orbit for the group Aut(Pn1)×···×Aut(Pnk) (in the set up of
the proof of Theorem 1.7 these schemes Γ’s correspond to the case in which each fi : P1 →Pni is
an embedding with f (P1) a smooth conic). Hence the case k = 3 of Question 3.2 is true by [9,
i
Theorem 1.8].
4. Proof of Theorem 1.10
Lemma 4.1. Fix an integer c>0 and u∈P1. Let E =cu⊂P1 be the degree c effective divisor of
P1 with u as its support. Let g : E → Pn be any morphism. Then there is a non-negative integer
e≤c and a morphism h:P1 →Pn such that h∗(OPn(1))∼=OP1(e) and h|E =g.
Proof. Every line bundle on E is trivial. We fix an isomorphism between g∗(OPn(1)) and OE(c).
After this identification, g is induced by n+1 sections u ,...,u of O (c) such that at least one
0 n E
of them has a non-zero restriction at {u}. The map H0(OP1(c)) → H0(OE(c)) is surjective and
its kernelis the sectionassociatedto the divisorcu. Hence there arev0,...,vn ∈H0(OP1(c)) with
v = u for all i. Not all sections v ,...,v vanish at 0. If they have no common zero, then
i|E i 0 n
they define a morphism P1 → Pn extending g and we may take e = c. Now assume that they
have a base locus and call F the scheme-theoretic base locus of the linear system associated to
v ,...,v . We have deg(F) ≤ c. Set e := c−deg(F) and S := F . The sections v ,...,v
0 m red 0 n
induce a morphism f : P1\S → Pn with f|E = g. See v0,...,vn as elements of |OP1(c)| and set
ui :=u−F ∈|OP1(e)|. Byconstructionthelinearsystemspannedbyu0,...,unhasnobasepoints,
hence it induces a morphism h : P1 → Pn such that h∗(OPn(1)) ∼= OP1(e). We have h|P1\S = f
and hence h =g. (cid:3)
|E
Proof of Theorem 1.10: Let E ⊂ P1 be as in Lemma 4.1 and let E ,...,E be the connected
1 α
components of E. By assumption there is p ∈ hν (E )i such that p ∈ h{p ,...,p }i. Note
i d1,...,dk i 1 α
that if Theorem 1.10 is true for each (E ,p ), then it is true for E. Hence it is sufficient to prove
i i
Theorem 1.10 under the additional assumption that E is connected, so from now on we assume
E connected.
Moreover,since r (p)=1≤2−1+ d if c=1, we may also assume that
X Pi i
degE =c≥2.
k
Finally, since the real-valued function x7→x −1+ d is increasing for x≥1, with no loss
(cid:16) Pi=1 i(cid:17)
of generality we may assume that, for any G(Z,
p∈/ hν (G)i.
d1,...,dk
Fix u ∈ P1 and let E = cu ⊂ P1 be the degree c effective divisor of P1 with u as its support.
Since Z is curvilinear and deg(Z) = c, we have Z ∼= E as abstract zero-dimensional schemes.
Let g : E → Pn1 ×···×Pnk be the composition of an isomorphism E → Z with the inclusion
Z ֒→Pn1 ×···×Pnk:
g :E →Z ֒→Pn1 ×···×Pnk.
8 E.BALLICOANDA.BERNARDI
Setg :=π ◦g. IfweapplyLemma4.1toeachg ,wegetthe existenceofanintegerc ∈{0,...,c}
i i i i
and of a morphism hi :P1 →Pni such that hi|Z =gi and hi∗(OPni(1))∼=OP1(ci). The map
h=(h ,...,h ):P1 →Pn1 ×···×Pnk (6)
1 k
has multi-degree (c ,...,c ). The curve
1 k
D :=h(P1)
isanintegralrationalcurvecontainingZ. Sincep∈hν (Z)i,wehaver (p)≤r (p).
d1,...,dk X νd1,...,dk(D)
Thus it is sufficient to prove that r (p)≤2+c −1+ k d . Since c ≤c for all i, it
νd1,...,dk(D) (cid:16) Pi=1 i(cid:17) i
k
is sufficient to prove that r (p)≤2−c+ c d .
νd1,...,dk(D) Pi=1 i i
Set D˜ :=ν (D), Z˜ :=ν (Z), m:=dim(hD˜i) and
d1,...,dk d1,...,dk
f =ν ◦h:P1 →PN.
d1,...,dk
By assumption Z˜ is linearly independent in hD˜i∼=Pm and in particular c≤m+1.
(a) Assume that the map h defined in (6) is birational onto its image. The curve D˜ ⊂ PN just
definedisarationalcurveofdegreea:= k c d containedintheprojectivespacePm :=hD˜i
and non-degenerate in Pm. Note that a≥Pmi=1anidithat p∈hZ˜i.
(1) First assume that a = m. In this case D˜ is a rational normal curve of Pm. If c ≤
k
⌈(a+1)/2⌉, then Sylvester’s theorem implies that r (p)=a+2−c=2−c+ c d .
Now assume c > ⌈(a+1)/2⌉. Since Z˜ is connectedD˜and curvilinear and p ∈/ hGPiif=o1r ainiy
G(Z˜, Sylvester’s theorem implies r (p)≤c.
D˜
(2) Now assume m < a. There is a rational normal curve C ⊂ Pa and a linear subspace
W ⊂Pa suchthatdim(W)=a−m−1, C∩W =∅andhis the compositionofthe degree
a complete embedding j :=P1 ֒→Pa and the linear projection ℓ:Pa\W →Pm from W.
TheschemeE′ :=j(E)isadegreeccurvilinearschemeandℓmapsE′ isomorphicallyonto
Z˜. SinceZ˜ islinearlyindependent,thenhE′i∩W =∅andℓmapsisomorphicallyhE′ionto
hZ˜i. Thusthereisauniqueq ∈hE′isuchthatℓ(q)=p. TakeanyfinitesetS ⊂j(P1)with
q ∈hSi. SinceC∩W =∅,ℓ(S)isawell-definedsubsetofD˜ withcardinality≤♯(S). Hence
r (p)≤r (q). As in step (a1) we see that either r (q)=a+2−c (case c≤⌈(a+1)/2⌉)
D˜ C C
or r (q)≤c (case c>⌈(a+1)/2⌉).
C
(b) Now assume that h is not birational onto its image, but it has degree k ≥ 2. Note that k
k
divides c for all i. In this case we will prove that r (p) ≤ 2−c+ c d /k. Let
i νd1,...,dk(D) Pi=1 i i
h′ :P1 →h(P1)denotethenormalizationmap. Thereisadegreekmaph′′ :P1 →P1suchthat
histhecompositionofh′◦h′′ andtheinclusionh(P1)⊂Pn1 ×···×Pnk. WehaveZ =h′(E′),
where E′ = cu′ and u′ = h′′(u). We use E′ and h′ instead of E and h and repeat verbatim
step (a).
(cid:3)
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(EdoardoBallico)Dipartimentodi Matematica,Univ. Trento, Italy
E-mail address: [email protected]
(AlessandraBernardi)Dipartimentodi Matematica,Univ. Trento, Italy
E-mail address: [email protected]