Complete intersections on general hypersurfaces
aa r X i v : . [ m a t h . AG ] J a n COMPLETE INTERSECTIONS ON GENERALHYPERSURFACES
ENRICO CARLINI, LUCA CHIANTINI, AND ANTHONY V. GERAMITA
Abstract.
We ask when certain complete intersections of codi-mension r can lie on a generic hypersurface in P n . We give acomplete answer to this question when 2 r ≤ n + 2 in terms of thedegrees of the hypersurfaces and of the degrees of the generatorsof the complete intersection. Introduction
Many problems in classical projective geometry ask about the natureof special subvarieties of some given family of varieties, e.g. how manyisolated singular points can a surface of degree d in P have? when isit true that the members of a certain family of varieties contain a line?contain a linear space of any positive dimension? The reader can easilysupply other examples of such questions.This is the kind of problem we consider in this paper: what types ofcomplete intersection varieties of codimension r in P n can one find onthe generic hypersurface of degree d ?In case r = 2 it was known to Severi [Sev06] that for n ≥ P by Noether (and Lefschetz)[Lef21, GH85] for general hypersurfaces of degree ≥
4. These ideaswere further generalized by Grothendieck [Gro05].Our approach to the problem mentioned above uses a mix of projec-tive geometry and commutative algebra and is much more elementaryand accesible than, for example, the approach of Grothendieck. We areable to give a complete answer to the question we raised for completeintersections of codimension r in P n which lie on a general hypersurfaceof degree d whenever 2 r ≤ n + 2.The paper is organized in the following way: in the next section(Section 2) we lay out the question we want to consider and explainwhat are the interesting parameters for a response. Mathematics Subject Classification.
Primary ; Secondary .
In Section 3 we collect some technical information we will need aboutvarieties of reducible forms and their joins. In order to find the di-mensions of these joins (using “Terracini’s Lemma”) we calculate thetangent space at a point of any variety of reducible forms. We alsorecall some information about artinian complete intersection quotientsof a polynomial ring.In Section 4, we use the technical facts collected in Section 3 toreformulate our original question. We illustrate the utility of this re-formulation to discuss complete intersections of codimension r in P n ona general hypersurface when 2 r < n + 1. We further use our approachto give a new proof for the existence of a line on the general hypersexticof P .In Section 5 we state and prove our main theorem which gives acomplete description of all complete intersections of codimension r in P n which lie on a generic hypersurface when 2 r ≤ n + 2.2. Question
The objects of study of this paper are complete intersection sub-schemes of projective space. Recall that Y ⊂ P n is a complete inter-section scheme if its ideal is generated by a regular sequence, moreprecisely, I ( Y ) = ( F , . . . , F r ), F i ∈ S = C [ x , . . . , x n ], and F , . . . , F r form a regular sequence in S . If deg F i = a i for all i , we will say thatsuch a Y is a CI ( a , . . . , a r ) and we will assume a ≤ . . . ≤ a r ; noticethat Y is unmixed of codimension r in P n . With this notation we canrephrase the statementthe degree d hypersurface X contains a CI ( a , . . . , a r )in terms of ideals as follows I ( X ) = ( F ) ⊂ ( F , . . . , F r ) for forms F i forming aregular sequence and such that deg F i = a i for all i .Clearly not all choices of the degrees are of interest for us, e.g. if a i > d for all i , then no CI ( a , . . . , a r ) can be found on a degree d hypersurface. On the other hand, any hypersurface of degree d containsa CI ( a , . . . , a r ) if a i = d , for some i . Simply cut that hypersurfacewith general hypersurfaces of degrees a j , j = i .So, one need only consider CI ( a , . . . , a r ) where none of the a i = d . Lemma 2.1.
Let a ≤ . . . ≤ a i < d < a i +1 ≤ . . . ≤ a r with r ≤ n . Thefollowing are equivalent facts: • there is a CI ( a , . . . , a r ) on the general hypersurface of degree d in P n ; OMPLETE INTERSECTIONS ON GENERAL HYPERSURFACES 3 • there is a CI ( a , . . . , a i ) on the general hypersurface of degree d in P n .Proof. Let I ( X ) = ( F ), where X is a general hypersurface of degree d in P n . Let I ( Y ) = ( F , . . . , F r ) be the ideal of a CI ( a , . . . , a r ), withdegrees a i as above.Then X ⊃ Y if and only if F = i X j =1 F j G j and hence, if and only if X ⊃ Y ′ , where Y ′ is the complete intersectiondefined by F , . . . , F i . (cid:3) From this Lemma it is clear that the basic question to be consideredis: (Q):
For which degrees a , . . . , a r < d does the genericdegree d hypersurface of P n contain a CI ( a , . . . , a r ) ? If, rather than restricting to the generic case, we asked if some hyper-surface of degree d contains a CI ( a , . . . , a r ), then the answer is trivial.Indeed, the ideal of any CI ( a , . . . , a r ) ( a i < d ) always contains degree d elements. 3. Technical facts If λ = ( λ , . . . , λ s ) is a partition of the integer d (i.e. P si =1 λ i = d and λ ≥ . . . ≥ λ s >
0) we write λ ⊢ d . For each λ ⊢ d we define asubvariety X λ ⊂ P ( S d ) ≃ P N (where N = (cid:0) d + nn (cid:1) −
1) as follows: X λ := { [ F ] ∈ S d | F = F · · · F s , deg F i = λ i } . We call X λ the variety of reducible forms of type λ . The dimension of X λ is easily seen to be [ P si =1 (cid:0) λ i + nn (cid:1) ] − s . (For other elementary propertiesof X λ see [Mam54] and for the special case λ = . . . = λ s = 1 see[Car06], [Car05] or [Chi02] for the n = 2 case).If x , . . . , x r are independent points of P N we will call the P r − spanned by these points the join of the points x , . . . x r and write J ( x , . . . , x r ) := h x , . . . , x r i . More generally, if X , . . . , X r are varieties in P N then the join of X , . . . , X r is J ( X , . . . , X r ) := [ { J ( x , . . . , x r ) | x i ∈ X i , { x , . . . , x r } independent } In case X = · · · = X r = X we write J ( X , . . . , X r ) := Sec r − ( X ) E. CARLINI, L. CHIANTINI, AND A.V. GERAMITA and call this the ( r − st (higher) secant variety of X .Joins and secants of projective varieties are important auxiliary vari-eties which can help us better understand the geometry of the originalvarieties (see e.g. [CGG02, CGG03, CC06, Cil01, Ger96, LM04, ˚Adl88,CJ96]). One of the most fundamental questions we can ask about joinsand secants is: What are their dimensions?
This is, in general, an extremely difficult question to answer. Thefamous
Lemma of Terracini (which we recall below) is an importantobservation which will aid us in answering this question.
Lemma 3.1 (Lemma of Terracini) . Let X , . . . , X r be reduced subva-rieties of P N and let p ∈ J = J ( X , . . . , X r ) be a generic point of J .Suppose that p ∈ J ( p , . . . , p r ) , then the (projectivized) tangent spaceto J at p , i.e. T p ( J ) , can be described as follows: T p ( J ) = h T p ( X ) , . . . , T p r ( X r ) i . Consequently, dim J = dim h T p ( X ) , . . . , T p r ( X r ) i . We want to apply this Lemma in the case that the X i are all ofthe form X λ ( i ) , λ ( i ) ⊢ d, i = 1 , . . . , r . A crucial first step in such anapplication is, therefore, a calculation of T p i ( X λ ( i ) ) where p i ∈ X λ ( i ) . Proposition 3.2.
Let λ ⊢ d , λ = ( λ , . . . , λ s ) and let p ∈ X λ be ageneric point of X λ .Write p = [ F · · · F s ] where deg F i = λ i , i = 1 , . . . , s and let I p ⊂ S = C [ x , . . . , x n ] be the ideal defined by: I p = ( F · · · F s , F F · · · F s , . . . , F · · · F s − ) . Then the tangent space to X λ at the point p is the projectivization of ( I p ) d and hence has dimension dim T p ( X λ ) = dim C ( I p ) d − . Proof.
Consider the map of affine spacesΦ : S λ × · · · × S λ s → S d defined by Φ(( A , . . . , A s )) = A · · · A s . Let P ∈ S λ × · · · × S λ s be the point P = ( F , . . . , F s ). A tangentdirection at P is given by any vector of the form v = ( F ′ , . . . , F ′ s ) andthe line through P in that direction is L v := ( F + µF ′ , . . . , F s + µF ′ s ) , µ ∈ C . OMPLETE INTERSECTIONS ON GENERAL HYPERSURFACES 5
A simple calculation shows that the tangent vector to Φ( L v ) at thepoint Φ( P ) = p is exactly P si =1 F · · · F i ′ · · · F s and that proves theproposition. (cid:3) In view of Terracini’s Lemma, the following corollary is immediate.
Corollary 3.3.
Let λ (1) , . . . , λ ( r ) all be partitions of d where λ ( i ) = ( λ i , λ i ) . Let I = ( F , F , F , F , . . . , F r , F r ) be an ideal of S generated by generic forms where deg F ij = λ ij , for ≤ i ≤ r, j = 1 , . If J = J ( X λ (1) , . . . , X λ ( r ) ) then dim J = dim C I d − . Remark . It is useful to note the following facts:(i) In Proposition 3.2 we are using the fact that C has character-istic 0. The problem is that the differential is not necessarilygenerically injective in characteristic p .(ii) Observe that the generic point in X λ , λ = ( λ , . . . , λ s ) ⊢ d canalways be written as the product of s irreducible forms withthe property that any ℓ -subset of these s forms ( ℓ ≤ n + 1) isa regular sequence.(iii) This last can be extended easily to joins of varieties of reducibleforms. I.e. the generic point in such a join can be written asa sum of elements with the property that each summand is apoint enjoying the property described in (ii) above. Moreover,every ℓ -subset ( ℓ ≤ n +1) of the set of all the irreducible factorsof all of these summands is also a regular sequence.(iv) Fr¨oberg (see [Fr¨o85]) has a conjecture about the multiplica-tive structure of rings S/I , where S = C [ x , . . . , x n ] and I isan ideal generated by a set of generic forms. This conjecturegives the Hilbert functions of such rings. However, apart fromthe cases n = 1 (proved several times by various authors, see[Fr¨o85, GS98, IK99]) and n = 2 (proved by [Ani86]) this con-jecture has resisted attempts to prove it. E. CARLINI, L. CHIANTINI, AND A.V. GERAMITA
Notice that in terms of the geometric problem in Corollary3.3, one need only consider Fr¨oberg’s conjecture for a stronglyrestricted collection of degrees.We will need some specific information about the Hilbert function ofsome artinian complete intersections in polynomial rings. The followinglemma summarizes the facts we shall use.
Lemma 3.5.
Let r > and F , . . . , F r , G , . . . , G r be generic forms in C [ y , . . . , y r − ] having degrees < deg F = a ≤ deg F = a ≤ . . . ≤ deg F r = a r ≤ d/ and d/ ≤ deg G r = d − a r ≤ . . . ≤ deg G = d − a for a non-negative integer d .Consider the quotient A = C [ y , . . . , y r − ] / ( F , . . . , F r , G r , . . . , G , G ) and its Hilbert function H A . The following facts hold: (i) H A is symmetric with respect to c = ( r − d + a − r +12 ; (ii) if H A ( i ) ≥ H A ( i + 1) then H A ( j ) is non-increasing for j ≥ i ; (iii) the multiplication map on A i given by G (the class of G in A ) has maximal rank.If one of the following holds r = 2 and a ≥ , or r = 3 and a ≥ , or r = 3 and a = 2 , d = 4 , or r > and a ≥ , we also have that: (iv) if i ≤ a , then H A ( i ) < H A ( i + 1) . (v) if a < i ≤ c , then H A ( a ) < H A ( i ) . (vi) if c < i , then H A ( a ) > H A ( i ) if and only if c − a < i − c .Proof. As A is a Gorenstein graded ring (i) follows immediately, while(iii) is a consequence of a theorem of Stanley [Sta80] and Watanabe[Wat87].To prove (ii) we can use the Weak Lefschetz property, i.e. multipli-cation by a general linear form has maximal rank, e.g. see [MMR03].The condition on H A , coupled with the Weak Lefschetz property, yieldsthat every element of A i +1 is the product of a fixed linear form witha form of degree i . Now consider an element of A i +2 , call it M , then OMPLETE INTERSECTIONS ON GENERAL HYPERSURFACES 7 since A is a standard graded algebra, M = P r − i =1 y i C i , where y i is theclass of y i in A , and C i is the class of a form of degree i + 1. By whatwe have seen, C i = LD i where L is the form we had earlier and the D i are forms of degree i . Rewriting we get M = L P r − i =1 y i D i . But P r − i =1 y i D i is in A i +1 hence A i +2 = LA i +1 and hence the dimensioncannot increase. Proceeding by induction we prove the statement.As for (iv), it is enough to give the proof for i = a as there areno generators of degree smaller than a . Let ¯ A be a quotient obtainedwhen all the forms F i and G i have the same degree a = a = . . . = a r = d − a r = . . . = d − a . Notice that it is enough to show the result for ¯ A .In fact, whenever we pass from ¯ A to another quotient A by increasingthe degrees of s forms we obtain(1) H A ( a ) = H ¯ A ( a ) + sH ¯ A ( a + 1) + s (2 r − − s ≤ H A ( a + 1)and the inequality H ¯ A ( a ) < H ¯ A ( a + 1) is preserved; these Hilbertfunction estimates use the fact that the forms F i and G i do not havelinear syzygies. By straightforward computations one gets H ¯ A ( a ) = (cid:18) a + 2 r − a (cid:19) − r + 1and H ¯ A ( a + 1) = (cid:18) a + 2 r − a + 1 (cid:19) − (2 r − . Thus the inequality H ¯ A ( a ) < H ¯ A ( a + 1) is equivalent to(2) (cid:18) a + 2 r − a + 1 (cid:19) − (2 r − r − > . Notice that if (2) holds for the pair ( a, r ) then it holds for all thepairs ( a + i, r ) with i ≥
0. By direct computations we verify that theinequality is satisfied for ( a, r ) = (5 , , (3 ,
3) and for a = 2 and r > r = 2 and a ≥ , or r = 3 and a ≥ , or r > a ≥ . To complete the proof of (iv) it is enough to evaluate (1) for r = 3 , a = 2in the case d = 4, i.e. s > H A ( a ) > H A ( i ) , a < i E. CARLINI, L. CHIANTINI, AND A.V. GERAMITA then H A is definitely non-increasing and hence it cannot be symmetricwith respect to c by (iv).To get (vi) it is enough to use symmetry and (v). (cid:3) Equivalences
In this section we give some equivalent formulations of our basicquestion (Q), formulated at the end of Section 2.Clearly, if X ⊂ P n is a hypersurface of degree d and Y ⊂ X isa CI ( a , . . . , a r ), then the ideal inclusion I ( X ) = ( F ) ⊂ I ( Y ) =( F , . . . , F r ) yields F = F G + . . . + F r G r for forms G i of degrees d − a i . But the converse is not true in general.If F = F G + . . . + F r G r and the forms F i do not form a regularsequence, then ( F , . . . , F r ) is not the ideal of a complete intersection.To produce an equivalence we need to use joins: Lemma 4.1.
The following are equivalent: (i) a generic hypersurface of degree d of P n contains a CI ( a , . . . , a r ) ,where a i < d for all i ; (ii) the join of the varieties of reducible forms X ( a i ,d − a i ) , i = 1 , . . . , r fills the space of degree d forms in n + 1 variables, i.e. J ( X ( a ,d − a ) , . . . , X ( a r ,d − a r ) ) = P ( S d ) . Proof.
The implication (i) ⇒ (ii) simply follows from the ideal inclusionargument above yielding the presentation F = P F i G i for the genericdegree d form, where [ F i G i ] ∈ X ( a i ,d − a i ) for all i . The implication(ii) ⇒ (i) is easily shown using the description of the generic element ofthe join, see Remark 3.4 (iii). (cid:3) Remark . Notice that there is an equality of varieties X ( i,j ) = X ( j,i ) for all non negative integers i and j . Hence, by Lemma 4.1, the condi-tion J ( X ( a ,d − a ) , . . . , X ( a r ,d − a r ) ) = P ( S d )is equivalent to the statementa generic hypersurface of degree d of P n contains a CI ( b , . . . , b r ), where b i = a i or b i = d − a i for all i . OMPLETE INTERSECTIONS ON GENERAL HYPERSURFACES 9
It follows from these observations that we can further restrict the rangeof the degrees in our basic question (Q), i.e. it is enough to consider a ≤ . . . ≤ a r ≤ d . Now we exploit Terracini’s Lemma and the tangent space descriptiongiven in Corollary 3.3 in order to produce another equivalent formula-tion of question (Q).
Lemma 4.3.
The following are equivalent: (i) the generic hypersurface of degree d of P n contains a CI ( a , . . . , a r ) ,where a i < d for all i ; (ii) let F , . . . , F r and G , . . . , G r be generic forms in S = C [ x , . . . , x n ] of degrees a ≤ . . . ≤ a r < d and d − a , . . . , d − a r respectively,then H ( S/ ( F , . . . , F r , G r , . . . , G ) , d ) = 0 where H ( · , d ) denotes the Hilbert function in degree d of thering.Proof. The condition on the join in Lemma 4.1 can be read in term oftangent spaces as equivalent to h T P ( X ( a ,d − a ) ) , . . . , T P r ( X ( a r ,d − a r ) ) i = P ( S d )for generic points P = [ F G ] , . . . , P r = [ F r G r ]. Using the descriptionof the tangent space to the variety of reducible forms this is equivalentto saying ( F , G ) d + . . . + ( F r , G r ) d = S d where S d is the degree d piece of the polynomial ring S and the forms F i and G i are generic of degrees a i and d − a i respectively. (cid:3) As a straightforward application we get the following result:
Proposition 4.4.
The generic degree d hypersurface of P n contains no CI ( a , . . . , a r ) , a i < d for all i , when r < n + 1 .Proof. We use Lemma 4.3. Consider in S = C [ x , . . . , x n ] generic forms F , . . . , F r and G , . . . , G r of degrees a i and d − a i respectively. If welet I be the ideal ( F , . . . , F r , G , . . . , G r ), then we want to show that H ( S/I, d ) = 0 and for that it is enough to show that S/I is not anartinian ring. As I has height 2 r and 2 r < n + 1 the quotient cannotbe zero dimensional and the conclusion follows. (cid:3) Remark . Using Lemma 4.3 we can also recover many classical re-sults in an elegant and simple way. More precisely, we can easily studythe existence of complete intersections curves, e.g. lines and conics, onhypersurfaces.
Example 4.6.
As an example we prove the following without usingSchubert calculus:
The generic hypersextic of P contains a line.Proof. Let S = C [ x , . . . , x ] and consider the ideal I = ( L , . . . , L , G , . . . , G )where the forms L i are linear forms and the forms G i have degree 5.We want to show that H ( S/I,
6) = 0. Clearly
S/I ≃ C [ x , x ] / ( ¯ G , . . . , ¯ G ) . It is well known [GS98, IK99, Fr¨o85] that 4 general binary forms ofdegree 5 generate C [ x , x ] and we are done. (cid:3) For more on this topic see Remark 5.5.5.
The Theorem
We are now ready to prove the main theorem of this paper, a descrip-tion of all the possible complete intersections of codimension r that canbe found on a general hypersurface of degree d in P n when 2 r ≤ n + 2. Theorem 5.1.
Let X ⊂ P n be a generic degree d hypersurface, with n, d > . Then X contains a CI ( a , . . . , a r ) , with r ≤ n + 2 , and the a i all less than d , in the following (and only in the following) instances: • n = 2 : then r = 2 , d arbitrary and a and a can assume anyvalue less than d ; • n = 3 , r = 2 : for d ≤ we have that a and a can assumeany value less than d ; • n = 4 , r = 3 : for d ≤ we have that a , a and a can assumeany value less than d ; • n = 6 , r = 4 or n = 8 , r = 5 : for d ≤ we have that a , . . . , a r can assume any value less than d ; • n = 5 , or n > , r = n + 1 or r = n + 2 : we have onlylinear spaces on quadrics, i.e. d = 2 and a = . . . = a r = 1 .Proof. Recall that from Lemma 2.1 and Remark 4.2 it is sufficient toconsider the existence of a CI ( a , . . . , a r ) on the generic hypersurfaceof degree d when a ≤ . . . ≤ a r ≤ d/ OMPLETE INTERSECTIONS ON GENERAL HYPERSURFACES 11
When 2 r < n + 1, by Proposition 4.4, we know that no completeintersection exists. Hence we have only to consider the cases 2 r = n + 1and 2 r = n + 2In order to use Lemma 4.3 we consider the generic forms F , . . . , F r and G , . . . , G r of degrees a i and d − a i respectively. If we let S = C [ x , . . . , x n ] and I = ( F , . . . , F r , G r , . . . , G ) we want to check whether H ( S/I, d ) = 0 or not.The case 2 r = n + 1. In this case, S/I is an artinian Gorenstein ringand e = r ( d −
2) + 1 is the first place where one has H ( S/I, e ) = 0.Thus, the generic degree d hypersurface contains a CI ( a , . . . , a r ) ifand only if H ( S/I, d ) = 0 and this is equivalent to the inequality d ≥ r ( d −
2) + 1which is never satisfied unless d = 2 and a = . . . = a r = 1.The case 2 r = n + 2 will be proved using Lemma 3.5. In order to dothis, we divide the proof into three parts: • the hyperplane case : a = 1 any r ; • the plane case : a = 2 , , r = 2, and hence n = 2; • the four space case : a = a = a = 2 and d = 4 for r = 3, andhence n = 4, ; • the general case : one of the following holds(3) r = 2 and a ≥ , or r = 3 and a ≥ , or r = 3 , a = 2 and d = 4 , or r > a ≥ . The hyperplane case.
We need to study CI (1 , a , . . . , a r ) on thegeneric degree d hypersurface of P r − . As one of the generators ofthe complete intersection is a hyperplane, we can reduce to a smallerdimensional case. In algebraic terms, for a generic linear form L , weconsider the surjective quotient map S −→ S/ ( L )to get the following: the generic element of S d can be decomposed as a prod-uct of forms of degrees , a , . . . , a r , i.e. it has theform P ri =1 F i G i with deg F = 1 and deg F i = a i , i = 2 , . . . , r if and only if the generic element of ( S/ ( L )) d ≃ ( C [ x , . . . , x n − ]) d can be decomposed as a product of forms of degrees a , . . . , a r , i.e. it has the form P ri =2 ¯ F i ¯ G i with deg ¯ F i = a i , i = 2 , . . . , r . Hence, we have to study CI ( a , . . . , a r ) on the generic degree d hy-persurface of P r − , i.e. codimension r ′ = r − P n ′ , n ′ = 2 r −
3. As 2 r ′ = n ′ + 1 this situation was treated beforeand the only case where the complete intersections exist is for d = 2and a i = 1 for all i . The plane case.
We have to study CI ( a , a ) on the generic degree d curve of P for a = 2 , ,
4, any a , d such that a ≤ a ≤ d . Now, S = C [ x , x , x ] and we consider forms F , F , G and G of degrees,respectively, a , a , d − a and d − a . We want to study the ring A = S/ ( F , F , G )and to compare H ( A, a ) and H ( A, d ) in order to apply (iii) of Lemma3.5 to show that H ( A/ ( ¯ G ) , d ) = 0 . Using Lemma 3.5 (i), we see that the last non-zero value of H A occursfor d + a − . In particular, for a = 2, H ( A, d ) = 0 and a CI (2 , a ) exists for any a and d , 2 ≤ a ≤ d . If a = 3, then H ( A, d ) = 1 and the same conclusionholds for CI (3 , a ). Finally, if a = 4, then H ( A, d ) = H ( A,
1) and itis easy to see that H ( A, ≤ H ( A, a = 2 , , a , d such that a ≤ a ≤ d , the generic degree d plane curve containsa CI ( a , a ). The four space case.
We address the case of CI (2 , ,
2) on the genericdegree 4 threefold in P . Hence, we consider S = C [ x , . . . , x ] andgeneric quadratic forms F , F , F , G , G and G . Let A be the quo-tient ring S/ ( F , F , F , G , G )and notice that by the vanishing of the lefthand side of (2) in the proofof Lemma 3.5 we have that H ( A,
2) = H ( A, H ( A,
2) = H ( A,
4) and hence the required CI exists. The general case.
Consider the ring A = C [ x , . . . , x r − ] / ( F , . . . , F r , G r , . . . , G , G )and the multiplication map given by the degree d − a form ¯ G m : A a → A d . OMPLETE INTERSECTIONS ON GENERAL HYPERSURFACES 13
Clearly, with this notation, one has that the generic degree d hyper-surface contains a CI ( a , . . . , a r ) if and only if H ( S/I, d ) = H ( A/ ( G ) , d ) = 0and this is equivalent to the surjectivity of m . We also recall that byLemma 3.5 (iii) m has maximal rank. Hence, to study the surjectivitywe only have to compare H ( A, a ) = dim A a and H ( A, d ) = dim A d .When d = 2, all the degrees a i are equal to 1, and this was treatedin the hyperplane case.Now we consider the d > c = ( r − d + a − r +12 . If d ≤ c then dim A a < dim A d by Lemma 3.5(v), thus m cannot besurjective. Standard computations yield d ≤ c ⇔ d ≤ ( r − d + a − r + 1 ⇔ − a r − ≤ d if r > r = 2 the inequality d ≤ c never holds. Thus we get that,when one of the conditions (3) holds, m is not surjective if r > d > r = 3 and a > . In the case c < d we have to be more careful and the distances α = c − a and β = d − c have to be considered. When one of theconditions (3) holds, by 3.5(vi), m is surjective if and only if α ≤ β .Thus we solve the inequality β − α ≥
0. This is equivalent to d ≤ r − r > r = 2 we always have β − α ≥ r = 2 : m is surjective ,r = 3 : d > , m is not surjective; d ≤ , m is surjective; r = 4 , d > , m is not surjective; d ≤ , m is surjective; r ≥ d > , m is not surjective; d ≤ , m is surjectiveand using the treatment of the hyperplane, the plane and the four spacecases we obtain the final result. (cid:3) Remark . The fact that a general hypersurface of degree d ≥ P cannot contain a complete intersection of any type with a , a , a < d is also a consequence of a result about vector bundles proved by MohanKumar, Rao and Ravindra (see [MKRR06]).In P n the existence statement for d = 2 is classical. The d = 3 casesin P and P can be obtained using Theorem 12.8 in [Har92] (see alsoProposition 5.6 in this paper). In P , for d = 3 and d = 4 the existencealso follows from the analysis of arithmetically Cohen-Macaulay ranktwo bundles on hypersurfaces, contained in [AC00] and [Mad00].In P , for the case d = 5, when min { a i } = 2, the result also followsfrom the existence of a canonical curve on the generic quintic threefoldof P , as was essentially proved in [Kle00]. Remark . For n = 2 the theorem above states that the generic degree d plane curve contains a CI ( a, b ) for any a, b < d , but it does not saythat this is a set of ab points. The complete intersection scheme couldvery well not be reduced. Actually, we can show reducedness and hencethe following holds the generic degree d plane curve contains ab completeintersection points for any a, b < d .Remark . Again a remark in the case n = 2. Taking a = a = a thetheorem above states that Sec ( X ( a,d − a ) ) is the whole space. Now, quitegenerally, the points of the variety of secant lines either lie on a truesecant line or on a tangent line to X ( a,d − a ) . We claim that Proposition3.2 allows us to conclude that the points of the tangent lines are in factalready on the true secant lines. In fact, if p = [ F G ] ∈ X ( a,d − a ) , thenany point q of a tangent line to p can be written as [ αF G ′ + βF ′ G ]for forms G ′ , F ′ of degrees d − a, a respectively, and scalars α and β .Thus, q lies on the secant line to X ( a,d − a ) joining [ F G ′ ] and [ GF ′ ].In conclusion, we can rephrase the equality Sec ( X ( a,d − a ) ) = P ( S d ) interms of polynomial decompositions let a < d , any degree d form in three variables F canbe written as F = F G + F G for suitable forms F i of degree a and G i of degree d − a . This answers a question raised during a correspondence between Zi-novy Reichstein and the first author.
Remark . The restriction 2 r ≤ n + 2 in Theorem 5.1 is related tothe fact that Fr¨oberg’s conjecture is only known to be true, in general,when the number of forms does not exceed one more than the numberof variables. However, there are other partial results on this conjecturethat we can use to extend our theorem. E.g. in [HL87] Hochester OMPLETE INTERSECTIONS ON GENERAL HYPERSURFACES 15 and Laksov showed that a piece of Fr¨oberg’s conjecture holds. Moreprecisely they showed that if an ideal is generated by generic forms ofthe same degree d then the size of that ideal in degree d + 1 is exactlywhat is predicted by Fr¨oberg’s conjecture. Using this we can prove thefollowing: Proposition . The generic degree d > hypersurface in P n containsa complete intersection of type ( a , . . . , a r ) where a i = 1 or a i = d − for all i, if and only if (cid:18) n − r + dd (cid:19) ≤ ( n − r + 1) r. When a = . . . = a r = 1 this is the well known result on the non-emptyness of the Fano variety of ( n − r ) -planes on the generic degree d hypersurface of P n (e.g. see [Har92] Theorem 12.8).Proof.
Using Lemma 4.3 we have to show the vanishing, in degree d ,of the Hilbert function of A = C [ x , . . . , x n ] / ( L , . . . , L r , F , . . . , F r )for generic linear forms L i and generic degree d − F i . Clearly,as the linear forms are generic, we have A ≃ C [ x , . . . , x n − r ] / ( F , . . . , F r ) . Hence A d = 0 if and only if ( F , . . . , F r ) contains all the degree d forms.Using the result by Hochester and Laksov this is equivalent to (cid:18) n − r + dd (cid:19) ≤ ( n − r + 1) r. (cid:3) Example 5.7.
The variety X (1 , of reducible quartic hypersurfaces of P and its secant line variety provide interesting examples for severalreasons.First note that X (1 , ⊂ P is a variety of dimension 3 + 19 = 22.From Corollary 3.3 it is easy to deduce that dim Sec ( X (1 , ) = 33.Thus, X (1 , is a defective variety whose virtual defect e is, e = 2 dim X (1 , + 1 − dim Sec ( X (1 , ) = 12 .
1) Consider the Noether-Lefschetz locus of quartic hypersurfaces in P with Picard group = Z . The quartic hypersurfaces which contain a lineare clearly in the Noether-Lefschetz locus. If ℓ is a line defined by the linear forms L , L then the form F , of degree 4, defines a hypersurfacecontaining ℓ if and only if F = L G + L G where deg G i = 3 . I.e. if and only if [ F ] ∈ Sec ( X (1 , ). Since, as we observed,dim Sec ( X (1 , ) = 33this forces the secant variety to be a component of the Noether-Lefschetzlocus.We wonder how often joins of other varieties of reducible forms givecomponents of the appropriate Noether-Lefschetz locus.2) Since X (1 , is defective for secant lines we have, by a theoremof [CC06], that for every two points on X (1 , there is a subvarietyΣ, containing those two points, whose linear span has dimension ≤ − e , where e is the defect of X (1 , .We now give a description of such Σ’s for the variety X (1 , .Let [ H F ] , [ H F ] be two points of X (1 , and let ℓ be the line in P defined by H = 0 = H . Consider Σ ⊂ X (1 , , the subvariety ofreducible quartics whose linear components contain ℓ . Clearly dim Σ =1+19 = 20. Notice that the linear span of Σ, < Σ > , is contained in thesubvariety of all quartics containing ℓ and that variety has dimension34 − < Σ > ≤
29 = 2(20) + 1 −
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Dipartimento di Matematica, Politecnico di Torino, Torino,Italia
E-mail address : [email protected] (L. Chiatini) Dipartimento di Scienze Matematiche e Informatiche,Universit`a di Siena, Siena, Italia
E-mail address : [email protected] (A.V. Geramita) Department of Mathematics and Statistics, Queen’sUniversity, Kingston, Ontario, Canada, K7L 3N6 and Dipartimento diMatematica, Universit`a di Genova, Genova, Italia
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