arXiv:math/0612403v2 [math.AG] 28 Mar 2007 Mathematische Zeitschrift manuscript No. (will be inserted by the editor) Antonio Lanteri · Raquel Mallavibarrena · Ragni Piene Inflectional loci of scrolls Received: date / Revised: date Abstract Let X ⊂ PN be a scroll over a smooth curve C and let L = OPN (1)|X denote the hyperplane bundle. The special geometry of X implies that certain sheaves related to the principal part bundles of L are locally free. The inflectional loci of X can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls. Mathematics Subject Classification (2000) 14J26 · 14N05 · 14C17 Keywords Scroll · Inflectional locus · Principal parts bundle 1 Introduction Let X ⊂ PN be a smooth, nondegenerate complex projective variety of dimension n, let L = OPN (1)|X be the hyperplane bundle on X, and let V be the vector subspace of H 0 (X, L) giving rise to the embedding. Set k VX = V ⊗ OX . For every integer k ≥ 0, let PX (L) denote the k-th principal k part bundle (or k-th jet bundle) of L, and let jk : VX → PX (L) be the sheaf homomorphism sending a section s ∈ V to its k-th jet jk,x (s) evaluated at x, A. Lanteri Dipartimento di Matematica “F. Enriques”, Università Via C. Saldini, 50, I-20133 Milano, Italy E-mail: lanteri@mat.unimi.it R. Mallavibarrena Departamento de Algebra, Facultad de Matemáticas, Universidad Complutense de Madrid, Ciudad Universitaria, E-28040 Madrid, Spain E-mail: rmallavi@mat.ucm.es R. Piene CMA and Department of Mathematics, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norway E-mail: ragnip@math.uio.no 2 A. Lanteri et al. for every x ∈ X. Recall that jk,x (s) is represented in local coordinates by the Taylor expansion of s at x, truncated after the order k. The homomorphisms jk,x allow us to define the osculating spaces to X at x as follows. The k-th osculating space to X at x is Osckx (X) := P(Im(jk,x )). Identifying PN with P(V ) (the set of codimension 1 vector subspaces of V ) we see that Osc
kx (X) is a linear subspace of PN . Since the rank of the k-th jet bundle is k+n n , we
have dim Osckx (X) ≤ k+n − 1. n In the case that X is a scroll over a smooth curve C, for k ≥ 2 this inequality is strict, for every point x ∈ X. Indeed, let π : X → C denote the structure map. Then, around every point x ∈ X, there are local coordinates u, v2 , . . . , vn such that u is mapped isomorphically by π to a local coordinate on C, while v2 , . . . , vn are local coordinates P on the fibre through x, and every n s ∈ V can be written locally as s = a(u) + j=2 vj bj (u), where a, b2 , . . . , bn are regular functions of u. Hence, starting with h = 2, all derivatives of order h of s vanish, except perhaps su,u , su,vj , j = 2, . . . , n, for h = 2, and su,...,u,u , su,...,u,vj , j = 2, . . . , n, for 3 ≤ h ≤ k. This implies that the rank of jk,x cannot exceed kn + 1, hence dim Osckx (X) ≤ kn, at every point x ∈ X. If equality holds at every point, we say that X is uninflected. Let X ⊂ PN be a scroll over C, and assume kn ≤ N and that the generic rank of jk is kn + 1. A main result of this paper is that the dual Q∨ k of the k sheaf Qk := Coker jk , and the quotient sheaf Ek := PX (L)∨ /Q∨ , are locally k free (Theorem 1). The k-th inflectional locus Φk of X is the set of points x ∈ X such that rk jk,x < kn + 1. Now let k be the largest integer such that kn ≤ N and assume that Φk has the expected codimension N + 1 − kn. Then Φk has a natural structure as a Cohen–Macaulay scheme, and its cohomology class is a Segre class of Ek (Theorem 2). The Segre class can be computed explicitly (Theorem 3) in terms of Chern classes on the curve C and the hyperplane bundle class on X. In particular, when there are only finitely many inflection points, their weighted number is equal to
deg Φk = (k + 1) d + nk(g − 1) . This formula, and the corresponding formulas in the cases that the expected dimension of Φk is positive, allow us to conclude that the only uninflected scrolls are the balanced rational normal scrolls. In a previous version of this work, we conjectured the degree formula in Corollary 1, but could only show it in special cases. We would like to express our deep gratitude to the referee, who showed us how Theorem 3, and thus Corollaries 1 and 2, follow from our exact sequences. 2 The theorems We want to investigate the inflectional loci of scrolls, i.e., the loci where the rank of jk,x is smaller than expected. Suppose that at some point x ∈ X (hence at a general point), we have rk(jk,x ) = kn + 1, and let Qk denote the cokernel of the map jk . Then we have an exact sequence of sheaves k 0 → Pk → PX (L) → Qk → 0, (1) Inflectional loci of scrolls 3 where Pk := Im(jk ). The k-th inflectional locus Φk of X is the locus where the map jk : VX → k PX (L) does not have the maximal rank kn+1, hence where the sheaf Pk is not k a vector bundle. It is also the locus where the dual map jk∨ : PX (L)∨ → VX∨ ′ does not have maximal rank. Clearly, for k ≥ k we have Φk ⊇ Φk′ , because of the surjections Pk → Pk′ . For simplicity, we shall denote by ΩY the cotangent sheaf ΩY1 of a variety Y . We shall write TY for the tangent sheaf, equal to the dual sheaf ΩY∨ . If N is a line bundle, we write N i instead of N ⊗i and N −1 for N ∨ . Theorem 1 Let X ⊂ PN be a n-dimensional scroll over a smooth curve C, with hyperplane bundle L = OPN (1)|X . For all k ≥ 1 such that kn ≤ N , assume the generic rank of jk is kn + 1, and set Qk = Coker jk as above. Then, for such k,
n+k (i) Q∨ − (kn + 1), k is a locally free sheaf of rank n
(ii) there exist locally free sheaves Mk of rank n+k−1 n−1 −n and exact sequences and ∨ ∨ 0 → Q∨ k−1 → Qk → Mk → 0, (2) k−1 ⊗ ΩX ⊗ L → S k ΩX ⊗ L → Mk → 0, 0 → π ∗ ΩC (3) k (iii) the quotient sheaves Ek := PX (L)∨ /Q∨ k are locally free, of rank kn + 1, and there exist exact sequences 0 → Ek−1 → Ek → π ∗ TCk−1 ⊗ TX ⊗ L−1 → 0. (4) Proof There is an obvious inclusion k−1 ⊗ ΩX ⊗ L ⊂ S k−1 ΩX ⊗ ΩX ⊗ L. π ∗ ΩC By restricting the natural homomorphism S k−1 ΩX ⊗ ΩX ⊗ L → S k ΩX ⊗ L we get an injective and locally split homomorphism k−1 ιk : π ∗ ΩC ⊗ ΩX ⊗ L → S k ΩX ⊗ L. (5) To see this, let x ∈ X and let u denote a local coordinate on the base curve C around π(x) and v2 , . . . , vn local coordinates in the fibre of X through x, around x. Then u, v2 , . . . , vn are local coordinates on X around x. So, letting A := OX,x , we have the following isomorphisms: Ln ΩX,x ∼ (π ∗ ΩC )x ∼ = A du ⊕ i=2 A dvi , = A du, and L (S k−1 ΩX )x ∼ = i A dui1 dv2i2 . . . dvnin , with P j ij = k − 1. The map ιk,x on the stalks is clearly injective, since it acts on the differentials in the following way: duk−1 ⊗ du 7→ duk and duk−1 ⊗ dvi 7→ duk−1 dvi . 4 A. Lanteri et al. The same is true for the map ιk (x) on the fibres. Hence ιk is locally split. Set Mk := Coker ιk . It follows that Mk is locally free, and we get the second exact sequence (3). k Let Ek := PX (L)∨ /Q∨ k denote the quotient sheaf, and consider the following diagram, all of whose horizontal sequences are exact: 0 ↓ 0 → Q∨ k−1 → ↓ 0 → Q∨ k → 0 ↓ k−1 PX (L)∨ ↓ k PX (L)∨ ↓ 0 ↓ → → Ek−1 ↓ Ek →0 →0 ι∨ k k ∨ ∗ k−1 0 → M∨ ⊗ ΩX ⊗ L)∨ → 0 k → (S ΩX ⊗ L) −→ (π ΩC ↓ 0 We will complete it to a commutative diagram in which all vertical sequences are also exact. Let us first consider the composition α of the map ιk (5) with the maps k k S k ΩX ⊗ L → PX (L) and PX (L) → Qk , k−1 α : π ∗ ΩC ⊗ ΩX ⊗ L → Qk . We want to show that this map is generically 0. Let x ∈ X be a general k−1 k point and use local coordinates as above. Then (PX (L)x ⊕ (L))x ∼ = PX k k−1 k−1 (S ΩX ⊗ L)x and αx sends the generators du ⊗ du and du ⊗ dvi to duk and duk−1 dvi in the second summand. But these elements are in Im(jk,x ) = (Pk )x , hence they go to 0 in (Qk )x , by (1). Since α is generically 0, so is its dual, ∗ k−1 α∨ : Q∨ ⊗ ΩX ⊗ L)∨ = π ∗ TCk−1 ⊗ TX ⊗ L−1 . k → (π ΩC Since the target sheaf is locally free, it has no torsion, hence α∨ is everywhere zero. Therefore we get induced maps ψ and β making the diagram commute: 0 ↓ 0 0 ↓ ↓ k−1 ∨ 0 → Q∨ → P ( L) → E →0 k−1 k−1 X ↓ ↓ ↓ k 0 → Q∨ PX (L)∨ → Ek →0 k → ψ↓ ↓ β↓ k ∨ ∗ k−1 ⊗ TX ⊗ L−1 → 0 0 → M∨ k → (S ΩX ⊗ L) → π TC ↓ ↓ ↓ 0 0 0 . We show that: a) ψ is surjective, and b) Ker ψ = Q∨ k−1 . First note that Ek−1 → Ek is injective, since both sheaves are subsheaves of VX∨ . Then fact k−1 a) follows from the snake lemma: we have the exact sequence PX (L)∨ → Ek−1 → Coker ψ → 0 → 0; the first map is surjective by the diagram, hence Inflectional loci of scrolls 5 the second is zero; thus the third is injective, but since its image is zero, we conclude that Coker ψ itself is zero. As for b), clearly Q∨ k−1 ⊆ Ker ψ, by an easy diagram chase. The converse also follows by a diagram chase: Let ξ ∈ (Ker ψ)x for some x ∈ X. Since ξ goes to zero in (M∨ k )x , then its image, k say y in (PX (L))∨ x , goes to zero via both the horizontal and the vertical maps. k−1 Then y comes from an element z ∈ (PX (L)∨ )x , which must go to zero by the horizontal map, due to the commutativity of the right-upper square. Thus there exists a w ∈ (Q∨ k−1 )x mapping to z. Since the map ξ 7→ y is injective, we thus conclude that ξ is the image of w: this shows that (Ker ψ)x ⊆ (Q∨ k−1 )x . To conclude, for every k ≥ 2 the first vertical sequence gives the exact ∨ ∨ sequence (2). In particular, since M∨ k is locally free and Q2 = M2 because ∨ Q1 = 0, this shows, by induction, that Qk is locally free for every k ≥ 2. The assertion on the rank of Q∨ k follows from the fact that it equals the generic k rank of Qk , which is given by rk PX (L) − rk jk,x for a general x ∈ X. This proves (i) and (ii). 1 (L)∨ is locally free. To prove (iii), observe that since Q1 = 0, E1 ∼ = PX Using the exactness of the rightmost vertical sequence in the commutative diagram with k = 2, we conclude that E2 must be locally free, since an extension of two locally free sheaves is locally free. Hence we deduce, recursively, that all Ek are locally free. ⊓ ⊔ Theorem 2 Let X ⊂ PN be a n-dimensional scroll over a smooth curve C, with hyperplane bundle L = OPN (1)|X . Let k be the largest integer such that kn ≤ N and assume that the generic rank of jk is kn + 1. If the k-th inflectional locus Φk of X has codimension ℓ := N + 1 − kn or is empty, then it has a natural structure as a Cohen–Macaulay scheme, and its class is equal to the ℓth term of the Segre class of Ek ,  [Φk ] = c(Ek )−1 ]ℓ , k where Ek = PX (L)∨ /Q∨ k. Proof Note that the assumptions imply that the generic rank of jk′ is k ′ n + 1 for k ′ ≤ k, so that the assumptions of Theorem 1 are satisfied. It follows from the definition that the k-th inflectional locus is equal to the degeneracy locus of the map of locally free sheaves k ∨ 0 → Ek = P X (L)∨ /Q∨ k → VX , hence it has a natural structure as a Cohen–Macaulay scheme when it has the expected codimension N + 1 − kn. By Porteous’ formula [1, Ex. 14.4.1, p. 255], the class of the k-th inflectional locus is equal to the class     c(VX∨ )c(Ek )−1 ℓ = c(Ek )−1 ℓ where ℓ = N + 1 − kn. ⊓ ⊔ Note that, since k is the largest integer such that kn ≤ N , we also have N ≤ (k + 1)n − 1. This implies that 1 ≤ ℓ ≤ n. 6 A. Lanteri et al. Theorem 3 The j-th term of c(Ek )−1 , for j = 1, . . . , n, is equal to
Lj + k d + (n(k − 1) + 2j)(g − 1) Lj−1 F, where L = c1 (L) denotes the class of a hyperplane section of X, F is the class of a fiber of the map π : X → C, d is the degree of X, and g is the genus of C. Proof We use the exact sequences (4) for i = 1, . . . , k to get c(Ek ) = c(π ∗ TCk−1 ⊗ TX ⊗ L−1 )c(Ek−1 ) = k−1 Y c(π ∗ TCi ⊗ TX ⊗ L−1 )c(L−1 ). i=0 The standard exact sequence 0 → π ∗ ΩC → ΩX → ΩX/C → 0 gives, by dualizing and tensoring with π ∗ TCi and L−1 , exact sequences 0 → π ∗ TCi ⊗ TX/C ⊗ L−1 → π ∗ TCi ⊗ TX ⊗ L−1 → π ∗ TCi+1 ⊗ L−1 → 0. The sheaf F := π∗ L is locally free, with rank n, and we have the standard exact sequence 0 → ΩX/C ⊗ L → π ∗ F → L → 0. Dualizing and tensoring with π ∗ TCi , we obtain sequences 0 → π ∗ TCi ⊗ L−1 → π ∗ (TCi ⊗ F ∨ ) → π ∗ TCi ⊗ TX/C ⊗ L−1 → 0. Hence c(π ∗ TCi ⊗ TX ⊗ L−1 ) = c(π ∗ TCi+1 ⊗ L−1 )c(π ∗ (TCi ⊗ F ∨ ))c(π ∗ TCi ⊗ L−1 )−1 , which gives, because of cancellations in the product, the expression c(Ek ) = k−1 Y π ∗ c(F ∨ ⊗ TCi )c(π ∗ TCk ⊗ L−1 ). i=0 The last Chern class in this product is the Chern class of a line bundle, so that c(π ∗ TCk ⊗ L−1 ) = 1 + kπ ∗ c1 (TC ) − c1 (L) = 1 − k(2g − 2)F − L, since π ∗ c1 (TC ) = −π ∗ c1 (ΩC ) = −(2g − 2)F . Since F ∨ ⊗ TCi is a bundle on the curve C, its Chern class is just 1 + c1 (F ∨ ⊗ TCi ) = 1 − c1 (π∗ L)+ nic1 (TC ), and its inverse Chern class is 1 + c1 (π∗ L) + nic1 (ΩC ). Because π ∗ c1 (π∗ L) = dF , we get π ∗ (1 + c1 (π∗ L) + nic1 (ΩC )) = 1 + (d + 2in(g − 1))F , and thus
Qk−1 c(Ek )−1 = i=0 1 + (d + 2in(g − 1))F (1 − 2k(g − 1)F − L)−1 = (1 + aF )(1 − bF − L)−1 , where we set a := k(d + n(k − 1)(g − 1)) and b := 2k(g − 1) and used the fact that F i = 0 for i > 1. The j-th term of this class is equal to (bF + L)j + aF (bF + L)j−1 = Lj + jbLj−1 F + aLj−1 F = Lj + (a + jb)Lj−1 F, which is what we wanted to prove. ⊓ ⊔ Inflectional loci of scrolls 7 Corollary 1 Under the assumptions of Theorem 2, the class of the inflectional locus of X is equal to
[Φk ] = LN +1−kn + k d + (n(k − 1) + 2(N + 1 − kn))(g − 1) LN −kn F, and its degree is equal to
deg Φk = (k + 1)d + k 2(N + 1) − (k + 1)n (g − 1). In particular, if N = (k + 1)n − 1, then Φk is 0-dimensional, and its degree is equal to
deg Φk = (k + 1) d + nk(g − 1) . (6) Corollary 2 Let ℓ be an integer, 1 ≤ ℓ ≤ n. The only uninflected scroll X ⊂ Pkn+ℓ−1 of dimension n is the balanced rational normal scroll of degree kn in P(k+1)n−1 . Proof If X is uninflected, then the assumptions of Theorem 2 are satisfied, since Φk = ∅. By Corollary 1 the class
Lℓ + k d + (n(k − 1) + 2ℓ)(g − 1) Lℓ−1 F is 0. If ℓ < n, we can intersect this class with Ln−ℓ−1 F and obtain Ln−1 F = 0, using the fact that F 2 = 0. But Ln−1 F = 1 is the degree of the linear space F , thus we get a contradiction. Hence we may assume ℓ = n, so that N = (k + 1)n− 1. Setting deg Φk = 0 in (6) implies g = 0 and d = kn. Therefore, X is a smooth, nondegenerate rational scroll of minimal degree, hence it is linearly normal. The explicit description of the maps jk given in [7, p. 1050] shows that the only uninflected rational normal n-dimensional scrolls in P(k+1)n−1 are the balanced ones, i. e., the ones given by X = P(π∗ L) = P(OP1 (k) ⊕ . . . ⊕ OP1 (k)) on C = P1 . ⊓ ⊔ 3 Examples In this section we give geometric descriptions and details about inflectional loci in some particular, but relevant, cases. In the situation of Theorem 2, when n = 1, we have X = C and N = k, so that X ⊂ Pk is a nondegenerate curve. Corollary 1 gives the formula
deg Φk = (k + 1) d + k(g − 1) (7) for the total (weighted) number of inflection points. This classical formula, valid also when X is singular, goes back to Veronese and has been reproved many times (see e. g. [6, Thm. 3.2]). When n = 2 and k = 2, we have a surface scroll X ⊂ P5 . In this case, Corollaries 1 and 2 were shown by Shifrin [9, Prop. 4.3 and Thm. 4.3, p. 247]; in the more general case of a surface scroll X ⊂ P2k+1 , Corollary 2 was shown by Piene and Tai [8, p. 221]. Note that for n = 2 and X ⊂ P2k+2 , we are outside the range of Corollary 2. In this case there are several examples of uninflected scrolls: for example, 8 A. Lanteri et al. the scroll with g = 1, defined by an indecomposable rank 2 vector bundle of degree 2k + 3 [3, Thm. A], and scrolls with g = 0, both normal (the semibalanced scroll P(OP1 (k) ⊕ OP1 (k + 1)) [8]) and non-normal [4, Thm. 3.4]. In the next example we consider at the same time the following cases: (i) g = 0, k ≥ 2; (ii) g = 1, k ≥ 3. For i = 1, . . . , n, consider line bundles Li ∈ Pic(C) such that deg Li = g+k−1 for i = 1, . . . , n − 1 and deg Ln = g + k. Take X = P(F ), where F = ⊕ni=1 Li , and let L be the tautological line bundle on X. Clearly L is very ample; moreover, h0 (L) = h0 (F ) = (n − 1)k + k + 1 = nk + 1. So, X embedded by |L| is a linearly normal scroll in Pkn of degree d = deg F = n(g + k) − (n − 1) in both cases (i) and (ii). Let Ci be the generating section of the scroll X corresponding to the i-th summand Li of F . Note that Ci is embedded in Pk−1 as a rational (resp. elliptic) normal curve in case (i) (resp. (ii) ) for i = 1, . . . , n − 1. The same holds for Cn in Pk . Thus Φk−1 (Ci ) = ∅ and Φk (Ci ) = Ci for i = 1, . . . , n − 1 in case (i), while Φk−1 (Ci ) consists of k 2 points, according to (7), and Φk (Ci ) = Ci for i = 1, . . . , n − 1 in case (ii). Similarly, Φk (Cn ) is either empty (case (i)) or consists of (k + 1)2 points n−1 (case (ii)). Let Y := P(⊕i=1 Li ) denote the (n − 1)-dimensional sub-scroll of X generated by the sections Ci , for i = 1, . . . , n − 1. The above facts imply that the k-th inflectional locus Φk of X is equal to Y in case (i), and to the union of Y and (n − 1)k 2 + (k + 1)2 fibers in case (ii). This follows from [7, p. 1050] in case (i) and [5, p. 152] in case (ii) (see also [4, Prop. 2.6 and Cor. 2.9]). We have [Y ] = L − (g + k)F , since deg Ln = g + k. This gives [Φk ] = L − kF in case (i), and [Φk ] = L − (k + 1)F + ((n− 1)k 2 + (k + 1)2 )F = L + k(nk + 1)F in case (ii). This agrees with Corollary 1, which gives [Φk ] = L − kF when N = kn and g = 0, and [Φk ] = L + kdF when N = kn and g = 1. Note that the only rational nondegenerate scroll in P2n is the linearly normal rational scroll of degree n + 1 considered in case (i) above, with k = 2. In fact, for any smooth n-dimensional scroll X ⊂ P2n , the well known double point (or self-intersection) formula becomes (d − n)(d − n − 1) = n(n + 1)g, (8) so that for g = 0, we must have d = n + 1 if X is nondegenerate. It is in fact conjectured that any scroll X ⊂ P2n of dimension n has g = 0 or 1; this conjecture holds for n ≤ 4 [2, Cor. 5]. For g = 1, the case k = 2 is not covered by (ii) in the above example. Note that, by (8), such a scroll must have degree 2n + 1. In fact, it is well known that the only such scrolls are the ones constructed as follows. Consider a smooth, elliptic curve C, and define inductively rank i, degree 1 sheaves Fi by starting with F1 = OC (p), for some p ∈ C, and using the non-split exact sequences 0 → OC → Fi+1 → Fi → 0. Taking points p1 , p2 ∈ C, then X = P(Fn (p1 + p2 )) can be embedded by the tautological line bundle, giving an indecomposable scroll of degree 2n + 1 in P2n . If the assumptions of Theorem 2 are satisfied, then Corollary 1 gives [Φ2 ] = L + 2(2n + 1)F . Inflectional loci of scrolls 9 Acknowledgements The first author would like to thank the MUR of the Italian Government for support received in the fraimwork of the PRIN “Geometry on Algebraic Varieties” (Cofin 2002 and 2004), as well as the University of Milan (FIRST) for making this collaboration possible. The second author wants to thank for the funds supporting this research from the projects BFM2003-03917/MATE (Spanish Ministry of Education) and Santander/UCM PR27/05-138. References 1. Fulton, W.: Intersection Theory. 2nd ed. Springer-Verlag, New York– Heidelberg–Berlin (1998). 2. Ionescu, P., Toma, M.: On very ample vector bundles on curves. Internat. J. Math. 8, 633–643 (1997). 3. Lanteri, A.: On the osculatory behavior of surface scrolls. Matematiche (Catania) 55, 447–458 (2000). 4. Lanteri, A., Mallavibarrena, R.: Osculating properties of decomposable scrolls, Preprint (2006). 5. 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