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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 9, September 2010, Pages 4481–4495 S 0002-9947(10)05054-3 Article electronically published on April 6, 2010

HYPERSURFACES CUTTING OUT A PROJECTIVE VARIETY

ATSUSHI NOMA

Abstract. Let X be a nondegenerate projective variety of degree d and codi- mension e in a PN defined over an algebraically closed field. We study the following two problems: Is the length of the intersection of X and a line L in PN at most d − e +1if L ⊆ X? Is the -theoretic inter- section of all of degree at most d − e + 1 containing X equal to X? To study the second problem, we look at the locus of points from which X is projected nonbirationally.

§0. Introduction Let X ⊆ PN (N = n + e) be a projective variety of dimension n,degreed,and codimension e over an algebraically closed field k. We always assume that X is nondegenerate in PN ; i.e., X is not contained in any hyperplane in PN .Letm be a positive integer. We say that X is m-regular if its ideal IX/PN satisfies i N H (P , IX/PN ⊗OPN (m − i)) = 0 for all i>0 (see [16], Lecture 14). By Em(X), we denote the scheme-theoretic intersection of all hypersurfaces in PN , containing X,ofdegreeatmostm. We consider the following conditions on X: N (Am) l(X ∩ L):=length(OX∩L) ≤ m for each line L in P with L ⊆ X; (Bm) X =Em(X); (Cm) X is m-regular.

The purpose here is to study (Am)and(Bm)form = d − e + 1. To study (Bm), we also look at the structure of the locus, denoted by B(X)andC(X) (see (0.1)), of points from which X is projected nonbirationally. First we briefly look at the three conditions. It is well-known that (Cm) implies (Bm) since the m-regularity of X implies that the homogeneous ideal of X is gen- erated in degree ≤ m ([16], Lecture 14, Proposition). Also it is clear that (Bm) implies (Am). On the other hand, the condition (Cm)form = d − e + 1 is the famous conjecture on Castelnuovo-Mumford regularity, and sometimes the implica- tion (Am) ⇒ (Cm)form close to d − e, with few trivial exceptions, is also included in the conjecture (see [6]; [9], §4). The conjecture is true for n = 1 ([9], Theorems 1.1 and 3.1), and the first part of it is true for a smooth X of char(k)=0 ([15]). For n ≥ 3, there are nice approaches to the conjecture but it is still open (see [5] and [13] for information). As evidence of the regularity conjecture, it is natural to expect (Am)and(Bm)form = d − e +1.

Received by the editors October 15, 2007. 2010 Mathematics Subject Classification. Primary 14N05, 14N15. Key words and phrases. Secant line, projection, , defining equation. This work was partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.

c 2010 American Mathematical Society Reverts to public domain 28 years from publication 4481

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One of the key ideas to study (Am)and(Bm)form = d − e + 1 is based on [17], Theorem 1, where it was shown that X =Ed(X)asaset,andX =Ed(X)as aschemeifX is smooth. To obtain these results, Mumford considered the image N n+1 πΛ(X) by the linear projection πΛ : P \ Λ → P from a general (e − 2)-plane N Λ ⊆ P and observed that the pull-back of the hypersurface πΛ(X) separates a point v ∈ PN \ X from X if Λ is suitably chosen according to v.Inthispaper,we will also consider linear projections πΛ with Λ ∩ X = ∅. Now we will state our results. First we deal with (Ad−e+1). We say that a line L is secant to X if X ∩ L is finite of length > 0. We say that a secant line L to N X is standard if dim πL(X \ L)=dimX for the linear projection πL of P \ L to PN−2 with center L. Theorem 1 (char k ≥ 0). For a standard secant line L to X, we have l(X ∩ L) ≤ d − e +1. For a nonstandard secant line, the same result is expected. But at this stage, no proof about it would be known for char k ≥ 0. The result here is a generalization of results in char k = 0, [2], [3], [14], and [18]. When X issmooth,wehaveasharp bound including another invariant of X ([19]). Next we deal with (Bd−e+1). To state our result, we introduce some notation: (0.1) B(X):={v ∈ PN \ X | l(X ∩ v, x ) ≥ 2 for general x ∈ X}, C(X):={u ∈ X \ Sing X | l(X ∩ u, x ) ≥ 3 for general x ∈ X}, where Sing X denotes the singular locus of X. In other words, these are the loci of points from which X is projected nonbirationally onto its image: in the former, points off X, and in the latter, points on X \ Sing X.Whene = 1, it is clear that N (Bd) holds and B(X)=P \ X and C(X)=X \ Sing X if d ≥ 3. Thus we consider (Bd−e+1)fore ≥ 2. Theorem 2 (char k =0). Assume e ≥ 2. (1) (Calabri and Ciliberto ([4], Corollary 2); and Sommese, Verschelde, and Wampler ([22])) As sets, X ⊆ Ed−e+1(X) ⊆ X ∪ B(X). (2) As schemes, X and Ed−e+1(X) are equal outside B(X), C(X),andSing X. As an application of Theorem 2, we have another proof of Theorem 1 in char(k)= 0, since a standard secant line L to X is not contained in B(X)(seeRemark3.6). As another application, we will prove (Bd−e+1) for a special case: N Corollary 3 (char k =0). Suppose that X (⊆ P ) with e ≥ 2 is contained in the Pm ⊆ PM m+l − ≥ image vl( ) (M = l 1) of an lth (l 2) Veronese embedding vl of a projective space Pm in PM (⊇ PN ) for some m>0.ThenB(X) and C(X) are empty. Consequently X =Ed−e+1(X) as sets; and X =Ed−e+1(X) as schemes if X is smooth. Finally we will study the structure of B(X)andC(X). We show that B(X)isa closed subset of PN \ X and that C(X) is a closed subset of X \ Sing X in (4.1) and (4.2). The set B(X) was first studied by Beniamino Segre [20] and [21], and later by Calabri and Ciliberto [4]. Segre [20] proved that the closure of B(X) is the union of a finite number of linear subspaces of dimension at most n−1 (see Theorem 4.3). Based on Segre’s result, we say more about B(X)andC(X). Conventionally, we mean dim ∅ = −1.

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Theorem 4 (char k =0). If e ≥ 2,thendim B(X) ≤ min{n − 1, dim Sing X +1}. In particular, if X is smooth (i.e., dim Sing X = −1) and e ≥ 2,thenB(X) is a finite set. Theorem 5 (char k =0). Assume e ≥ 2.LetZ be an of C(X). Then the closure of Z is a linear subspace of dimension l ≤ min{n − 1, dim Sing X +2}. As a consequence of Theorems 2, 3, 4 and 5, we have the following. Corollary 6 (char k =0). Suppose that X is smooth, n ≥ 2 and e ≥ 2.Then B(X) is a finite set and C(X) is the union of a finite number of linear subspaces of dimension ≤ 1. Consequently the intersection of all hypersurfaces containing X, of degree ≤ d − e +1,isequaltoX as a scheme, except for a finite union of linear subspaces of dimension ≤ 1. Moreover we will show that the inequality in Theorem 5 is sharp by giving an example in (4.10). Also, in Theorem 4.11, we study the singular locus of X contained in the boundary of C(X). We organize this paper as follows. In §1, we summarize some results of inner projections which we will use later. In §2, we prove Theorem 1. In §3, we prove Theorem 2 and Corollary 3. In §4, we look at the structure of B(X)andC(X)and prove Theorems 4.4, which is a strong form of Theorem 4, and Theorem 5. The author would like to thank the organizers of the conference at KAIST in January 2006, especially Professor Sijong Kwak, for their hospitality. Also the author would like to thank Professor Fyodor Zak for information regarding the results of Segre and Calabri-Ciliberto. Notation. We use standard terminology from , e.g, [11]. By a point, we always mean a closed point. A general point means a point off a finite union of suitable proper closed subvarieties. For a point x of X,byTx(X), we denote the embedded tangent space to X at x in PN .By Y,Z ,wedenotethe linear span of subschemes Y and Z of PN , the smallest linear subspace containing both Y and Z. By Sing X (resp. Sm X), we denote the singular locus (resp. smooth locus) of X. §1. Inner projection In this section, we summarize some results of inner projections. 1.1. Let X ⊆ PN be as in §0. Let Λ ⊆ PN be a linear subspace of dimension l N N−l−1 with Λ ∩ X = ∅ but Λ ⊇ X. The linear projection πΛ : P \ Λ → P from N−l−1 Λ induces a morphism πΛ,X : X \ Λ → P .LetV be the linear space of linear forms on PN , and let W ⊆ V be the subspace of linear forms vanishing on N N−l−1 Λ, i.e., P = P(V )andP = P(W ). Then πΛ,X is defined by the surjection ε : W ⊗OX →IX∩Λ/X ⊗OX (1) induced from V ⊗OX →OX (1). We can identify PN−l−1 to be a subspace of PN disjoint from Λ. Let σ : Xˆ → X be the of X by the ideal sheaf IX∩Λ/X of OX , and let E be the exceptional Cartier N−l−1 divisor. Then we have the morphismπ ˆΛ,X : Xˆ → P withπ ˆΛ,X ◦ σ = πΛ,X as ⊗O →O − ⊗ ∗O a rational map, since ε induces a surjectionε ˆ : W Xˆ Xˆ ( E) σ X (1). Thus the closure X¯ of πΛ(X \ Λ) is exactlyπ ˆΛ,X (Xˆ) and hence

(1.1.1) X¯ = πΛ(X \ Λ) ∪ πˆΛ,X (E), as a set.

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¯ { ∈ ¯| −1 ≥ }∪ 1.1.2. Assume dim X =dimX.ThenY := x¯ X dimπ ˆΛ,X (¯x) 1 πˆΛ,X (E) ¯ ˆ \ −1 → ¯ \ is a proper closed subset of X. Thus the induced morphism X πˆΛ,X (Y ) X Y ˆ \ −1 | fromπ ˆΛ,X is finite, and hence for the open set U := σ(X πˆΛ,X (Y )) of X, πΛ,X U is finite.

1.1.3. On the other hand, assume that e ≥ 2andΛ={z} for a smooth point z of X. Then (1) dim X¯ =dimX,sinceX is not a cone with vertex z nor linear; (2) \{ } 0 ⊗O ⊗ k → 2 πˆz,X(E)=πz(Tz(X) z ), since the map H (ˆε E)isε (z):W mz/mz for the maximal ideal mz of OX,z;(3)0≤ deg X¯ ≤ d − 1byB´ezout’s theorem or O − ⊗ ∗O  ∈ by computing the degree of Xˆ ( E) σ X (1); (4) For a point x (= z) X, ∩ −1 l(X z,x )=l(ˆπz,X(πz,X(x))) + 1 by a local computation (see for example, [7], p.269, (8.4.3)). In particular, if l(X ∩ z,x )=2forapointx (= z) ∈ X,then πz,X is an embedding at x by (1.1.2).

1.2. Returning to the original situation in (1.1), we look at the behavior of tangent spaces under the projection. By π : X \ Λ → X¯ we denote the induced morphism from πΛ,X .Letx be a smooth point of X with x ∈ Λ. Assumex ¯ := πΛ(x) ∈ Sm X¯. O O By comparing the bundles of the principal part with respect to X (1) and X¯ (1) (see [12], (IV.A)), we have πΛ(Tx(X) \ Λ) ⊆ Tx¯(X¯), and the equality holds if and only if π is smooth at x. If the equality holds, then Tx¯(X¯), Λ = Tx(X), Λ ,and hence

(1.2.1) dim X¯ =dimπΛ(Tx(X) \ Λ) = n − dim(Tx(X) ∩ Λ) − 1 and  (1.2.2) Tx (X) ⊆ Tx(X), Λ for each x ∈ Xx¯ ∩ Sm X,

where Xx¯ is the closure of πΛ,X (¯x). In particular, if char(k) = 0, by the generic smoothness of π|Sm X\Λ (e.g. [11], III.10.7), the above holds for general x ∈ Sm X and its imagex ¯ = πΛ(x).

§2. Multisecant lines: Proof of Theorem 1 In this section we will prove Theorem 1. The key is to find a hypersurface F of degree ≤ d, containing X, and meeting L in at least e − 1 distinct points off X ∩ L.

2.1. Proof of Theorem 1. First we claim that if M ⊆ PN is a general e- dimensional linear subspace containing L,thenM ∩ X is finite and containing at least e − 1distinctpointsoffL. To prove this, consider the linear projection N N−2 πL : P \ L → P from L. The closure X¯ of πL(X \ L) has dimension n by assumption. Since X is nondegenerate, then so is X¯ ⊆ PN−2.Thusd¯ := deg X¯ ≥ e − 1 (see for example [10], (18.12); [8], (I.4.2)). Since M¯ := πL(M \ L)isageneral (e − 2)-dimensional linear subspace of PN−2,byB´ezout’s and Bertini’s Theorems, X¯ ∩M¯ is d¯distinct points, contained in a nonempty open subset U of X¯ over which the induced morphism πL,X is a finite morphism (see (1.1.2) for U). This implies the claim. Let M ⊆ PN be a general e-dimensional linear subspace containing L.Bythe first part, as sets, (M \ L) ∩ X = {x1,...,xm} and L ∩ X = {y1,...,yt} for some distinct points xj and yi with (2.1.1) m (≥ d¯) ≥ e − 1.

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Take a general (e−2)-dimensional linear subspace Λ ⊆ M. We may assume that Λ is disjoint from X, L, and lines yi,xj and xj,xk (i =1,...,t; j = k =1,...,m). Consequently, Λ,xj ∩L is a point, say zj ,and{zj } are m distinct points off N n+1 X ∩ L. Now consider the linear projection πΛ : P \ Λ → P from Λ. The image  n+1  X := πΛ(X) ⊆ P is a hypersurface of degree d (≤ d), since the projection n+1 πΛ,X : X → P is finite. Let F be the hypersurface obtained by the pulling-  n+1  back of X ⊆ P by πΛ.Inotherwords,F istheconeoverX with vertex Λ if we consider Pn+1 to be a subspace of PN , disjoint from Λ. Thus deg F = d ≤ d, F ⊇ X ∩ ⊇ ∩ ∪{ } ⊇ −1 and F L (X L) z1,...,zm .MoreoverF L, since the closure of πΛ (πΛ(L)) is L, Λ = M and M ∩ X is finite. Consequently d = l(F ∩ L) ≥ l(X ∩ L)+m, and hence (2.1.2) l(X ∩ L) ≤ d − m ≤ d − m ≤ d − e +1.  Remark 2.2. Under the same notation and assumptions as in Theorem 1, if l(X ∩ L)=d − e + 1, then the closure X¯ of πL(X \ L) is a variety of minimal degree, i.e., d¯ =¯e + 1 (see [10], (19.9); [8], (I.2.2), (I.5.10)). Furthermore, if char(k)=0,thenπ : X \ L → X¯ is separable and hence X¯ is birational to X. Indeed, if l(X ∩ L)=d − e + 1, then it follows from (2.1.2) that m = e − 1and d = d. Hence m = d¯= e − 1 by (2.1.1), as required. Remark 2.3. If a secant line L meets X only at Sm X and e ≥ 2, by a Bertini- type theorem, X ∩ H is irreducible and reduced for a general hyperplane H ⊇ L, and hence L is standard (see [19], Lemma for the proof; see also [14] and [18], Lemma 2.1). On the other hand, if X has bad singularity at X ∩ L, for a general hyperplane H ⊇ L of PN , X ∩ H is not necessarily irreducible (see [18], Example 2.2(2)). Moreover, if X is integral and regular in codimension 1, but not normal at x ∈ X ∩ L,thenX has Serre’s condition S1 but not S2 at x (see [1], VII (2.2) and (2.13)), and hence X ∩ H is not S1 at x, i.e., X ∩ H is not reduced. Thus to obtain Theorem 1, the method of taking hyperplane sections used in [2], [14], and [19] does not work in case X ∩ L ⊆ Sm X.

§3. Hypersurfaces containing projective variety: Proof of Theorem 2 Let X ⊆ PN be as in §0. In this section, we always assume char(k)=0and will prove Theorem 2. This assumption is necessary only to apply the trisecant lemma, which asserts for general points x, y of X, l(X ∩ x, y )=2ife ≥ 2and char(k) = 0: In fact, the set of points x = y ∈ X with l(X ∩ x, y ) ≥ 3 is a closed subset of X × X \ ΔX , and also a proper subset by the trisecant lemma (see for example [11], IV 3.8) for curves which are obtained by hyperplane sections of X. Theorem 2 is reduced to proving Propositions 3.1 and 3.2. For scheme-theoretic part (2), see [17], p.34, Lemma. Proposition 3.1 (char k =0). Let v be a point of PN \ X.Supposev ∈ B(X) if e ≥ 2. Then there exists a hypersurface F of degree ≤ d − e +1 such that F ⊇ X but v ∈ F . Proof. When e = 1, this is clear. By induction on e, suppose e ≥ 2. Let x be a general point of X,sothatx ∈ Sm X and l(X ∩ v, x )=1byv ∈ B(X). Consider N N−1 the projection πx : P \{x}→P from x and let X¯ be the closure of πx(X\{x}). By (1.1.1) and (1.1.3), X¯ is a nondegenerate projective variety of degree d¯≤ d − 1,

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and codimensione ¯ = e − 1, and X¯ = πx(X \{x}) ∪ πx(Tx(X) \{x}). Thus v¯ := πx(v) ∈ X¯,sincel(X ∩ v, x ) = 1. We assume, for a moment, thatv ¯ ∈ B(X¯) fore ¯ = e − 1 ≥ 2, and will complete the induction. By the induction, we have a hypersurface F¯ ⊆ PN−1,ofdegree≤ d¯− e¯ +1(≤ d − e + 1) such that F¯ ⊇ X¯ but v¯ ∈ F¯.LetF be the hypersurface obtained by the pulling-back of F¯ by πx.Then deg F =degF¯ ≤ d − e +1, F ⊇ X and v ∈ F ,sinceF istheconeoverF¯ with vertex x, as required. To conclude the proof, we will show thatv ¯ ∈ B(X¯)fore ≥ 3. By contradiction, we assume that for a general point y ∈ X withy ¯ := πx(y) ∈ Sm X¯, (3.1.1) l( v,¯ y¯ ∩X¯) ≥ 2.

By the generality of x and y and the trisecant lemma, the projection πx,X : X \ N−1 {x}→P is an embedding at y (see (1.1.3)(4)), and consequently Ty¯(X¯)= πx(Ty(X)) (see (1.2)). Moreover, v ∈ Tx(X), v ∈ Ty(X), x ∈ Ty(X),v ,and y ∈ Tx(X),v since v ∈ B(X)andx and y are general in nondegenerate X,and consequently

(3.1.2)v ¯ ∈ Ty¯(X¯)=πx(Ty(X)) and y,¯ v¯ ∩πx(Tx(X) \{x})=∅. By (3.1.1) and (3.1.2), there is a pointz ¯ ∈ v,¯ y¯ ∩X¯ distinct fromy ¯ which is an image πx(z)ofsomez ∈ X. This means that v, x, y contains z ∈ X off v, x and v, y ,sincex and y are general and v ∈ B(X). Thus for the linear projection N N−1 πv : P \{v}→P from v, the image πv(X) has a secant line meeting at the three distinct points πv(x), πv(y)andπv(z). This contradicts the trisecant lemma, since πv(x)andπv(y) are general points of πv(X) because of the generality of x and y in X.  Proposition 3.2 (char k =0). Let u be a smooth point of X with embedded tangent N N space Tu(X) ⊆ P and let w be a point of P \ Tu(X).Supposeu ∈ C(X) if e ≥ 2. Then there exists a hypersurface F containing X,ofdegree≤ d − e +1, such that N F is smooth at u with w ∈ Tu(F ) ⊆ P . Proof. When e = 1, this is clear. By induction, suppose e ≥ 2. Let x be a general point of X.Thenx ∈ Sm X with l(X∩ x, u )=2andx ∈ Tu(X),w . Consider the N N−1 projection πx : P \{x}→P from x, and let X¯ be the closure of πx(X \{x}). N−1 The projection πx,X : X \{x}→P is an embedding at u (see (1.1.3)), and henceu ¯ := πx(u) ∈ Sm X¯,and¯w := πx(w) ∈ Tu¯(X¯)=πx(Tu(X)) (see (1.2)). Moreover X¯ is a nondegenerate projective variety of degree d¯ ≤ d − 1, dimension n¯ = n, and codimensione ¯ = e − 1, with X¯ = πx(X \{x}) ∪ πx(Tx(X) \{x})(see (1.1.1) and (1.1.3)). We assume, for a moment, thatu ¯ ∈ C(X¯)fore ≥ 3, and will complete the induction. By induction, we have a hypersurface F¯ ⊆ Pn+e−1 ¯ containing X¯,ofdegree≤ d − e¯+1(≤ d − e + 1), smooth atu ¯ withw ¯ ∈ Tu¯(F¯). The ¯ hypersurfaces F obtained by the pulling-back of F¯ by πx is of degree ≤ d − e¯ +1, smooth at u with w ∈ Tu(F ), as required. To conclude our proof, we will show thatu ¯ ∈ C(X¯)fore ≥ 3. By contradiction, assume that for general y ∈ X withy ¯ := πx(y) ∈ Sm X¯, l( y,¯ u¯ ∩X¯) ≥ 3. By the generality of x and y and the trisecant lemma, πx,X is an embedding at y, and hence Ty¯(X¯)=πx(Ty(X)). Moreover, y ∈ Tu(X),x , y ∈ Tx(X),u ,and x ∈ Ty(X),u . Hencey ¯ ∈ Tu¯(X¯),y ¯ ∈ u,¯ πx(Tx(X) \{x}) ,and¯u ∈ Ty¯(X¯). Consequently there is a pointz ¯ ∈ X¯ ∩ y,¯ u¯ distinct fromy ¯ andu ¯ withz ¯ = πx(z) for some z ∈ X. This means that z lies on u, x, y , but off u, x and u, y by

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N N−1  u ∈ C(X). Thus for the projection πu : P \{u}→P from u, the closure X of πu(X \{u}) has a general secant line πu(x),πu(y) meeting at the three distinct points, which contradicts the trisecant lemma. 

Remark 3.3. If v ∈ B(X), then πx(v) ∈ πx(X \{x}) for any x ∈ X.Thusthe points of B(X) cannot separate from X by the hypersurfaces obtained in (3.1). On the other hand, if u ∈ C(X), then πx(u) ∈ Sing πx(X \{x}) for any x ∈ X.Thus at the points of C(X), the hypersurfaces in (3.2) cannot separate the tangent space of X from PN . Example 3.4. Let X ⊆ PN (N = n + e) be a projective variety of dimension n,degreed, and codimension e over an algebraically closed field k of char k =0. Assume X is a variety of minimal degree, or of delta genus Δ(X) = 0, i.e., d = e+1 (see [10], (19.9); [8], (I.2.2), (I.5.10)). Assume e ≥ 2, or equivalently d ≥ 3. Then B(X)=∅.Moreover,X has no 3-secant line and hence C(X)=∅. Proof. Suppose B(X) = ∅, and we will show that e = 1. Consider the projection N N−1 ¯ πv : P \{v}→P from a point v ∈ B(X). Set X¯ = πv(X)andd =degX¯. Then d¯ ≤ d/2(=(e +1)/2). Since X¯ is nondegenerate in PN−1,wehaved¯ ≥ N − n = e by [10], (18.12) or [8],(I.4.2). From these two inequalities, we obtain e ≤ 1, and hence e = 1, as required. The second part follows from the first part and Theorem 2, noting that d − e + 1 = 2. (Or directly, if there is a 3-secant line L, consider the projection from L, and follow the same argument as above. Then we have a contradiction N = n.)  3.5. Proof of Corollary 3. We have only to show that B(X)andC(X)are empty. First we will prove B(X)=∅. By contradiction, assume there is a point M v ∈ B(X). For general x1 = x2 ∈ X, consider the lines in P joining xi and v, and observe on the lines that there exist points yi ∈ X different from xi.LetLi m m (i =1, 2) be the line in P joining the preimages of xi and yi in P .Notethat v ∈ vl(Li) .IfL1 = L2,thenX = vl(L1) with B(X) = ∅, and hence X is conic by Example 3.4, a contradiction. Thus L1 = L2. On the other hand, it is easy to see m for two distinct lines 1,2 in P that

(1) vl(1) ∩ vl(2) = ∅ if 1 and 2 are disjoint, and that (2) vl(1) ∩ vl(2) = vl(1 ∩ 2)if1 ∩ 2 = ∅.

Therefore v ∈ vl(L1 ∩ L2). This implies that vl(L1) has a 3-secant line v, xi = xi,yi , which contradicts Example 3.3, as required. The second part C(X)=∅ follows from the same argument above for u ∈ C(X) instead of v ∈ B(X).  Remark 3.6. Proposition 3.1 implies Theorem 1 in case char k = 0: Indeed, since L is standard, L \ X ⊆ B(X) by the argument as in the proof of (4.4). By (3.1), we obtain a hypersurface F containing X but not L.Consequentlyl(X ∩ L) ≤ l(F ∩ L) ≤ d − e +1.

§4. Study of B(X) and C(X): Proof of Theorems 4 and 5 Let X ⊆ PN be as in §0. In this section, we assume char k = 0. We will study the structure of B(X)andC(X) in (0.1), and prove Theorems 4 and 5. As a consequence, we will obtain Corollary 6. First we will show that B(X)andC(X) are algebraic sets. Lemma 4.1. B(X) is a closed subset of PN \ X.

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Proof. For a hyperplane H ⊆ PN ,wesetU := PN \ (X ∪ H). We have only to show that B(X) ∩ U is closed in U. To this purpose, consider the morphism  : U × X → U × PN−1 defined by (u, x) → (u, u, x ∩H), that is, the family N−1 ∼ N−1 of the projections πu,X : X → P from u ∈ U to H = P .Notethat is projective, since U × X → U is projective and U × PN−1 → U is separated (see [11], Ex.II.4.9). Moreover  is finite, since  is quasi-finite (see [11], Ex.III.11.2). N−1 Hence W := {(u, x¯) ∈ U × P | dimk(u,x¯) ∗OU×X ⊗ k(u, x¯) ≥ 2} is the set of ∈ × PN−1 −1 ∼ −1 points (u, x¯) U whose fibre  (u, x¯)(= πu (¯x)) is of length at least 2. Moreover W is closed in U × PN−1 by [11], Ex.II.5.8, and hence the first projection p1 : W→U is projective. Therefore a point u ∈ U is contained in B(X) ∩ U if W −1 W and only if the fibre u := p1 (u)isdenseinπu(X), i.e., u = πu(X). Then Wu = πu(X) if and only if dim Wu = n,sinceX is irreducible and πu is finite. Thus B(X) ∩ U = {u ∈ U| dim Wu ≥ n}, and hence B(X) ∩ U is closed in U,by [11], Ex.II.3.22 (d) and the properness of p1, as required.  Lemma 4.2. C(X) is a closed subset of the smooth locus Sm X of X. Proof. For a hyperplane H ⊆ PN , we have only to show that C(X) \ H is closed in N−1 X0 := Sm X \ H. Recall that the linear projection πz,X : X \{z}→P from ∼ N−1 N N−1 z ∈ X0 to H = P ⊆ P is extendable to the morphismπ ˆz,X : Xˆz → P by taking the blowing-up σz : Xˆz → X of X at z (see (1.1)). To construct the family N N ofπ ˆz,X over z ∈ X0, consider the family of linear projections (P \ H) × P  − ∼ PN 1 = H ⊆ PN ,(z,x) → z,x ∩H whose base locus is the diagonal of PN \ H. N−1 Its restriction 2 : X := X0 × X  P is a rational map whose base locus is X→X the diagonal ΔX0 of X0. By taking the blowing-up σ : by the ideal sheaf of Δ with reduced structure, we have an extension ˆ : X→ PN−1 of  .Since X0 ∼ 2 2 J X |{z}×X = J{ } by a local computation, σ is the family of blowing-ups ΔX0 / z /X  N−1 σz for z ∈ X0.Thusˆ := (p1 ◦ σ) × ˆ 2 : X→X0 × P is the required family −1 −1 ∈ × PN−1 such that ˆ (z,x¯)=ˆπz,X(¯x)for(z,x¯) X0 .Notethatˆ is projective,  N−1 since X→X0 is projective and X0 × P → X0 is separated. To prove C(X) \ H is closed in X0,weconsider W { ∈ × PN−1| O ⊗ k ≥ −1 ≥ } := (z,x¯) X0 dimk(z,x¯) ˆ ∗ X (z,x¯) 2 or dimπ ˆz,X(¯x) 1 . N−1 By [11], Ex. II.5.8, and Ex. II.3.22 (d), W is closed in X0 × P ,andthe first projection p1 : W→X0 is projective. Moreover W is the set of points ∈ × PN−1 −1 ∼ −1 (z,x¯) X0 whose fibre ˆ (z,x¯)(= πˆz,X(¯x)) is of length at least 2, since −1 N−1 ˆ is finite around (z,x¯) if dim ˆ (z,x¯) = 0. A point (z,x¯) ∈ X0 ×P belongs to W ∩ ≥ −1 ≥ ∩ ≥ if and only if l(X z,x¯ ) 3, since l(ˆπz,X(¯x)) 2 if and only if l(X z,x¯ ) 3 ∈ \ W −1 (see (1.1.3)). Thus z X0 belongs to C(X) H if and only if the fibre z := p1 (z) is equal to X¯. The last condition is equivalent to dim Wz ≥ n. Therefore C(X) \ H is closed in X0, as required.  Next we study the structure of B(X). Recall the following result. Theorem 4.3 (Beniamino Segre [20]; see also [4]). Let X ⊆ PN (N = e + n) be a nondegenerate, projective variety of dimension n and codimension e.Ife ≥ 2 and B(X) = ∅, then every irreducible component of the closure of B(X) is a linear subspace of dimension at most n − 1. Moreover dim B(X)=n +1 if and only if e =1.

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Based on this result and the idea of the proof, we will prove Theorem 4.4 and Theorem 5. Theorem 4 is an immediate consequence of Theorem 4.4. Theorem 4.4. Let X be as in §0. Assume n ≥ 2 and e ≥ 2.LetΛ be an irreducible component of the closure of B(X),ofdimensionl ≥ 0.Then (1) dim X ∩ Λ=l − 1,and (2) X ∩ Λ ⊆ Sing X. Consequently dim Λ ≤ dim Sing X +1, and in particular, dim B(X) ≤ dim Sing X + 1. Proof. When l = 0, the assertion is trivial, so we assume l ≥ 1. By (4.3), Λ is N−l−1 linear. First we observe the linear projection πΛ,X : X \ Λ → P from Λ, in particular, its fibre and image. For a general point x ∈ X \ Λ, let Xx¯ be the −1 ≥ closure of πΛ,X (¯x)over¯x := πΛ(x). Then dim Xx¯ = l. Indeed, dim Xx¯ l,since a line joining x and general v ∈ Λ ∩ B(X) contains a point of X different from x. −1 ⊆ ⊆ Hence dim Xx¯ = l,sinceπΛ,X (¯x) Λ,x and Λ X. Consequently the closure ¯ X¯ of πΛ,X (X \ Λ) has dimension n − l.Moreoverd := deg X¯ ≥ 2: If not, X¯ is linear and nondegenerate in PN−l−1, and hence X¯ = PN−l−1, which implies e =1, a contradiction. Now (1) is clear, since dim Xx¯ = l and Xx¯ ⊆ Λ,x with Λ ⊆ X. Next we will show (2) in case n =2.Thenl = 1 by assumptions l ≥ 1 and (4.3). N−2 ¯ Hence dim Xx¯ =1andX¯ ⊆ P is a nondegenerate curve of degree d ≥ 2. Let H be a general hyperplane containing Λ. The first step is to show that (4.4.1) Sing(X ∩ H) ⊇ X ∩ Λ. N−2 Since H¯ := πΛ(H \ Λ) is a general hyperplane in P ,byB´ezout’s theorem, ¯ ∩ ¯ ¯ \ X H is d distinct points, sayx ¯1,...,x¯d¯, which lie on πΛ,X (X Λ) (see (1.1.1)). ∩ ¯ ≥ Consequently X H contains d ( 2) distinct curves Xx¯i , each of which contains X ∩ Λ by Lemma 4.5 below. This implies (4.4.1). Now to obtain (2), we assume, to the contrary, that there is a point z ∈ Λ ∩ Sm X.ThenH ⊇ Tz(X)bythe generality of H ⊇ ΛandΛ= Tz(X). Hence X ∩H is smooth at z. This contradicts (4.4.1). We will show (2) in case n>2. By contradiction, assume that there is a point z ∈ Λ ∩ Sm X. Take a general line L through z, contained in Λ, not contained in X,sothatL ∩ B(X) is dense open in L by (4.1). Let H be a general hyperplane   containing L. We claim that the reduced induced structure X := (X ∩ H )red  ∼ −  is irreducible and nondegenerate in H = PN 1 such that z ∈ Sm X and the closure of B(X) contains L. If the claim is proved, by induction on n,wehavea projective surface X such that the closure of B(X) contains L with L∩Sm X = ∅, which contradicts the case n = 2. Thus we have only to show the claim. By the same argument as in the first paragraph, a general fibre of the linear projection N−2 πL,X : X \ L → P has dimension 1. Hence the image of πL,X has dimension n − 1(≥ 2). By Bertini’s theorem [24], (I.6.3), X ∩ H is irreducible and generically reduced. (To be precise, apply Bertini’s theorem for the normalization X˜ of X and the pull-back H˜ of H, and take the push-forward of X˜ ∩ H˜.) Moreover, since  X is integral, (X \ L) ∩ H satisfies Serre’s condition S1 by [7], (3.4.6), and hence (X \ L) ∩ H is reduced (see [1], (VII.2.2)). On the other hand, since X ⊆ PN is N−2 nondegenerate, so is πL,X (X \ L) ⊆ P , and also so is its hyperplane section (see [7], p.116). Consequently (X \ L) ∩ H is nondegenerate. Moreover, X ∩ H

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  is smooth at z ∈ Λ ∩ Sm X,sinceH ⊇ Tz(X) by the generality of H ⊇ L and     Tz(X) = L. Finally it is clear that L ⊆ X and L\X ⊆ B(X ). In sum, X satisfies the property we have claimed. This complete the proof.  Lemma 4.5. Let X ⊆ PN be a nondegenerate projective variety of dimension n and codimension e ≥ 2.LetΛ ⊆ PN be a linear subspace of dimension l (1 ≤ l

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1 1 on P , with projection τ : P → P and the tautological bundle OP(1). Here {zi} is a formal basis. Let s, t be the homogeneous coordinates of P1.Letφ : P → P5 4 3 3 4 0 be the morphism defined by z0,z1,s z2,s tz2,st z2,t z2 ∈ H (OP(1)). Let X˜i 2 − 2 2 2 − 3 (i =1, 2) be divisors defined by Gi for G1 = z1 s t z0z2 and G2 = z1 s tz0z2 in 0 H (OP(2)). Then the images Xi = φ(X˜i) are nondegenerate projective surfaces of degree 8. The line L := φ(P(OP1 z0 ⊕OP1 z1)) is contained in the closure of B(Xi). Moreover X1 ∩ L = X2 ∩ L as sets, and l(X1 ∩ L)=4andl(X2 ∩ L) = 2. On the ˜ ∩ P1 ˜ ∩ P1 ∈ P1 other hand, l(X1 y)=l(X2 y)=2foreachy . Remark 4.7. In Theorem 4.4, to obtain the inequality dim B(X) ≤ dim Sing X +1 for e ≥ 2, there is an easier argument using Bertini’s theorem as follows. Assume to the contrary that l := dim B(X) ≥ dim Sing X + 2. By (4.3), l ≤ n − 1. For m = N − l +1,let M ⊆ PN be a general m-dimensional linear subspace. By Bertini’s theorem, X := X ∩ M is a smooth projective variety of dimension n−l+1, nondegenerate in M (see [10], (18.10) or [7], (3.5.8)). Moreover the closure of B(X) contains a line L with L ⊆ X,sinceB(X) ⊇ B(X) ∩ M.LetM  ⊆ M be a general (e + 1)-dimensional linear subspace containing L.SinceL ∩ X ⊆ Sm X, by a Bertini-type theorem (see [14], (2.1); [18], (2.1)), X := X ∩ M  is a smooth nondegenerate projective curve in M  with dim B(X) ≥ 1. Since L ∩ B(X)is dense in L (or apply Theorem 4.3), X lies on the 2-plane spanned by L and a general point x ∈ X, and consequently e = 1, a contradiction.

4.8. Proof of Theorem 5. For a general (smooth) point x of X,letΛx be the linear span Z, Tx(X) . Now, according to Segre [20], we will show that

(4.8.1) dim Λx = n +1.

N−1 To this purpose, consider the linear projection πz,X : X \{z}→P from a point z ∈ Z.Sincez ∈ Sm X, πz,X is generically quasi-finite (see (1.1.3)). By the generic smoothness of πz,X, the line x, z meets X at a point y ∈ Sm X distinct from x and z.MoreoverTy(X) ⊆ Tx(X),z (see (1.2.2)). Let Y be an irreducible component of the closure of the set of the points y ∈ x, z ∩X for moving z ∈ Z and fixed x ∈ X.Ify ∈ Y is general, the corresponding point z is also general in Z, and hence,

Ty(Y ) ⊆ Ty(X) ⊆ Tx(X),z ⊇Tx(X).

From this, by considering the projection of Y from Tx(X), we observe that Tx(X),y does not depend on y ∈ Y (see (1.2.1)). Consequently Tx(X),y = Tx(X),Y = Λx, which implies (4.8.1). Let Λ be the intersection of Λx for general points x ∈ X,andsetl := dim Λ. If l = n +1,thenΛ=Λx and hence Λ contains X, which contradicts e ≥ 2. Thus N−l−1 l ≤ n.LetπΛ,X : X \ Λ → P be the linear projection from Λ to a subspace N−l−1 N P of P disjoint from Λ, and let X¯ be the closure of πΛ,X (X \Λ). For general ∈ −1 x X,letXx¯ be the closure of πΛ,X (¯x)over¯x := πΛ(x). We claim that l

for general v ∈ Λ. If l = n, i.e., codim(Λ, Λx)=1,thendim(Tx(X) ∩ Λ) = n − 1, and hence X ⊆ Λ,x , which contradicts e ≥ 2. Thus l

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dim X¯ = n − l − 1, and hence X is the cone over X¯ with vertex Λ, which means Z ⊆ Λ ⊆ Sing X, a contradiction. Therefore dim(Tx(X) ∩ Λ) = l − 1. By (1.2.1), dim X¯ = n − l, and dim Xx¯ = l.Thefactthatl((X \{z}) ∩ z,x ) ≥ 2forz ∈ Z implies that the l-dimensional part of Xx¯ is a hypersurface of Λ,x ,ofdegree≥ 2, not containing Λ. Thus we have (4.8.2). Next we will show that Λ is the closure of Z.IfΛ⊆ X, by (4.8.2), a general point of Λ lies on B(X), and hence Z ⊆ Λ ∩ X ⊆ Sing X by (4.4), a contradiction. Thus Λ ⊆ X.SinceZ is an irreducible component of C(X), if dim Z

l =dimXx¯ ≤ dim Tx¯(X¯), Λ −dim X + dim(Xx¯ ∩ Sing X)+1

=2+dim(Xx¯ ∩ Sing X) ≤ 2 + dim Sing X. This completes the proof of Theorem 5.  4.9. Proof of Corollary 6. If X is smooth, then B(X) is a finite set and C(X)is a finite union of linear subspaces of dimension ≤ 1. The rest follows from Theorem 2.  Here we will show that the inequality of Theorem 5 is sharp, by giving an example of X whose C(X) contains a linear subspace of dimension dim Sing X +2.

Example 4.10. For integers l ≥ 0, n ≥ l+2 and an ≥···≥al+1 >al = ···= a0 = P P E P1 0, let be the projective bundle P1 ( )over , associated with the vector bundle E n O P1 P → P1 = i=0 P1 (ai)on , with projection τ : and the tautological bundle ∗ OP(1). Assume an ≥ 2ifn = l + 2. For a general member X˜ ∈|OP(μ) ⊗ τ OP1 (1)| N with an integerμ ≥ 2, let X be the image of X˜ by the morphism φ : P → P n |O | ⊆ PN (N = n +1+ i=0 ai) defined by P(1) .LetL be the l-dimensional linear ˜ ˜ P l O ⊆ P subspace which is the image φ(L) of the subbundle L = P1 ( i=0 P1 (ai)) . Then (1) L ⊆ X; (2) Sing X is a subset of L, of codimension 2 if l ≥ 1, and Sing X = ∅ if l =0; (3) C(X) contains L \ Sing X. Hence dim C(X) ≥ dim L = l ≥ dim Sing X +2if l ≥ 1 and, dim C(X) ≥ 0and dim Sing X = −1ifl =0.ConsequentlydimC(X) = dim Sing X +2forl ≥ 1. Proof. Note that X \ L is smooth, since φ gives an embedding of P \ L˜ into PN and since X˜ is smooth by the generality of X˜. To see (1) and (2), let us look at X˜ ∩ L˜ and the induced morphism X˜ ∩ L˜ → L.Lets, t be the homogeneous coordinates of 1 P and let zi be the formal basis of OP1 (ai)inE.ThenX˜ is defined as a subscheme of P by  μ0 μn 0 ∗ · ··· ∈ O ⊗ O 1 ≥ G = gμ0···μn z0 zn H ( P(μ) τ P (1)) (μi 0) μ ,··· ,μ ≥0,μ +···+μ =μ 0 n 0 n  n ··· ∈ k for some homogeneous polynomials gμ0 μn [s, t]ofdegree1+ i=l+1 aiμi. ∼ 1 Moreover X˜ ∩ L˜ is defined by G|L˜ in L˜ = P × L.SinceL˜ is defined by zl+1 = ··· = zn =0inP, the degree of G|L˜ with respect to s and t is one, and we may

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write G|L˜ = h1s + h2t for some h1 and h2 ∈ k[z0,...,zl]. Let W be the subscheme of L defined by h1 = h2 = 0. Then codim(W, L)=2ifl ≥ 1, and W = ∅ if l =0, −1 since X˜ is general and hence h1 and h2 are general. This together with φ (L)=L˜ implies that X˜ → X is one-to-one and unramified at every point of L \ W and also X contains L.ThusX \ W ⊆ Sm X. To show (2), we will prove W ⊆ Sing X. For a point x ∈ W , take general −1 ∼ 1 1 pointsx ˜1 =˜ x2 of φ (x) = P so that τ|X˜ : X˜ → P is unramified atx ˜i. P˜n −1 Pn P˜n ∼ P˜n Set i = τ (τ(˜xi)) and i = φ( i )(= i ). To look at the dimension of the Zariski tangent space Θx,X to X at x,considerΘx,X as a subspace of Θx,PN . The space Θx,X contains the image of Θ ˜ (i =1, 2) by the tangent map of φ. x˜i,X ∼ − ⊆ n Then dim Θ ˜ = dim Θx˜i,P 1andΘ P˜n Θ ˜ . Since Θ P˜n = Θx,P and x˜i,X x˜i, i x˜i,X x˜i, i i Pn ∩ Pn N ≥ − − Θx, 1 Θx, 2 =Θx,L as subspaces of Θx,P , we have dim Θx,X 2(n 1) l = n +(n − l − 2). If n ≥ l + 3, this means that x ∈ Sing X.Whenn = l +2,wetake −1 another general pointx ˜3 ∈ φ (x), and the same argument implies that x ∈ Sing X. Now (3) is easy. In fact, for p ∈ P1, X˜ ∩ τ −1(p) is a hypersurface of degree μ ≥ 2 − ∼ in τ 1(p)(= Pn), and hence l(X ∩ x, y ) ≥ 3 for every x ∈ L \ W and general y ∈ X. 

Finally we look at the relation between Sing X and the boundary of C(X).

Theorem 4.11. Let X ⊆ PN be a nondegenerate, projective variety of dimension n and codimension e ≥ 2.LetΛ be an irreducible component of the closure of C(X), of dimension l, which is necessarily linear by Theorem 5. (1) Assume l ≥ 2. (Hence necessarily n ≥ l +1>l≥ 2.) Then dim Sing X ≥ n − 2,ordim(Λ ∩ Sing X) ≥ l − 2. (2) Assume l ≥ 3. (Hence necessarily n ≥ l +1 >l≥ 3.) Then dim(Λ ∩ Sing X) ≥ l − 3.

Proof. (1). Assume dim(Λ ∩ Sing X) ≤ l − 3. We will show dim Sing X ≥ n − 2. By the assumption, there exists a 2-dimensional subspace M of Λ with M ⊆ Sm X,and consequently M ⊆ C(X) by (4.2). Consider the linear projection πM,X : X \ M → N−3 P from M and let X¯ be the closure of πM,X(X \ M). Let x be a general point −1 ∩ \ of X and setx ¯ := πM,X(x). Let Xx¯ be the closure of πM,X(¯x)= M,x (X M) overx ¯.SinceM ⊆ C(X), we have dim Xx¯ = 2 or 3, and hence dim X¯ = n − 2or n − 3. In the latter, X is the cone over X¯ with vertex Λ and hence Λ ⊆ Sing X,a contradiction. Thus dim X¯ = n − 2. Hence dim Tx¯(X¯),M = n +1andTx (X) ⊆  Tx¯(X¯),M for each x ∈ Xx¯ ∩ Sm X (see (1.2.2)). By the theorem of tangencies ([23], (I.1.7)), dim(Xx¯∩Sing X) ≥ dim Xx¯−dim Tx¯(X¯),M +n−1 = 0. This implies that M,x ∩(X\M)∩Sing X = ∅,sinceM ⊆ Sm X and Xx¯ \M = M,x ∩(X\M). By the generality of x ∈ X, Sing X dominates X¯, and consequently dim Sing X ≥ n − 2, as required. (2). Assume dim(Λ ∩ Sing X) ≤ l − 4togetacontradiction.Thereexistsa 3-dimensional subspace M  of Λ with M  ⊆ Sm X, and consequently M  ⊆ C(X).  N−4  Consider the linear projection πM ,X : X \ M → P from M and the closure  −1 ¯ \ ∈ X of πM ,X (X M ). For a general point x X,letXx¯ be the closure of πM ,X (¯x) overx ¯ := πM ,X (x). By the same argument as in (1), dim Xx¯ =3,dimX¯ = n − 3,   and dim(Xx¯ ∩ Sing X) ≥ 1. Since Xx¯ ⊆ M ,x ,wehaveXx¯ ∩ Sing X ∩ M = ∅, which contradicts M  ⊆ Sm X. 

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Remark 4.12. In Example 4.10, by (4.11), we can relax the assumption n ≥ l +2 to n ≥ l+1 if l ≥ 2. Indeed, W ⊇ Sing X without the assumption. Since dim W = l−2, we have dim(Λ∩Sing X) ≥ l−2 by (4.11)(1). Since W is irreducible by the generality of X˜ and hence by the generality of h1 and h2,wehaveW = Sing X =Λ∩ Sing X.

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Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, Yokohama 240-8501, Japan E-mail address: [email protected]

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