M-regularity of \mathbb{Q}-twisted sheaves and its application to linear systems on abelian varieties
aa r X i v : . [ m a t h . AG ] F e b M-REGULARITY OF Q -TWISTED SHEAVES AND ITS APPLICATIONTO LINEAR SYSTEMS ON ABELIAN VARIETIES ATSUSHI ITO
Abstract.
G. Pareschi and M. Popa give criterions for global generations and surjectivityof multiplication maps of global sections of coherent sheaves on abelian varieties in the theoryof M-regularity. In this paper, we generalize some of their criterions via the M-regularity of Q -twisted sheaves. As an application, we show that the M-regularity of a suitable Q -twistedsheaf implies property ( N p ) and jet-ampleness for ample line bundles on abelian varieties. Introduction
Throughout this paper we work over an algebraically closed field K . In [PP03], G. Pareschiand M. Popa introduce the notion of M-regularity as follows:For a coherent sheaf F on an abelian variety X defined over K , set V i ( F ) = { α ∈ b X | h i ( X, F ⊗ P α ) > } , where b X is the dual abelian variety of X and P α is the algebraically trivial line bundle on X corresponding to α ∈ b X . Then F is said to be GV if codim b X V i ( F ) > i for any i >
0. It issaid to be
M-regular if codim b X V i ( F ) > i for any i >
0. It is said to be
IT(0) if V i ( F ) = ∅ for any i > Theorem 1.1 ([PP03, Theorem 6.3], [PP11, Theorem 7.34]) . Let A be an ample line bundleon an abelian variety X and E , F be coherent sheaves on X . (1) If F ⊗ A − is M-regular, then F is globally generated. (2) If F ⊗ A − is M-regular, then the natural map H ( A n ) ⊗ H ( F ) → H ( A n ⊗ F ) issurjective for any n > . (3) If E , F are locally free and E ⊗ A − , F ⊗ A − are M-regular, then the natural map H ( E ) ⊗ H ( F ) → H ( E ⊗ F ) is surjective. Recently Z. Jiang and G. Pareschi [JP20] extend the notions such as GV, M-regular, IT(0)to a Q -twisted sheaf F h xl i , where x ∈ Q and l ∈ Pic X/ Pic X is the class of an ample linebundle L (see § Q -twisted sheaves). In [JP20], the authors also definean invariant 0 < β ( l )
1, which is characterized as β ( l ) < x ⇐⇒ I o h xl i is IT(0)for x ∈ Q , where I o ⊂ O X is the maximal ideal corresponding to the origin o ∈ X .The first purpose of this paper is to generalize Theorem 1.1 to the Q -twisted setting asfollows: Theorem 1.2 (Propositions 3.1, 4.4) . Let L be an ample line bundle, E be a locally free sheaf,and F be a coherent sheaf on an abelian variety X . Take x ∈ Q such that x > β ( l ) . (1) If F h− xl i is M-regular, then F is globally generated. Mathematics Subject Classification.
Key words and phrases.
Abelian variety, M-regularity, Linear system. (2) If x < and F h− x − x l i is M-regular, then the natural map H ( L ) ⊗ H ( F ) → H ( L ⊗F ) is surjective. (3) If there exist rational numbers s, t > such that Eh− sl i , F h− tl i are M-regular and st/ ( s + t ) > β ( l ) , then the natural map H ( E ) ⊗ H ( F ) → H ( E ⊗ F ) is surjective. We note that
F h ml i is M-regular if and only if so is F ⊗ L m for m ∈ Z . Hence Theorem 1.1(1) is nothing but the case A = L, x = 1 of Theorem 1.2 (1). Similarly, Theorem 1.1 (2)follows from the case L = A n , x = 1 /n of Theorem 1.2 (2), and Theorem 1.1 (3) follows fromthe case A = L, s = t = 2 of Theorem 1.2 (3).The second purpose of this paper is to study linear systems on abelian varieties by usingTheorem 1.2 as in [PP04]. For an ample line bundle A on an abelian variety X , • A n is basepoint free if n > • A n is projectively normal if n > • the homogeneous ideal of X embedded by | A n | is generated by quadrics if n > A n satisfies property ( N p ) if n > p + 3 for p >
0. See § N p ). We just note here that ( N )holds for an ample line bundle L if and only if L defines a projectively normal embedding, and( N ) holds if and only if ( N ) holds and the homogeneous ideal of the embedding is generatedby quadrics.Lazarsfeld’s conjecture is proved by Pareschi [Par00] in char( K ) = 0. In [PP04], Pareschiand Popa strengthen Lazarsfeld’s conjecture when A has no base divisor, that is, whencodim X Bs | A | >
2, by using Theorem 1.1:
Theorem 1.3 ([PP04, Theorem 6.2]) . Let p > be an integer such that char( K ) does notdivide p + 1 and p + 2 . Let A be an ample line bundle with no base divisor on an abelianvariety X . (1) If n > p + 2 , then A n satisfies ( N p ) . (2) More generally, if n > ( p + r + 2) / ( r + 1) , then A n satisfies ( N rp ) for r > . Property ( N rp ) for p, r > A n satisfies( N rp ) if n > ( p + r + 3) / ( r + 1) without the assumption on the base divisors of A . See § N rp ). We just note here that ( N p ) is equivalent to ( N p ) and ( N rp ) is aproperty of “being off” by r from ( N p ).On the other hand, the following theorem by Jiang-Pareschi and F. Caucci generalizesLazarsfeld’s conjecture to ample line bundles which is not necessarily multiples of anotherline bundles: Theorem 1.4 ([JP20, Section 8],[Cau20a, Theorem 1.1]) . Let L be an ample line bundle onan abelian variety X and p > . Let I o ⊂ O X be the maximal ideal sheaf corresponding to theorigin o ∈ X . Then (1) I o h xl i is IT(0) for < x ∈ Q , and I o h l i is IT(0) if and only if L is basepoint free. (2) If I o h p +2 l i is IT(0), then L satisfies ( N p ) . Theorem 1.4 gives a quick and characteristic-free proof of Lazarsfeld’s conjecture as follows:If L = A n for some A and n >
1, then I o h p +2 l i = I o h np +2 a i is IT(0) if n/ ( p + 2) > A n satisfies ( N p ) if n > p + 3 by Theorem 1.4 (2). Furthermore, theproof of Theorem 1.4 shows that L satisfies ( N rp ) for r > I o h r +1 p + r +2 l i is IT(0).Applying Theorem 1.2, we obtain the following theorem, which contains Theorem 1.3 andthe case p > Theorem 1.5.
Let L be an ample line bundle on an abelian variety X and p > . (1) If I o h p +2 l i is M-regular, then L satisfies ( N p ) . (2) More generally, if I o h r +1 p + r +2 l i is M-regular, then L satisfies ( N rp ) for r > . -REGULARITY AND LINEAR SYSTEMS ON ABELIAN VARIETIES 3 By [PP04, Remark 3.6], I o h l i is M-regular if and only if L has no base divisor. HenceTheorem 1.5 gives a characteristic-free proof of Theorem 1.3: If A has no base divisor and L = A n , then I o h xa i is M-regular for x > I o h p +2 l i = I o h np +2 a i is M-regular if n/ ( p + 2) >
1. Thus Theorem 1.3 (1) holds in any characteristic. Similarly, Theorem 1.5 (2)recovers Theorem 1.3 (2) in any characteristic.Other than ( N p ), we can study jet-ampleness via M-regularity as in [PP04]. Recall that aline bundle L is called k - jet ample for k > H ( L ) → H ( L ⊗ O X / I p I p · · · I p k +1 )is surjective for any (not necessarily distinct) k + 1 points p , . . . , p k +1 ∈ X . In particular,0-jet ampleness is equivalent to basepoint freeness and 1-jet ampleness is equivalent to veryampleness.For an ample line bundle A on an abelian variety, [BS97a] proves that A n is k -jet ampleif n > k + 2, and the same holds if A has no base divisor and n > k + 1 >
2. In [PP04,Theorem 3.8], the authors generalize this result using the theory of M-regularity. On theother hand, Caucci shows that L is k -jet ample if I o h k +1 l i is IT(0) [Cau20b, Theorem D],which generalizes the first statement of the above result in [BS97a]. The following theoremgeneralizes the result in [BS97a] and improves [Cau20b, Theorem D]. See Proposition 5.1 fora generalization of [PP04, Theorem 3.8]. Theorem 1.6.
Let L be an ample line bundle on an abelian variety X and k > be aninteger. If I o h k +1 l i is M-regular, then L is k -jet ample. Remark 1.7.
We note that the statement of Theorem 1.3 (1) does not hold for p = 0 as wewill see Example 7.1. Hence Theorem 1.5 (1) also fails for p = 0, that is, the M-regularityof I o h l i does not imply the projective normality of L in general. However, the M-regularityof I o h l i implies the very ampleness of L by Theorem 1.6. See § k = 0 as well since the M-regularity of I o h l i , which is equivalent to saythat L has no base divisor, is strictly weaker than the 0-jet ampleness = basepoint freeness.This paper is organized as follows. In §
2, we recall some notation. In §
3, we showTheorem 1.2 (1). In §
4, we show Theorem 1.2 (2), (3). In §
5, we prove Theorem 1.6. We alsogive an alternative proof of a characterization of polarized abelian varieties whose Seshadriconstants are one by M. Nakamaye [Nak96]. In §
6, we prove Theorem 1.5. In §
7, we studyprojective normality.
Acknowledgments.
The author would like to thank Dr. Federico Caucci for useful com-munication and valuable comments. The author was supported by JSPS KAKENHI GrantNumber 17K14162. 2.
Preliminaries
Throughout this paper, X is an abelian variety of dimension g . We denote the origin of X by o X or o ∈ X . Let L be an ample line bundle on X and l ∈ NS( X ) = Pic( X ) / Pic ( X ) bethe class of L in the Neron-Severi group. Then we have an isogeny ϕ l : X → b X := Pic ( X ) , p t ∗ p L ⊗ L − which depends only on the class l , where t p : X → X is the translation by p .For b ∈ Z , we denote the multiplication-by- b isogeny by µ b = µ Xb : X → X, p bp. It holds that µ ∗ b l = b l . Since ϕ l is a group homomorphism, we have ϕ bl = ˆ µ b ◦ ϕ l = ϕ l ◦ µ b ,where ˆ µ b is the multiplication-by- b isogeny on b X .For two line bundles L, L ′ , L ≡ L ′ means that L, L ′ are algebraically equivalent. A. ITO
Properties ( N p ) and ( N rp ) . Let L be a basepoint free ample line bundle L on a pro-jective variety Y and set S L := Sym H ( Y, L ). Take a minimal free resolution of R L := L n > H ( Y, L n ) as an S L -module · · · → E p → · · · → E → E → R L → E i ≃ M j S L ( − a ij ) . (2.1)Then L is said to satisfy property ( N p ) if E = S L and a ij = i + 1 for any 1 i p and any j .For example, ( N ) holds for L if and only if L defines a projectively normal embedding, and( N ) holds if and only if ( N ) holds and the homogeneous ideal of the embedding is generatedby quadrics.More generally, L is said to satisfy property ( N r ) if a j r for any j in (2.1). Inductively, L is said to satisfy property ( N rp ) if L satisfies ( N rp − ) and a pj p +1+ r for any j . For example,( N p ) is equivalent to ( N p ) and L satisfies ( N r ) if and only if H ( L ) ⊗ H ( L n ) → H ( L n +1 )is surjective for n > r + 1. If L is projectively normal, L satisfies ( N r ) if and only if thehomogeneous ideal of X embedded by | L | is generated by homogeneous polynomials of degreeat most r + 2.We note that this equivalence about ( N r ) and the degrees of generators of the homogeneousideal does not hold in general without assuming the projectively normality. For example, thereexists a polarized abelian surface ( X, A ) of type (1 ,
2) over the complex number field C suchthat A has no base divisor, A is very ample, and the homogeneous ideal of X embedded by | A | is generated by quadrics and quartics [Ago17, Remark 3] (see also [Bar87, § A has no base divisor, A satisfies ( N ) by Theorem 1.3. However, the homogeneous ideal of X is not generated by polynomials of degree at most 3.In particular, we need to assume that A is projectively normal in [PP04, Theorem 6.1 (b)].2.2. Generic vanishing, M-regularity and IT(0) of Q -twisted sheaves. Let l ∈ NS( X )be an ample class. For a coherent sheaf F on X and x ∈ Q , a Q -twisted coherent sheaf F h xl i is the equivalence class of the pair ( F , xl ), where the equivalence is defined by( F ⊗ L m , xl ) ∼ ( F , ( x + m ) l )for any line bundle L representing l and any m ∈ Z .As explained in Introduction, a coherent sheaf F on X is said to be GV (resp. M-regular ,resp.
IT(0) ) if codim b X V i ( F ) > i (resp. codim b X V i ( F ) > i , resp. V i ( F ) = ∅ ) for any i > V i ( F ) = { α ∈ b X | h i ( X, F ⊗ P α ) > } . In [JP20], these notions are extended to the Q -twisted setting. A Q -twisted coherent sheaf F h xl i for x = ab with b > µ ∗ b F ⊗ L ab . Thisdefinition does not depend on the representation x = ab nor the choice of L representing l .We note that this is true even in char( K ) > f : Y → X , F h xl i is GV, M-regular, or IT(0) ⇐⇒ so is f ∗ ( F h xl i ) := f ∗ F h xf ∗ l i (2.2)by [Cau20b, Proposition 1.3.3] or arguments in [Cau20a, Remark 3.2].By [JP20, Theorem 5.2], we also have an equivalence F h xl i is GV ⇐⇒ F h ( x + x ′ ) l i is IT(0) for any rational number x ′ > Example 2.1.
For a line bundle B on X , B h xl i is M-regular if and only if B h xl i is IT(0) ifand only if B + xL is ample by [PP08, Example 3.10 (1)]. Hence B h xl i is GV if and only if B + xL is nef. Example 2.2.
For an ample line bundle L on X , I o ⊗ L = I o h l i is GV, and I o ⊗ L is IT(0)if and only if L is basepoint free by Theorem 1.4 (1). By [PP04, Remark 3.6], I o ⊗ L isM-regular if and only if L has no base divisor. -REGULARITY AND LINEAR SYSTEMS ON ABELIAN VARIETIES 5 For a polarized abelian variety (
X, l ), Jiang and Pareschi introduce an invariant β ( l ) ∈ R .It is defined by using cohomological rank functions, which is also introduced by them, but β ( l ) is characterized by the notion IT(0) as follows: Lemma 2.3 ([JP20, Section 8],[Cau20a, Lemma 3.3]) . Let ( X, l ) be a polarized abelian varietyand x ∈ Q . Then β ( l ) < x if and only if I p h xl i is IT(0) for some (and hence for any) p ∈ X . By Lemma 2.3 and (2.3), it also holds that β ( l ) x if and only if I p h xl i is GV for x ∈ Q .We note that Theorem 1.4 is stated by using β ( l ) in [JP20],[Cau20a], but the originalstatement is equivalent to that in Theorem 1.4 by Lemma 2.3. Example 2.4. If β ( l ) < / L satisfies ( N ) by Theorem 1.4 (2). Contrary to Theorem 1.4(1), the converse does not hold in general. For example, let ( X, A ) be a general polarizedabelian variety of type (1 , . . . , , d ) with 3 d g and set L := A . Since h ( A ) = d g , A is not basepoint free, which is equivalent to β ( a ) = 1 by Theorem 1.4 (1). Hence we have β ( l ) = β ( a ) / /
2. On the other hand, L is projectively normal by [Rub98]. Example 2.5. If β ( l ) < / L satisfies not only ( N ) but also ( N ) as in the followingparagraph of Theorem 1.4 (see also the proof of Theorem 1.5 in § X embedded by | L | is generated by quadrics and cubics if β ( l ) < / X is generated by quadrics andcubics under the condition that L is projectively normal, which is weaker than the condition β ( l ) < /
2. At least, the answer is yes if dim X = 2 by [Ago17, Lemma 2.2]. The answer isyes as well when L = A for some ample line bundle A . In fact, A is very ample if and onlyif A has no base divisor by [Ohb87]. Thus if A is projectively normal, then A has no basedivisor and hence A satisfies ( N ) by Theorem 1.3 (2). Therefore the homogeneous ideal of X embedded by | A | is generated by quadrics and cubics. We note that Theorem 1.3 is truein any characteristic by Theorem 1.5, which we will prove in § Proposition 2.6 ([PP11, Proposition 3.1],[PP11, Theorem 3.2],[Cau20a, Proposition 3.4]) . Assume that F and G are coherent sheaves on X , and that one of them is locally free.(i) If F h xl i is IT(0) and Gh yl i is GV, then F h xl i ⊗ Gh yl i := ( F ⊗ G ) h ( x + y ) l i is IT(0).(ii) If F h xl i and Gh yl i are M-regular, then F h xl i ⊗ Gh yl i is M-regular. We also note that for an isogeny f : Y → X and a coherent sheaf F on Y , f ∗ F is GV, M-regular, or IT(0) ⇐⇒ so is F (2.4)since h i ( f ∗ F ⊗ P α ) = h i ( F ⊗ f ∗ P α ) = h i ( F ⊗ P ˆ f ( α ) ) holds for any α ∈ b X , where ˆ f : b X → b Y is the dual isogeny. We see a Q -twisted version of this fact in Lemma 4.6.2.3. Fourier-Mukai transforms.
Let P be the Poincar´e line bundle on X × b X . Let D b ( X )be the bounded derived category of coherent sheaves on X andΦ P = Φ X P : D b ( X ) → D b ( b X )be the Fourier-Mukai functor associated to P . We note that Φ P ( F ) is a locally free sheaf(concentrated in degree 0) for an IT(0) sheaf F . For an isogeny f : Y → X ,ˆ f ∗ ◦ Φ Y P ≃ Φ X P ◦ f ∗ , ˆ f ∗ ◦ Φ X P ≃ Φ Y P ◦ f ∗ (2.5)holds by [Muk81, (3.4)]. A. ITO (Skew) Pontrjagin products.
For coherent sheaves E , F on X , their Pontrjagin prod-uct
E∗F is defined as
E∗F = ( p + p ) ∗ ( p ∗ E ⊗ p ∗ F ) , where p i is the natural projection from X × X to the i -th factor for i = 1 ,
2. By definition,
E∗F = F ∗E holds. Similarly, their skew Pontrjagin product E ˆ ∗F is defined as E ˆ ∗F = p ∗ (( p + p ) ∗ E ⊗ p ∗ F ) . As in [Par00, Remark 1.2], E ˆ ∗F ≃ E∗ ( − X ) ∗ F and F ˆ ∗E ≃ ( − X ) ∗ ( E ˆ ∗F ), where − X = µ X − .We will use the following properties of (skew) Pontrjagin products. For simplicity, weassume locally freeness or IT(0) for some sheaves. In particular, all the objects are sheavesand ⊗ is the usual (non-derived) tensor product in the following proposition. Proposition 2.7 ([Muk81, (3.7),(3.10)],[Par00, Proposition 1.1],[PP04, Proposition 5.2]) . Let L be an ample line bundle, E be a vector bundle and F be a coherent sheaf on X .(i) If E , F are IT(0), then Φ P ( E∗F ) = Φ P ( E ) ⊗ Φ P ( F ) .(ii) Assume that h i (( t ∗ q E ) ⊗ F ) = 0 for any q ∈ X and i > . For p ∈ X , the natural map H ( t ∗ p E ) ⊗ H ( F ) → H (( t ∗ p E ) ⊗ F ) is surjective if and only if E ˆ ∗F is generated byglobal sections at p .(iii) If L ⊗F is IT(0), then L ∗F = L ⊗ ϕ ∗ l Φ P ((( − X ) ∗ F ) ⊗ L ) and L ˆ ∗F = L ⊗ ϕ ∗ l Φ P ( F ⊗ L ) . Criterion for global generations
In this section, we prove Theorem 1.2 (1), which is nothing but (iii) in the following propo-sition.
Proposition 3.1.
Let L be an ample line bundle on an abelian variety X . Let F be a coherentsheaf on X and let x = a/b, y = a ′ /b be rational numbers. Assume that x > β ( l ) .(i) If F ⊗ L − ab is M-regular, I µ − b ( p ) ·F is IT(0) for any p ∈ X , where µ − b ( p ) ⊂ X is thescheme-theoretic fiber over p and I µ − b ( p ) ·F is the image of the natural homomorphism I µ − b ( p ) ⊗ F → F .(ii) If F h yl i is M-regular, ( I p ·F ) h ( x + y ) l i is IT(0) for any p ∈ X .(iii) If F h− xl i is M-regular, F is globally generated.Proof. (i) Fix p ∈ X and consider the exact sequence0 → I µ − b ( p ) ·F ⊗ P α → F ⊗ P α → F ⊗ P α | µ − b ( p ) → α ∈ b X . Since F ⊗ L − ab is M-regular and ab > F is IT(0) by Proposition 2.6. Sincethe support of F ⊗ P α | µ − b ( p ) is zero dimensional, h i ( I µ − b ( p ) ·F ⊗ P α ) = 0 for any i > I µ − b ( p ) ·F is IT(0) if and only if the restriction map H ( F ⊗ P α ) → F ⊗ P α | µ − b ( p ) (3.1)is surjective for any α ∈ b X . Since F ⊗ P α ⊗ L − ab is also M-regular, it suffices to show thesurjectivity of (3.1) for α = o b X .By x = a/b > β ( l ), L ab ⊗ I µ b − ( p ) = L ab ⊗ µ ∗ b I p is GV. Hence V := { α ∈ b X | h ( L ab ⊗ I µ b − ( p ) ⊗ P α ) > } is a proper closed subset of b X .Since F ⊗ L − ab is M-regular, there exists an integer N > α , . . . , α N ∈ b X the natural map N M i = j H ( F ⊗ L − ab ⊗ P α j ) ⊗ P ∨ α j → F ⊗ L − ab is surjective by [PP03, Proposition 2.13]. We take general α j so that − α j V . -REGULARITY AND LINEAR SYSTEMS ON ABELIAN VARIETIES 7 Consider the following diagram: L Nj =1 H ( F ⊗ L − ab ⊗ P α j ) ⊗ H ( L ab ⊗ P ∨ α j ) ⊗ O X / / (cid:15) (cid:15) H ( F ) ⊗ O X (cid:15) (cid:15) L Nj =1 H ( F ⊗ L − ab ⊗ P α j ) ⊗ L ab ⊗ P ∨ α j / / F Since the bottom map is surjective, H ( F ) → F | µ − b ( p ) is surjective if so is N M j =1 H ( F ⊗ L − ab ⊗ P α j ) ⊗ H ( L ab ⊗ P ∨ α j ) → N M j =1 H ( F ⊗ L − ab ⊗ P α j ) ⊗ L ab ⊗ P ∨ α j | µ − b ( p ) . This map is surjective since so is H ( L ab ⊗ P ∨ α j ) → L ab ⊗ P ∨ α j | µ − b ( p ) for any j by − α j V .Hence (i) holds.(ii) If F h yl i is M-regular, so is µ ∗ b F ⊗ L a ′ b by definition. By (i), I µ − b ( p ) · µ ∗ b F ⊗ L a ′ b ⊗ L ab is IT(0) for any p ∈ X . Since µ b is flat, I µ − b ( p ) · µ ∗ b F = µ ∗ b I p · µ ∗ b F = µ ∗ b ( I p ·F ) holds. Hence µ ∗ b ( I p ·F ) ⊗ L a ′ b ⊗ L ab = µ ∗ b ( I p ·F ) ⊗ L ( a + a ′ ) b is IT(0), which means that ( I p ·F ) h ( x + y ) l i isIT(0).(iii) The global generation of F follows from the vanishing h ( I p ·F ) = 0 for any p ∈ X . Henceit suffices to show that I p ·F is IT(0) for any p ∈ X . This follows from (ii) for y = − x . (cid:3) Surjectivity of multiplication maps on global sections
Throughout this section, L is an ample line bundle on an abelian variety X . If L is basepointfree, we can define a vector bundle M L on X by the exact sequence0 → M L → H ( L ) ⊗ O X → L → . (4.1)We note that we do not assume the basepoint freeness of L otherwise stated. The followingproposition is essentially proved in [JP20, Proposition 8.1]: Proposition 4.1 ([JP20, Proposition 8.1]) . Let F be an IT(0) sheaf F on X and y ∈ Q > .Then F h− yl i is GV, M-regular, or IT(0) if and only if so is ϕ ∗ l Φ P ( F ) h y l i .In particular, if L is basepoint free, I o h xl i is GV, M-regular, or IT(0) if and only if so is M L h x − x l i for a rational number < x < . To show Proposition 4.1, we recall some results in [JP20]. For a coherent sheaf F on X , y ∈ Q and i >
0, Jiang and Pareschi define a rational number h i F ( yl ) > Q → Q : y h i F ( yl ), which is called the cohomological rank function of F . See[JP20] for the definition. We note that h i F ( yl ) can be defined in char( K ) > F h x l i is GV (resp. M-regular, resp.IT(0)) if and only if for all i > h i F (( x − t ) l ) = O ( t i ) (resp. = O ( t i +1 ), resp. = 0) for sufficiently small t ∈ Q > .(4.2) Proof of Proposition 4.1.
By [JP20, Proposition 2.3], it holds that h i F ( − yl ) = y g χ ( l ) h iϕ ∗ l Φ P ( F ) (cid:18) y l (cid:19) for any y ∈ Q > and hence h i F (( − y − t ) l ) = ( y + t ) g χ ( l ) h iϕ ∗ l Φ P ( F ) (cid:18) y + t l (cid:19) for any y, t ∈ Q > . Since 1 y + t = 1 y − y ( y + t ) t, A. ITO for a fixed y > h i F (( − y − t ) l ) = O ( t i ), or = O ( t i +1 ), or = 0 for sufficiently small t > h iϕ ∗ l Φ P ( F ) (cid:16) y + t l (cid:17) if and only if so is h iϕ ∗ l Φ P ( F ) (cid:16) ( y − t ) l (cid:17) . By (4.2), the firststatement of this proposition holds.When L is basepoint free, I o ( L ) is IT(0) and ϕ ∗ l Φ P ( I o ( L )) = M L ⊗ L − as shown in theproof of [JP20, Proposition 8.1]. Hence the second statement follows from the first one byconsidering F = I o ( L ) and y = 1 − x . (cid:3) Remark 4.2.
Assume that L is projectively normal. Then H ( L ) ⊗ H ( L ) → H ( L ) issurjective and hence h ( M L ⊗ L ) = 0 by (4.1). Since h i ( M L ⊗ L ⊗ P α ) = 0 for any α ∈ b X and i > M L ⊗ L is GV and hence so is I o h l i by Proposition 4.1, which is equivalentto β ( l ) / β ( l ) n/ ( n + 1) if H ( L ) ⊗ H ( L n ) → H ( L n +1 ) is surjectivefor n > X is simple and h ( L ) > (( n + 1) /n ) g · g !in char( K ) = 0, this multiplication map is surjective by [Blo19, Theorem 1.2], and hence β ( l ) n/ ( n + 1) holds. Lemma 4.3.
Let E be a locally free sheaf and F be a coherent sheaf on X . If E and F areIT(0), then E ˆ ∗F is also IT(0).Proof. Since E ˆ ∗F ≃ ( − X ) ∗ ( F ˆ ∗E ), it suffices to show that F ˆ ∗E is IT(0). For α ∈ b X and i >
0, we have h i (( F ˆ ∗E ) ⊗ P α ) = h i (( F ˆ ∗ P α ) ⊗ E ) = h i ( H ( F ⊗ P α ) ⊗ P ∨ α ⊗ E ) = 0 , where the first equality follows from [PP04, Proposition 5.5 (b)(i)], the second one from[Par00, Remarks 3.5 (c)], and the third one holds since E is IT(0). We note that we need h i ( t ∗ p F ⊗ P α ) = h i ( t ∗ p F ⊗ E ) = 0 for any p ∈ X and i > t ∗ p F and t ∗ p F ⊗ E are IT(0). (cid:3)
Theorem 1.2 (2), (3) follow from the following proposition:
Proposition 4.4.
Let E be a locally free sheaf and F be a coherent sheaf on X . Assume(a) E , F are IT(0), and(b) ϕ ∗ l Φ P ( E ) ⊗ ϕ ∗ l Φ P (( − X ) ∗ F ) (cid:10) x l (cid:11) is M-regular for some rational number x > β ( l ) .Then the natural map H ( t ∗ p E ) ⊗ H ( F ) → H ( t ∗ p E ⊗ F ) is surjective for any p ∈ X .Furthermore, the assumptions (a), (b) are satisfied if (1) E = L , and F h− x − x l i is M-regular for some β ( l ) x < , or (2) there exist rational numbers s, t > such that Eh− sl i and F h− tl i are M-regular and st/ ( s + t ) > β ( l ) .Proof. Assume that (a), (b) are satisfied. To show the surjectivity of H ( t ∗ p E ) ⊗ H ( F ) → H ( t ∗ p E ⊗F ) for any p ∈ X , it suffices to show that E ˆ ∗F is globally generated by Proposition 2.7(ii). By Theorem 1.2 (1), it is enough to show the M-regularity of E ˆ ∗F h− xl i . Since E , F are IT(0), so is E ˆ ∗F by Lemma 4.3. Hence E ˆ ∗F h− xl i is M-regular if and only if so is ϕ ∗ l Φ P ( E ˆ ∗F ) h x l i by Proposition 4.1. SinceΦ P ( E ˆ ∗F ) = Φ P ( E∗ ( − X ) ∗ F ) = Φ P ( E ) ⊗ Φ P (( − X ) ∗ F )by Proposition 2.7 (i), we have ϕ ∗ l Φ P ( E ˆ ∗F ) h x l i = ϕ ∗ l Φ P ( E ) ⊗ ϕ ∗ l Φ P (( − X ) ∗ F ) h x l i , which isM-regular by (b). Hence E ˆ ∗F h− xl i is M-regular and H ( t ∗ p E ) ⊗ H ( F ) → H ( t ∗ p E ⊗ F ) issurjective for any p ∈ X The rest is to show that (a), (b) are satisfied in the cases (1), (2).(1) Since x/ (1 − x ) > F h − x − x l i is M-regular, F is IT(0). Since L is an ample line bundleand hence IT(0), (a) is satisfied. -REGULARITY AND LINEAR SYSTEMS ON ABELIAN VARIETIES 9 Since ϕ ∗ l Φ P ( E ) = ϕ ∗ l Φ P ( L ) = H ( L ) ⊗ L − by [Muk81, Proposition 3.11 (1)], we have ϕ ∗ l Φ P ( E ) ⊗ ϕ ∗ l Φ P (( − X ) ∗ F ) (cid:28) x l (cid:29) = H ( L ) ⊗ ϕ ∗ l Φ P (( − X ) ∗ F ) (cid:28) − xx l (cid:29) . By Proposition 4.1, this is M-regular if and only if so is (( − X ) ∗ F ) h− x − x l i , which is equivalentto the M-regularity of F h− x − x l i by ( − X ) ∗ l = l . Hence (b) is satisfied.(2) Since s, t > Eh− sl i and F h− tl i are M-regular, (a) is satisfied.For (b), set x = st/ ( s + t ) > β ( l ). Then 1 /x = 1 /s + 1 /t holds and ϕ ∗ l Φ P ( E ) ⊗ ϕ ∗ l Φ P (( − X ) ∗ F ) (cid:28) x l (cid:29) = (cid:18) ϕ ∗ l Φ P ( E ) (cid:28) s l (cid:29)(cid:19) ⊗ (cid:18) ϕ ∗ l Φ P (( − X ) ∗ F ) (cid:28) t l (cid:29)(cid:19) . By Proposition 4.1, ϕ ∗ l Φ P ( E ) (cid:10) s l (cid:11) and ϕ ∗ l Φ P (( − X ) ∗ F ) (cid:10) t l (cid:11) are M-regular since so are Eh− sl i and (( − X ) ∗ F ) h− tl i . Hence (b) is satisfied by Proposition 2.6. (cid:3) Remark 4.5.
The M-regularity of L ˆ ∗F h− xl i in the case (1) of Proposition 4.4 can be provedslightly easier as follows:By Proposition 2.7 (iii), we have L ˆ ∗F h− xl i = L ⊗ ϕ ∗ l Φ P ( F ⊗ L ) h− xl i = ϕ ∗ l Φ P ( F ⊗ L ) h (1 − x ) l i . By Proposition 4.1, this is M-regular if and only if so is F ⊗ L h− − x l i = F h− x − x l i .Proposition 4.1 also implies the following lemma, which is a Q -twisted version of (2.4). Lemma 4.6.
Let f : Y → X be an isogeny and F be a coherent sheaf on Y . For x ∈ Q , f ∗ F h xl i is GV, M-regular or IT(0) if and only if so is F h xf ∗ l i .In particular, for a coherent sheaf F on X , an integer m > and x ∈ Q , µ m ∗ F h xl i is GV,M-regular or IT(0) if and only if so is F h m xl i .Proof. For a sufficiently large integer n , F ⊗ f ∗ L n is IT(0). Since f ∗ F h xl i = f ∗ ( F ⊗ f ∗ L n ) h ( x − n ) l i , F h xf ∗ l i = F ⊗ f ∗ L n h ( x − n ) f ∗ l i , we may assume that F is IT(0) and x < F and x with F ⊗ f ∗ L n and x − n respectively.Set y := − x >
0. By Proposition 4.1 and (2.2), f ∗ F h xl i = f ∗ F h− yl i is GV, M-regular orIT(0) if and only if so is ϕ ∗ l Φ X P ( f ∗ F ) h y l i if and only if so is f ∗ ϕ ∗ l Φ X P ( f ∗ F ) h y f ∗ l i . We have f ∗ ϕ ∗ l Φ X P ( f ∗ F ) = f ∗ ϕ ∗ l ˆ f ∗ Φ Y P ( F ) = ϕ ∗ f ∗ l Φ Y P ( F )by (2.5) and ˆ f ◦ ϕ l ◦ f = ϕ f ∗ l . Hence f ∗ F h xl i is GV, M-regular or IT(0) if and only if so is ϕ ∗ f ∗ l Φ Y P ( F ) h y f ∗ l i if and only if so is F h− yf ∗ l i = F h xf ∗ l i by Proposition 4.1.The last statement is nothing but the spacial case f = µ m since µ ∗ m l = m l . (cid:3) On jet ampleness
In this section, we study k -jet ampleness using Proposition 3.1. Throughout this section, L is an ample line bundle on an abelian variety X . First, we prove Theorem 1.6. Proof of Theorem 1.6. If I o h k +1 l i is M-regular, so is I p h k +1 l i for any p ∈ X . Furthermore,we have β ( l ) k +1 . Applying Proposition 3.1 (ii) to x = k +1 and F h yl i = I p h k +1 l i , we seethat I p I p h k +1 l i is IT(0) for any p ∈ X . We note that p can be p . Since IT(0) impliesM-regularity, we can apply Proposition 3.1 (ii) to I p I p h k +1 l i and hence I p I p I p h k +1 l i isIT(0) for any p ∈ X . Repeating this, we obtain that I p k +1 · · · I p I p h k +1 k +1 l i is IT(0), i.e. I p k +1 · · · I p I p ⊗ L is IT(0) for any (not necessarily distinct) p , . . . , p k +1 ∈ X . By [PP04,Lemma 3.3], this is equivalent to the k -jet ampleness of L . (cid:3) In [PP04], the authors introduce an invariant m ( L ), called the M-regularity index , as m ( L ) := max { m > | L ⊗ I p I p · · · I p m is M-regularfor any (not necessarily distinct) m points p , . . . , p m ∈ X } . [PP04, Theorem 3.8] states that if A, L , . . . , L k +1 − m ( A ) are ample line bundles on X and k > m ( A ) >
1, then A ⊗ L ⊗ · · · ⊗ L k +1 − m ( A ) is k -jet ample. In particular, A k +2 − m ( A ) is k -jet ample for k > m ( L ) >
1. The following is a generalization of this result:
Proposition 5.1.
Let n, k , . . . , k n be positive integers and A, L , . . . , L n be ample line bundleson X . If β ( l i ) k i for any i n , then A ⊗ L ⊗ · · · ⊗ L n is k -jet ample, where k = m ( A ) + P ni =1 k i − .Proof. For simplicity, set m = m ( A ). By definition, A ⊗ I p I p · · · I p m is M-regular for any p i ∈ X . By Proposition 3.1 (ii), A ⊗I p I p · · · I p m I p m +1 h k l i is IT(0) for any p m +1 ∈ X . SinceIT(0) implies M-regularity, A ⊗ I p I p · · · I p m I p m +1 I p m +2 h k l i is IT(0) for any p m +2 ∈ X byProposition 3.1 (ii). Repeating this, A ⊗ I p I p · · · I p m + k h k k l i = A ⊗ L ⊗ I p I p · · · I p m + k is IT(0) for any p i . Repeating this argument, A ⊗ L ⊗ · · · ⊗ L n ⊗ I p I p · · · I p m + P ki = A ⊗ L ⊗ · · · ⊗ L n ⊗ I p I p · · · I p k +1 is IT(0) for any p i ∈ X . By [PP04, Lemma 3.3], this isequivalent to the k -jet ampleness of A ⊗ L ⊗ · · · ⊗ L n . (cid:3) We note that A ⊗ L ⊗ · · · ⊗ L n in Proposition 5.1 is a tensor product of two or more ampleline bundles since we assume n >
1. Hence Proposition 5.1 do not contain Theorem 1.6.In the rest of this section, we see some relation between m ( L ), the M-regularity of I o h xl i ,and Seshadri constants. Corollary 5.2. If I o h m l i is M-regular for an integer m > , then m ( L ) > m holds.Proof. If m = 1, this holds from definition. Hence we may assume that m >
2. Then wealready see that I p m · · · I p I p ⊗ L is IT(0) for any p , . . . , p m ∈ X in the proof of Theorem 1.6.Hence m ( L ) > m holds. (cid:3) In general, the converse of Corollary 5.2 does not hold, that is, I o h m ( L ) l i is not M-regularin general. For example, let ( X, L ) be a general polarized abelian surface of type (1 , m ( L ) = 2. On the other hand, I o h l i is not M-regular byTheorem 1.6 since L is not very ample.In fact, I o h m ( L ) l i is not even GV, equivalently β ( l ) /m ( L ) does not hold in general.For example, let ( X, L ) be a polarized abelian surface of type (1 , d ) with Picard number one.Then L is k -jet ample if and only if d > ( k + 2) by [BS97b]. Hence we have m ( L ) > k + 1 if d > ( k + 2) by [PP04, Proposition 3.5]. Thus m ( L ) > ⌊√ d − ⌋ −
1. On the other hand, β ( l ) > / √ d holds by [Ito20a, Lemma 3.4]. Hence 1 /m ( L ), which is bounded from above byroughly 1 / √ d , is strictly smaller than β ( l ) for d ≫ I o h dim Xm ( L ) l i is IT(0) at least in char( K ) = 0. To see this, recallthat the Seshadri constant ε ( L ) of L is defined as ε ( X, L ) = ε ( L ) := max { t > | π ∗ L − tE is nef } , where π : ˜ X → X is the blow-up at o ∈ X and E ⊂ ˜ X is the exceptional divisor. By [PP11,Theorem 7.4], ε ( L ) > m ( L ) holds. On the other hand, β ( l ) · ε ( L ) > ε ( L ) > x − holds if I o h xl i is IT(0). The following is a refinementof these results: Proposition 5.3.
Assume char( K ) = 0 .(i) If I mo h xl i is M-regular for an integer m > and x ∈ Q > , then ε ( L ) > x − · m holds.(ii) If I mo h xl i is GV for an integer m > and x ∈ Q > , then ε ( L ) > x − · m holds.(iii) ε ( L ) > m ( L ) holds.(iv) If m ( L ) > , I o h gm ( L ) l i is IT(0), where g = dim X .Proof. (i) Since ε ( L ) >
0, this is clear if m = 0. Hence we may assume m >
1. Let π : ˜ X → X be the blow-up at o ∈ X and E ⊂ ˜ X be the exceptional divisor. What we need to show is theampleness of π ∗ L − x − · mE , i.e. the ampleness of π ∗ xL − mE . -REGULARITY AND LINEAR SYSTEMS ON ABELIAN VARIETIES 11 Let x = a/b . Then µ ∗ b I mo ⊗ L ab is M-regular. Hence µ ∗ b I mo ⊗ L ab is ample, that is, thetautological line bundle O (1) of P X ( µ ∗ b I mo ⊗ L ab ) := Proj X M n > Sym n ( µ ∗ b I mo ⊗ L ab ) ! is ample by [PP03, Proposition 2.13] and [Deb06, Corollary 3.2]. SinceProj X M n > Sym n ( µ ∗ b I mo ⊗ L ab ) ! ≃ Proj X M n > Sym n ( µ ∗ b I mo ) ! ≃ Proj X M n > Sym n ( µ ∗ b I o ) ! , P X ( µ ∗ b I mo ⊗ L ab ) → X is isomorphic to the blow-up π ′ : ˜ X ′ → X along the ideal µ ∗ b I o .Under this isomorphism, the tautological line bundle O (1) on P X ( µ ∗ b I mo ⊗ L ab ) correspondsto O ( − mE ′ ) ⊗ π ′∗ L ab , where E ′ ⊂ ˜ X ′ is the exceptional divisor of π ′ . Hence π ′∗ abL − mE ′ isample.Since π ′ is the blow-up along µ ∗ b I o , there exists a morphism ˜ µ b : ˜ X ′ → ˜ X such that π ◦ ˜ µ b = µ b ◦ π ′ and ˜ µ ∗ b O ( − E ) = O ( − E ′ ). Hence we have π ′∗ abL − mE ′ ≡ ˜ µ ∗ b ( π ∗ xL − mE )and the ampleness of π ∗ xL − mE follows from that of π ′∗ abL − mE ′ .(ii) If I mo h xl i is GV, then I mo h ( x + x ′ ) l i is IT(0) for any rational number x ′ >
0. Hence ε ( L ) > ( x + x ′ ) − · m holds by (1). By x ′ →
0, we have ε ( L ) > x − · m .(iii) By definition, I m ( L ) o h l i is M-regular and hence ε ( L ) > m ( L ) holds by (1).(iv) By [Ito20a, Proposition 3.1] and (3), we have β ( l ) g · ε ( L ) − < g · m ( L ) − . Hence (4)follows from Lemma 2.3. (cid:3) Example 5.4. (1) By definition, I o h l i is GV (resp. M-regular) if and only if the codimensionof { p ∈ X | h ( X, I p ⊗ L ) > } in X is at least one (resp. at least two). Since h ( X, I p ⊗ L ) = 0holds if and only if the rational map f | L | defined by | L | is an immersion at p , we have ε ( L ) > f | L | is generically finite, and ε ( L ) > f | L | is an immersion outside a codimension twosubset by Proposition 5.3 (i), (ii) in char( K ) = 0.(2) In [Nak96, Theorem 1.1, Lemma 2.6], Nakamaye proves that for a polarized abelian variety( X, L ) in char( K ) = 0, ε ( L ) > X, L ) ≃ ( E, L E ) × ( X ′ , L ′ ) := ( E × X ′ , p ∗ E L E ⊗ p ∗ X ′ L ′ )(5.1)for a principally polarized elliptic curve ( E, L E ) and a polarized abelian variety ( X ′ , L ′ ). Wecan recover this result from Proposition 5.3 as follows:We show this by the induction on g = dim X . If g = 1, this is clear since ε ( X, L ) = deg( L )by definition. Assume g > X, L ) ≃ ( E, L E ) × ( X ′ , L ′ ) as (5.1), then we have ε ( X, L ) = min { ε ( E, L E ) , ε ( X ′ , L ′ ) } (see [MR15, Proposition 3.4] for example). Since ε ( E, L E ) = deg L E = 1 and ε ( X ′ , L ′ ) > ε ( X, L ) = 1.Conversely, assume ε ( L )
1. Then m ( L ) = 0 by Proposition 5.3 (ii) and hence L has abase divisor.Assume that ( X, L ) is indecomposable, that is, (
X, L ) is not isomorphic to any productof polarized abelian varieties of positive dimensions. Then L is an indecomposable principalpolarization by [BL04, Theorem 4.3.1] since L has a base divisor. Replacing L with analgebraically equivalent line bundle, we may assume that L is symmetric. Then the morphism f | L | defined by | L | is the natural morphism π : X → X/ h− X i ⊂ P g − to the Kummervariety X/ h− X i by [BL04, Theorem 4.8.1]. Since π is an immersion outside two-torsionpoints in X and g >
2, we have 2 ε ( L ) = ε ( L ) > ε ( L ) X, L ) is decomposable, that is, (
X, L ) is isomorphic to a product ( X , L ) × ( X , L )with dim X i >
1. Then ε ( X, L ) = min { ε ( X , L ) , ε ( X , L ) } and hence we may assume ε ( X , L ) = ε ( X, L )
1. Since ε ( X , L ) > ε ( X , L ) = 1 holds and ( X , L ) has a decomposition as (5.1). Hence ε ( X, L ) = ε ( X , L ) = 1 and ( X, L ) has adecomposition as (5.1). 6.
On property ( N p )Throughout this section, L is an ample line bundle on an abelian variety X . If I o h xl i isM-regular for some rational number 0 < x <
1, then L is basepoint free by Theorem 1.4 (1).Furthermore, M L h x − x l i is M-regular by Proposition 4.1. Hence M ⊗ mL h mx − x l i is also M-regularfor any m > M ⊗ mL h mx − x l i is IT(0) if m > Proposition 6.1.
Assume that I o h xl i is M-regular for a rational number < x < . Then M ⊗ mL h mx − x l i is IT(0) for any integer m > .Proof. Let 1 − x = ab for integers b > a >
0. Then M L (cid:28) x − x l (cid:29) = M L (cid:28) b − aa l (cid:29) and µ ∗ a M L ⊗ L a ( b − a ) are M-regular(6.1)by Proposition 4.1. If M ⊗ L h x − x l i is IT(0), so is M ⊗ mL (cid:28) mx − x l (cid:29) = M ⊗ L (cid:28) x − x l (cid:29) ⊗ (cid:18) M L (cid:28) x − x l (cid:29)(cid:19) ⊗ m − for m > M ⊗ L h x − x l i is IT(0) and M L h x − x l i is GV. Hence it sufficesto show the case m = 2.By definition and (2.4), M ⊗ L h x − x l i = M ⊗ L h b − a ) a l i is IT(0) if and only if so is µ ∗ a M ⊗ L ⊗ L a ( b − a ) if and only if so is µ a ∗ ( µ ∗ a M ⊗ L ⊗ L a ( b − a ) ) = M ⊗ L ⊗ µ a ∗ L a ( b − a ) . Consider the exactsequence0 → M ⊗ L ⊗ µ a ∗ L a ( b − a ) → H ( L ) ⊗ M L ⊗ µ a ∗ L a ( b − a ) → L ⊗ M L ⊗ µ a ∗ L a ( b − a ) → M L ⊗ µ a ∗ L a ( b − a ) with (4.1). Since µ ∗ a M L ⊗ L a ( b − a ) is M-regular by (6.1), µ ∗ a M L ⊗ L a ( b − a ) is IT(0). Hence M L ⊗ µ a ∗ L a ( b − a ) = µ a ∗ ( µ ∗ a M L ⊗ L a ( b − a ) ) and L ⊗ M L ⊗ µ a ∗ L a ( b − a ) are IT(0) by (2.4) and Proposition 2.6. Thus h i ( M ⊗ L ⊗ µ a ∗ L a ( b − a ) ⊗ P α ) = 0 forany i > α ∈ b X , and M ⊗ L ⊗ µ a ∗ L a ( b − a ) is IT(0) if and only if the natural map H ( L ) ⊗ H ( M L ⊗ µ a ∗ L a ( b − a ) ⊗ P α ) → H ( L ⊗ M L ⊗ µ a ∗ L a ( b − a ) ⊗ P α )is surjective for any α ∈ b X . By Theorem 1.2 (2), this map is surjective if M L ⊗ µ a ∗ L a ( b − a ) ⊗ P α (cid:28) − x − x l (cid:29) = M L ⊗ µ a ∗ L a ( b − a ) ⊗ P α (cid:28) − b − aa l (cid:29) = µ a ∗ ( µ ∗ a M L ⊗ L a ( b − a ) ⊗ µ ∗ a P α ) (cid:28) − b − aa l (cid:29) is M-regular. By Lemma 4.6, this is equivalent to the M-regularity of µ ∗ a M L ⊗ L a ( b − a ) ⊗ µ ∗ a P α (cid:28) − a b − aa l (cid:29) = µ ∗ a M L h a ( b − a ) l i , which is nothing but (6.1). (cid:3) Proposition 6.2.
Assume that I o h xl i is M-regular for a rational number < x < . Then h i ( M ⊗ mL ⊗ B ) = 0 for an integer m > and a line bundle B on X if B − mx − x L is ample or m > and B − mx − x L is nef.In particular, if I o h p +2 l i is M-regular for an integer p > , then h i ( M ⊗ mL ⊗ L h ) = 0 forpositive integers m, h if h > m/ ( p + 1) or m > and h > m/ ( p + 1) . -REGULARITY AND LINEAR SYSTEMS ON ABELIAN VARIETIES 13 Proof.
The proof is essentially the same as that of [Cau20a, Proposition 3.5] other than weuse Proposition 6.1 when m > Q -twisted sheaf, M ⊗ mL ⊗ B is written as (cid:18) M L (cid:28) x − x l (cid:29)(cid:19) ⊗ m ⊗ B (cid:28) − mx − x l (cid:29) . By Proposition 4.1, M L h x − x l i is M-regular.If B − mx − x L is ample, B h− mx − x l i is IT(0) by Example 2.1. Thus M ⊗ mL ⊗ B is IT(0) byProposition 2.6 and hence the vanishing h i ( M ⊗ mL ⊗ B ) = 0 holds.When m > B − mx − x L is nef, (cid:16) M L h x − x l i (cid:17) ⊗ m is IT(0) by Proposition 6.1 and B h− mx − x l i is GV by Example 2.1. Hence M ⊗ mL ⊗ B is IT(0) by Proposition 2.6 and the vanishing h i ( M ⊗ mL ⊗ B ) = 0 holds.The last statement is just a special case when x = 1 / ( p + 2) and B = L h . (cid:3) Proof of Theorem 1.5. (1) Since β ( l ) / ( p + 2) < L is basepoint free. By [Cau20a,Proposition 4.1], ( N p ) holds for L if h ( M ⊗ p +1 L ⊗ L h ) = 0 for any h > K ) >
0. Since p + 1 >
2, here we use the assumption p >
1, and h > p + 1) / ( p + 1), this vanishingfollows from Proposition 6.2.For (2), ( N rp ) holds for L if h ( M ⊗ p +1 L ⊗ L h ) = 0 for any h > r + 1 by [PP04, Proposition6.3] when char( K ) does not divide p + 1 and by [Cau20a, Section 4] in any characteristic. Wecan prove (2) similarly. (cid:3) We give an example which does not follow from Theorem 1.3, Theorem 1.4.
Example 6.3.
Let (
X, L ) be a general polarized abelian variety of dimension g > , , . . . ,
3) in char( K ) = 0. Then L n satisfies ( N p ) if n > p +2)3 . To see this, it suffices toshow that I o h l i is M-regular by Theorem 1.5. In fact, we can show that I o h l i is M-regularbut not IT(0) as follows:Take an isogeny π : Y → X with kernel π − ( o ) ≃ Z / Z such that there exists a principallypolarization Θ such that π ∗ L ≡ I o h l i is M-regular or IT(0) if and only if so is π ∗ I o h π ∗ l i = I π − ( o ) h θ i by (2.2). Hence it suffices to show that I π − ( o ) ⊗
2Θ is M-regularbut not IT(0). We may assume that Θ is symmetric. Since h i ( I π − ( o )+ p ⊗ p ∈ Y and i >
2, it suffices to show that V := { p ∈ Y | h ( I π − ( o )+ p ⊗ > } ⊂ Y is not empty and the codimension is greater than 1.Since ( X, L ) is general, ( Y, Θ) is indecomposable, i.e. ( Y, Θ) is not a product of smallerdimensional principally polarized abelian varieties. Hence | | gives a double cover f : Y → Y / ( − Y ) ⊂ P g − . Thus p ∈ Y is contained in V if and only if the restriction map H ( Y, H ( P g − , O (1)) → | π − ( o )+ p is not surjective if and only if f ( π − ( o ) + p ) is contained in a line in P g − . Hence we have V = { p ∈ Y | f ( π − ( o ) + p ) is contained in a line } . Let ε ∈ π − ( o ) ≃ Z / Z be a generator. If 2 p = ε for p ∈ Y , then f ( p ) = f ( − p ) = f ( − ε + p )holds and hence f ( π − ( o )+ p ) = { f ( p ) , f ( ε + p ) , f ( − ε + p ) } is contained in a line. Thus µ − ( ε )is contained in V . By a similar argument, we have µ − ( π − ( o )) = µ − ( { o Y , ε, − ε } ) ⊂ V . (6.2)In particular, V is not empty and hence I o h l i is not IT(0).By [Wel84, Theorem (0.5)], V is at most one dimensional. If g >
3, the codimension of V ⊂ Y is at least g − > I o h l i is M-regular. When g = 2, we need to see dim V = 0 for the M-regularity of I o h l i . Assume dim V > Y is the jacobian J ( C ) of a smooth curve C of genus two and µ ( V ) ⊂ Y = J ( C ) is atheta divisor, i.e. µ ( V ) is the image of C by a ∈ C
7→ O C ( a ) ⊗ Q ∈ J ( C ) for a line bundle Q of degree − C by [Wel84, Theorem (0.5)]. On the other hand, π − ( o ) = { o Y , ε, − ε } iscontained in µ ( V ) by (6.2). Hence there exist a, b, c ∈ C such that O C ( a ) ⊗ Q = o Y , O C ( b ) ⊗ Q = ε, O C ( c ) ⊗ Q = − ε. Thus ε = O C ( b − a ) , − ε = O C ( c − a ) and hence O C ( b + c ) ≃ O C (2 a ). Since b = a, c = a by ε = o Y , O C (2 a ) is basepoint free of degree two on a curve C of genus two. Thus O C (2 a ) islinearly equivalent to the canonical line bundle ω C .Furthermore, we also have O C ( c + a ) ≃ O C (2 b ) by O C ( b + c ) ≃ O C (2 a ) and o Y = 3 ε = O C (3 b − a ). Thus O C (2 b ) is also basepoint free of degree two, and hence O C (2 b ) ≃ ω C . Then O C (2 b ) ≃ ω C ≃ O C (2 a ) ≃ O C ( b + c ) , which implies b = c . This contradicts ε = O C ( b − a ) = − ε = O C ( c − a ). Hence V cannot beone dimensional and we have the the M-regularity of I o h l i . Remark 6.4. (1) In [PP04, Conjecture 6.4], it is conjectured that if A is ample, m ( A ) > m ,and p > m , then A n satisfies ( N p ) for any n > p + 3 − m . The case m = 0 is nothing butLazarsfeld’s conjecture and the case m = 1 is nothing but Theorem 1.3. If we want to applyTheorem 1.5, we need to show the M-regularity of I o h p +3 − mp +2 a i . Since m ( A ) > m and p > m ,it is enough to see the M-regularity of I o h m ( A )+2 a i . However, the author does not know thisholds or not in general.(2) Let ( X, L ) be a general polarized abelian variety of dimension g and of type (1 , . . . , , d ).In [Ito20b], the author shows that for an integer m > I o h m l i is GV if and only if d > m g and IT(0) if d > m g + m g − + · · · + m + 1.On the other hand, L has no base divisor if d > g , and L is very ample if d > g by[DHS94, Remark 3, Corollary 25], and L satisfies ( N ) if d > when g = 2 by [GP98].Hence it might be natural to guess that I o h m l i is M-regular if d > m g . If this is true, we canrecover the above results in [DHS94], [GP98] from Theorems 1.5, 1.6.We also note that I o h m l i is not M-regular if d m g . More generally, I o h xl i is not M-regularif x / g p χ ( l ) for any polarized abelian variety ( X, L ). In fact, we have an inequality h I o ( yl ) > h O X / I o ( yl ) − h O X ( yl ) = 1 − χ ( l ) y g for y ∈ Q > by the exact sequence 0 → I o → O X → O X / I o → h I o (( x − t ) l ) = O ( t ) for sufficiently small t ∈ Q > if x / g p χ ( l ) and hence I o h xl i is not M-regular by (4.2).7. On Projective normality
Contrary to Lazarsfeld’s conjecture, the statement of Theorem 1.3 does not hold for p = 0,that is, A might not be projectively normal even if A has no base divisor as in the followingexample. Hence the statement of Theorem 1.5 does not hold for p = 0 as well. Equivalently,the M-regularity of I o h l i does not imply the projective normality of L in general. Example 7.1. If A is a symmetric ample line bundle on an abelian variety X , A is projec-tively normal if and only if the origin o ∈ X is not contained in the base locus of | A ⊗ P | forany line bundle P on X with P ≃ O X by [Ohb88] in char( K ) = 2 and [Ohb96] in char( K ) > A might not satisfy ( N ) even if A has no base divisor. More explicitly, if ( X, A ) isa general polarized abelian surface of type (1 , A has no base divisor but A is notprojectively normal by [Ohb93, Lemma 6].In [PP04, § K ) = 2 using the theory of M-regularity. For general L , which is not necessarily writtenas A , we can show the following lemma: -REGULARITY AND LINEAR SYSTEMS ON ABELIAN VARIETIES 15 Lemma 7.2.
Assume char( K ) = 2 and let L be an ample line bundle on an abelian variety X . Then the following are equivalent:(i) L is projectively normal,(ii) L ⊗ µ ∗ ( L − ) is globally generated at the origin o ∈ X ,(iii) Let X := µ − ( o ) be the set of two torsion points in X and f : X → P N be themorphism defined by | µ ∗ L ⊗ L − | . Then f ( X ) ⊂ P N spans a linear subspace ofdimension X − g − .Proof. (ii) ⇔ (iii): We note that the line bundle µ ∗ L ⊗ L − is basepoint free since µ ∗ L ⊗ L − ≡ L . Since L ⊗ µ ∗ ( L − ) = µ ∗ ( µ ∗ L ⊗ L − ) by projection formula, the natural map H ( L ⊗ µ ∗ ( L − )) → L ⊗ µ ∗ ( L − ) ⊗ O X / I o (7.1)can be identified with the natural map H ( µ ∗ L ⊗ L − ) → µ ∗ L ⊗ L − ⊗ O X /µ ∗ I o = µ ∗ L ⊗ L − ⊗ O X / I X . (7.2)Since (ii) and (iii) are equivalent to the surjectivity of (7.1) and (7.2) respectively, we havethe equivalence (ii) ⇔ (iii).(i) ⇔ (ii): By [Iye03, Proposition 2.1], L is projectively normal if and only if the natural map H ( L ) ⊗ H ( L ) → H ( L ) is surjective. We note that the proof of [Iye03, Proposition 2.1]works in any characteristic. Hence L is projectively normal if and only if L ˆ ∗ L = L ⊗ ϕ ∗ l Φ P ( L )is globally generated at o by Proposition 2.7.For simplicity, set E = ϕ ∗ l Φ P ( L ) and F = µ ∗ ( L − ). Since ϕ l ◦ µ = ϕ l , we have µ ∗ µ ∗ E = µ ∗ µ ∗ ϕ ∗ l Φ P ( L ) = µ ∗ ϕ ∗ l Φ P ( L ) = µ ∗ ( L − ) ⊕ h ( L ) = F ⊕ h ( L ) , where the third equality follows from [Muk81, Proposition 3.11 (1)]. Since we assume char( K ) =2, E is a direct summand of µ ∗ µ ∗ E = F ⊕ h ( L ) . Hence L ˆ ∗ L = L ⊗ E is globally generated at o if so is L ⊗ F = L ⊗ µ ∗ ( L − ), which shows (ii) ⇒ (i).On the other hand, we have ϕ l ∗ ϕ l ∗ F = ϕ l ∗ ϕ l ∗ µ ∗ ( L − ) = ϕ l ∗ ϕ l ∗ ( L − ) = ϕ l ∗ (Φ P ( L )) ⊕ h ( L ) = E ⊕ h ( L ) , where the third equality follows from [Muk81, Proposition 3.11 (2)]. Since the natural homo-morphism ϕ l ∗ ϕ l ∗ F → F is surjective, L ⊗ F = L ⊗ µ ∗ ( L − ) is globally generated at o if sois L ˆ ∗ L = L ⊗ E , which shows (i) ⇒ (ii). (cid:3) Remark 7.3. (1) If L = A for a symmetric A , we have µ ∗ A = A and hence L ⊗ µ ∗ ( L − ) = µ ∗ ( µ ∗ L ⊗ L − ) = µ ∗ ( µ ∗ A ⊗ A − ) = µ ∗ µ ∗ A = M P A ⊗ P in char( K ) = 2, where we take the direct sum of all P with P ≃ O X . Thus Lemma 7.2recovers Ohbuchi’s result when char( K ) = 2.(2) By Theorem 1.4, Remark 4.2, L is projectively normal if β ( l ) < / β ( l ) > /
2. Hence the projective normality of L is determined by β ( l ) when β ( l ) = 1 /
2. It might be interesting to find examples with β ( l ) = 1 / L = A . We note thatwe cannot use Theorem 1.2 (1) to show the globally generation of L ⊗ µ ∗ ( L − ) at o when β ( l ) = 1 /
2. In fact, L ⊗ µ ∗ ( L − ) (cid:28) − l (cid:29) = µ ∗ ( µ ∗ L ⊗ L − ) (cid:28) − l (cid:29) is not M-regular by Lemma 4.6 since µ ∗ L ⊗ L − h− l i = L h− l i is not M-regular.(3) Finally, we summarize relations between projective normality and related notions:(a) I o h l i is IT(0), i.e. β ( l ) < ,(b) I o h l i is M-regular,(c) I o h l i is GV, i.e. β ( l ) , (d) L is projectively normal,(e) L is very ample.For these five notions, we have the following relations: is projectively normal,is very ample.For these five notions, we have the following relations:( a ) IT(0) ( b ) M-reg ( c ) GV( d ) p.n. ( e ) v.a. ×× ××× × × In fact, (a) ⇒ (b) ⇒ (c) and (d) ⇒ (e) hold by definition. It is easy to see that the conversesof these implications do not hold in general by considering suitable L = A .(a) ⇒ (d), (b) ⇒ (e), and (d) ⇒ (c) follow from Theorem 1.4, Theorem 1.6, and Remark 4.2,respectivelyOn the other hand, (b) ⇒ (d) and (d) ⇒ (a) do not hold in general by Example 7.1 andExample 2.4, respectively. (c) ⇒ (e) does not hold in general since L = A for a principalpolarization A satisfies (c) but is not very ample. (e) ⇒ (c) also does not hold in generalsince a general polarized abelian 4-fold ( X, L ) of type (1 , , ,
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