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(N £ \o 43 3 A FROBENIUS VARIANT OF SESHADRI CONSTANTS MIRCEA MUSTATA AND KARL SCHWEDE Abstract. We define and study a version of Seshadri constant for ample line bundles in positive characteristic. We prove that lower bounds for this constant imply the global generation or very ampleness of the corresponding adjoint line bundle. As a consequence, we deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic. o <^ 1. Introduction Let L be an ample line bundle on an n- dimensional projective variety X (denned over an algebraically closed field k of positive characteristic). The Seshadri constant e(L;x) of L at a smooth point x G X measures the local positivity of L at x. Introduced by Demailly twenty years ago in [Dcm], it has generated a lot of interest: see, for example [Laz] and [B+] for some of the results and open problems involving this invariant. If ix : B1 X (X) —?• X is the blow-up of X at x, with exceptional divisor E, then the Seshadri constant e(L; x) is defined as s(L,x) = sup{£ > | T*(L)(-tE) is nef}. fv; We will be especially interested in an equivalent description in terms of separation of jets. Recall that one says that a power L m separates £-jets at x if the restriction map £+1^ H°(X, L m ) -> H°(X, L m ® O x /m is surjective, where m x is the ideal defining x. If s(L m ; x) is the largest £ such that V X H separates t-jets at x, then s(L m ;x) s(L m ;x) e{L; x) = hm = sup m->oo rn m >i m Part of the interest in Seshadri constants comes from its connection to statements about the base-point freeness or very ampleness of adjoint line bundles. If the ground field has characteristic zero and X is smooth, then it is an easy consequence of the Kawamata- Viehweg vanishing theorem that if e(L; x) > n, then ux ® L is globally generated at x, and if e(L; x) > In for every i6l, then ux ® L is very ample. One of the main results in our paper says that the same implications hold also in positive characteristic. 2010 Mathematics Subject Classification. 14C20, 13A35. Key words and phrases. Seshadri constant, Frobenius. The first author was partially supported by NSF research grant no: 1068190 and by a Packard Fellow- ship. The second author was partially support by NSF research grant no: 1064485. 2 M. MUSTATA AND K. SCHWEDE We prove this by studying another version of Seshadri constant in positive charac- teristic, which is designed to take advantage of the Frobenius morphism. Suppose that the ground field has characteristic p > 0. For a positive integer e and a smooth point i6l, we say that a power L m separates p e -Frobenius jets at x if the restriction map H°{X, L m ) -)• H°{X, L m <8> O x /mf ] ) is surjective, where mj is the ideal locally generated by the p e -powers of the generators of m x . We denote by sp(L m ; x) the largest e such that L m separates p e -Frobenius jets at x. With this notation, the Frobenius- Seshadri constant is defined as „s F {L m ;x) _ y pS F (L m ;x) _ ^ Ef(L; x) := sup = limsup rri>l TTl m— >oo Wl The following inequalities between the two versions of Seshadri constants follow easily from definition ^i^l< £F (L;x)<e(L;x), n and we give examples when either of the two bounds is achieved. One can further compare the two invariants in the case of torus-invariant points on smooth toric varieties, when both of them can be described in terms of the polytope P attached to the line bundle (and suitably normalized). In this case the usual Seshadri constant is obtained by comparing P with the simplex {u = (m 1; . . . ,u n ) G R> | U\ + . . . + u n < 1}, while the Frobenius Seshadri constant is obtained by comparing P with the cube [0, l] n (see Theorem 4.2 below for the precise statement). The Frobenius- Seshadri constant satisfies some of the basic properties of the usual Seshadri constant, though in some cases the proofs are a bit more subtle. For example, we show that £p(L; x) only depends on the numerical equivalence class of L and £p(L m ; x) = m ■ Ep{L, x) for every m > 1. The following theorem is our main result (see Theorem 3.1 below for these statements and other related ones). Theorem. Let L be an ample line bundle on the smooth projective variety X, defined over a field of positive characteristic. i) If £p{L\ x) > 1, then ujx <8> L is globally generated at x. ii) If Ef(L; x) > 2 for every i6l, then ux <8> L is very ample Note that since ep(L;x) > e( — , the above theorem implies the positive characteristic version of the facts we mentioned above. Namely, if e(L; x) > n, then ux ® L is globally generated at x, and if e(L; x) > 2n at every i6l, then ux <8> L is very ample. In light of the above theorem, it would be interesting to obtain lower bounds for either of the two versions of Seshadri constants at very general points of X. Recall that over an uncountable field of characteristic zero, it is expected that e(L; x) > 1 if x G X is a very general point. This is known if X is a surface [EL]. When n > 3, it is shown in [EKL] that e(L; x) > - when x is very general in X. However, the proofs of both these results make essential use of the characteristic zero assumption. The paper is structured as follows. In the next section we introduce the Frobenius- Seshadri constant and prove some basic properties. In fact, we define a slightly more A FROBENIUS VARIANT OF SESHADRI CONSTANTS 3 general version, in which we consider a finite set of smooth points. In the third section we prove our main result relating lower bounds on Seshadri constants to global generation and very ampleness properties of the corresponding adjoint line bundles. In the last section we describe the Frobenius- Seshadri constant at a torus-fixed point on a smooth toric variety. Acknowledgment. This project originated in discussions held during the AIM workshop "Relating test ideals and multiplier ideals" . We are indebted to AIM for organizing this event. We would also like to thank Bhargav Bhatt and Rob Lazarsfeld for discussions related to this work. 2. Definition and basic properties We begin by recalling the definition and some basic properties of the Seshadri con- stant of an ample line bundle. For details and a nice introduction to this topic, we refer to [Laz, §5]. Let X be an n-dimensional projective variety (assumed irreducible and reduced) over an algebraically closed field k. Consider an ample line bundle L on X and x G X a (closed) point. If it: Y — > X is the blow-up of X at x, with exceptional divisor E, then the Seshadri constant of L at x is e(L; x) := sup{a G R>o | 7r*(L)(— aE) is nef}. (2.1) It is easy to see that since L is ample, we have e(L; x) > 0. Furthermore, one has It | dim ( y )\ e(L;x) = inf [My ' Yix mult x (F)' where the infimum is over all positive-dimensional irreducible closed subsets Y of X containing x. We briefly recall some basic properties of Seshadri constants for the purpose of comparison and ease of reference. All this material can be found in [Laz, §5]. Proposition 2.1. Suppose that L is an ample line bundle on the projective variety X , and let x G X . (1) e(L m ;x) = m ■ e(L;x). [Laz, Example 5.1.4], cf. Proposition 2.8. (2) e(L;x) depends only on the numerical equivalence class of L. [Laz, Example 5.1.3], cf. Proposition 2.14- Proposition 2.2. Suppose, in addition, that X is smooth and the ground field k has characteristic zero. (1) Ife(L;x) > dim(X), then the line bundle ux <8> L is globally generated at x. [Laz, Proposition 5.1.19(i)], cf. Corollary 3.2(\). (2) Ife(L;x) > 2 dim A" then the rational map defined byux®L is birational onto its image. [Laz, Proposition 5.1.19(h)], cf. Corollary 3.2(H). (3) If e(L;x) > 2dimX for every point x G X, then u>x ® L is very ample. [Laz, Proposition 5.1.19(iii)], cf. Corollary 3.2(iii). 4 M. MUSTATA AND K. SCHWEDE The Seshadri constant can be alternatively described in terms of jet separation, as follows. One says that a line bundle A on X separates P. jets at x G X if the restriction map H°(X, A) -> H°(X, A ® O x /mi +1 ) (2.2) is surjective, where m x is the ideal defining x. Let s(A; x) be the largest l > such that A separates £ jets at x (with the convention s(A; x) — if there is no such £). It is proved in [Laz, Theorem 5.1.17] cf. [Dem], that if x G X is a smooth point, then s(L m ;x) s(i7";x) ,_ _. e(L;x) = hm — = sup — (2.3) m-i-oo rn m >i m (note that the proof therein is characteristic-free). We turn to the definition of the Frobenius version of the Seshadri constant. From now on we assume that the ground field has characteristic p > 0. We denote by F: X — y X the absolute Frobenius morphism, given by the identity on points, and which maps a section / of Ox to f p . This is a finite morphism since k is perfect, and it is flat on the smooth locus of X by [Kun]. If J is an ideal on X (always assumed to be coherent), we denote by j' pE ] the inverse image of J by F e : if J is locally generated by (/ii)ie/, then J^ is locally generated by (/if )%ei- The definition of the Frobenius-Seshadri constant is modeled on the above interpre- tation of the usual Seshadri constant in terms of separation of jets. The main difference is that we replace the usual powers of the ideal m x defining x by the Frobenius powers. While we are mostly interested in the Frobenius-Seshadri constant at one point, we will need the notion in the case of several points in the next section, hence we give the definition in this slightly more general setting. Suppose that Z is a finite set of smooth closed points on X (with the reduced scheme structure), defined by the ideal Iz- Given a positive integer e, we say that a line bundle A on X separates p e -Frobenius jets at Z if the restriction map H°(X, A) -> H°(X, A ® O x /lf ] ) (2.4) is surjective. Remark 2.3. If A separates p e -Frobenius jets at Z and B is globally generated, then A® B also separates p e -Frobenius jets at Z. Indeed, there is a section t G H°(X, B) that does not vanish at any point in Z, and we have a commutative diagram H°(X,A) ^^ H°(X,A®B) H\X,A®O x /I [ f) y H°(X,A®B®O x /I l f) in which the bottom horizontal map is an isomorphism by the assumption on t, and the left vertical map is surjective by the hypothesis on A. Therefore the right vertical map is surjective. A FROBENIUS VARIANT OF SESHADRI CONSTANTS 5 Let spi^L" 1 ; Z) be the largest e > 1 such that L m separates p e -Frobenius jets at Z (if there is no such e, then we put SF(L m ; Z) = 0). Now we come to the main definition of this paper. Definition 2.4 (Frobenius-Seshadri Constant). The Frobenius-Seshadri constant of L at Z is p s F (L m ;Z) _ 1 Sp(L; Z) := sup . m>i m The following lemma gives the analogue in our setting for the inequality s(L mr ; x) > r ■ s(L m ; x) for all positive integers m and r. Lemma 2.5. If e = SF(L m ; Z) and d r = - ' e ~^ for some positive integer r, then s F (L mdr ;Z) > re. Proof. If e = 0, then the assertion is clear, hence we may assume that e > 1. It is enough to show that for every x e Z, if Z x = Z \ {x} and T r x := H°(X, I [ £ ] ® L mdr ) } then the map induced by restriction ip r : T^ -). H°(X, L mdr ® Ox/raSP) (2.5) is surjective, where va x is the ideal defining x. We prove this by induction of r. Note that the case r = 1 follows from hypothesis. Suppose now that r > 2. By assumption, x is a smooth point of X. Let us choose local algebraic coordinates yi, . . . ,y n at x (that is, a regular system of parameters of Ox, x )- After choosing an iso- morphism L x ~ O x , x , we identify L mdr (g> Cx/ntz with Cx/trii 1 , which has a &;-basis given by y^ 1 ■ ■ ■ y® n , with < Oj < p re — 1 for all i. Let us write a-i = Oi,o + «i,iP e + • • • + aj jr _ip e(r_1) = a i)0 + p e a-, with < ajj < p e — 1 for all i and j. Observe that < a\ < p(' r_1 ) e — 1 for all i. We will prove that y^ 1 ■ ■ ■ y^" lies in the image of (p r by descending induction on S := Xir=i a i- This will complete the proof. The first non-trivial step in the induction starts with S = n(p ( - r ^ 1 ' )e — 1), in which case all a\ = p( r-1 ) e — 1. Claim. Each element in the product [y\ , . . . , y^) ■ \Yi=i Vi a ^ s congruent to an element in Im((^ r ) mod xtix . Proof of claim. To see this, consider the term n yf-U^- ( 2 - 6 ) If a' e = p( r_1 ) e — 1 (for example, this is the case when S = n(j/ r-1 -' e — 1)), then p e +p e a' i = p re and the monomial in (2.6) lies in mf? ■ On the other hand, if a' e < p( r_1 ) e — 1, then the monomial in (2.6) can be written as n IK" ( 2 - 7 ) 1=1 6 M. MUSTATA AND K. SCHWEDE with < b' i < p e ( r_1 ) — 1 for every i. We then write b\ = Y^jZi h,jP l ^ e for some by with < bij < p e — 1. Since Y^i=i fy = S + 1, it follows from the inductive hypothesis with respect to S that the monomial in (2.7) is congruent to an element in Im((p r ) mod mf . This completes the proof of the claim. □ We now return to the proof of the lemma. Surjectivity of ip\ implies that there is t x G T l x whose image t 1>x e O x , x - L x satisfies H7=iyT'° ~ h,x e (Vi , • • -,Vn) = m * • Furthermore, by the inductive assumption with respect to r, we can find £2 £ r^ -1 such that nT=i % * * s congruent to (p r -i(h) modulo m^ . In other words n^-^emf- 1)e ]co XiX .. i=\ In this case (F e )*(£ 2 ) = t{ lies in #°(7|f ~ 1)e+e] <g> L™''^- 1 ). Observe that t x tf eTJC H°(X,L mdr ) and also that <^ r (^i*2 ) is simply the residue of £i i:r • tf ^ = (i^ ) ffi modulo m[p re l. We certainly have that = nr=i #* - *m> • € + *m> • rnu ^ - *m • m=i »f °< = (niu yf '° - m • m=i vf K + ti, x ■ (niu yf K - C The claim then implies that niLi ^ * s congruent to an element in Im(<£v) mod m p . This proves the lemma. □ Proposition 2.6. Let L be an ample line bundle on the projective variety X , and Z a reduced finite set of smooth points on X . i) We have ef(L;Z) = sup,^ 2 — !-, where the supremum is over all m, e > 1 such that L m separates p e -Frobenius jets at Z ii) Given any 5 > 0, there is e$ > 1 such that for every e divisible by e$, there is m such that L m separates e-Frobenius jets at Z and P — > Ep(L; Z) — 5. iii) We have Ef(L; Z) = limsup^^oo p F — — — . Proof. The assertion in i) follows from definition, since whenever L separates p e -Frobenius jets at Z, we have SF(L m ; Z) > e, hence p -^— < p -^- ! — =_, f n order to prove ii), note that by definition we may choose m such that p °~ 1 > £f(L; Z) — 5, where e = Si?(L m °; Z). If e = re , then it follows from Lemma 2.5 that if m = mo p sjjfV , then SF(L m ; Z) > e. Since 2 - = - = p °~ , the assertion in ii) follows. We deduce from ii) that there is a sequence ( I rn i ■ 7\ (me)e>i with m^ — y 00 such that lim^oo — ■ — — = £f{L; Z), which implies iii). □ Remark 2.7. We will see below in Example 2.11 that we cannot replace the limsup in Proposition 2.6 iii) by a limit. This stands in contrast with the ordinary Seshadri constant. A FROBENIUS VARIANT OF SESHADRI CONSTANTS 7 Proposition 2.8. If L is an ample line bundle on the projective variety X , and Z is a reduced set of smooth points on X , then e F (L r ; Z) = r ■ e F (L; Z) for every positive integer r. Proof. The inequality "<" follows from definition since the left side is the supremum over a smaller set. Hence we prove the opposite inequality. Let us fix jo such that IP is globally generated for every j > jo- In particular, LP has sections that do not vanish at any of the points in Z, hence s F (L m+ i; Z) > s F (L m ; Z) for all such j and all m. Given S > 0, let us choose m such that if e = s F (L m ; Z), then £-=- > e F (L; Z) — 5. For every i > 1, let di = p eZ\ - Using Lemma 2.5, we deduce that for every j > j we have s F (L mdi+j ; Z) > s F (L md >; Z) > ie. Let Oj = \{mdi + jo)/r~\, where \u\ denotes the smallest integer > u. Note that raj > mdi + jo which implies p s F (L ra i;Z) _ i p ie - I p e — 1 diTTl a-i ~ cii rn \(mdi+j )/r]' Applying limsup^oo to both sides yields s F (L™,;Z) _ -j e _ 1 e F (L r , Z) > limsup ^ > ^ r > (e F (L; Z) - 8) ■ r. i— >oo Oj m Since this holds for every 5 > 0, we deduce the inequality ">" in the proposition. □ Remark 2.9. Based on the analogy with the usual Seshadri constant, it is natural to expect that if L\ and L2 are ample line bundles on X, and Z is a finite set of smooth points on X, then e F {L\ eg) L2; Z) > e F (Li; Z) + e F (L2] Z). However, we do not know whether this is, indeed, the case. Remark 2.10. Suppose that L is an ample line bundle on the n-dimensional projective variety X and Z is a finite set of smooth points on X, defined by the ideal Iz- Let m be such that L m ° is globally generated and H l (X, L m ) = for alH > 1 and m > m . If m ^ %, then s F (L m ; Z) > e if and only if i7 1 (X, /^ ® -f/ m ) = 0. Note that in any case we have H l (X, I [ f ] ® L m ) = for z > 2. It follows that if m > m and s F (L m ; Z) > e, then Ig (gi L m+m ° is 0-regular with respect to L m ° in the sense of Castelnuovo-Mumford regularity. In particular, we see that 1% <g> /, m+m o j s globally generated (we refer to [Laz, §1.8] for the basic facts about Castelnuovo-Mumford regularity). From now on, we will mostly consider Frobenius-Seshadri constants at a single smooth point x G X, in which case we simply write s F (L; x). Example 2.11. Consider the case when X = P n and L = 0(1). Note that for every x G P™, we have s F (0(m);x) > e if and only if m > n(p e — 1). This implies that £i?(0(m);x) = -. Note that in this case s((9(m);x) = m, hence £(0(1); x) = 1. This example also shows that the limsup in Proposition 2.6 iii) might not be limit: indeed, if we consider m e = n(p e — 1) — 1, then pSp(0(m e );x) _ 1 „e-l _ j -^ = — r >• — as e — > 00. m e n(p e — 1) — 1 np 8 M. MUSTATA AND K. SCHWEDE Proposition 2.12. If L is an ample line bundle on the n- dimensional projective variety X , then for every smooth point x G X we have ^^<e F (L;x)<e(L;x). n Proof. Note that if m^ is the ideal defining x, then we have m n(p«-i)+i c m b e ] c mf. (2.8) The second containment in (2.8) implies s(L m ; x) > p s ^ Lm "' x ) — 1 and we obtain e(L; x) > Sf{L'i x) using the definition of the Frobenius-Seshadri constant and (2.3). Given S > 0, let m be such that — — > e(L; x) — 5. Given any positive integer e, let d = \n(p e — l)/s(L m °; x)~\. Therefore we have s{L mod ; x)>d- s{L mo ; x) > n{p e - 1). The first containment in (2.8) implies Si?(L m ° d ; x) > e, and thus s F {L™od; X ) _ 1 e _ i e F (L;x)>- ; > — F ^. . r-. (2.9) v _ m d ~ m \n{p e - l)/s{L m °;x)] v ' When e goes to infinity, the right-hand side of (2.9) converges to — — > -(e(L; x) —5). We conclude that Sf(L; x) > Me(L; x) — 5), and since this holds for every 5 > 0, we get e F (L;x)> e -^. " D We note that interpret the Frobenius-Seshadri constant £f(L; x) as corresponding to di \ x , e(L; x) rather than to e(L; x). This is justified by Example 2.11, but also by the results in connection to adjoint linear systems that we discuss in the next section. Remark 2.13. If L is ample and globally generated, then the usual Seshadri constant at any smooth point iGl satisfies e(L;x) > 1 (see [Laz, Example 5.1.18]). It follows from Proposition 2.12 that in this case we have Ef(L;x) > -. As Example 2.11 shows, this is optimal. Proposition 2.14. If Li and L2 are numerically equivalent ample line bundles on the projective variety X , and x G X is a smooth point, then £p(Li; x) = Ef{L2', x). Proof. Let us write L 2 — Li® P, where P is a numerically trivial line bundle. Note that if A is a very ample line bundle on X, then there is m such that A m <g> P l is globally generated for every m > m and every % G Z. Indeed, by Fujita's vanishing theorem (see [Fuj]) there is m\ such that H^[X 1 A m ® Lf) = for all j ; > 1, m > m\, and all nef line bundles V . In particular, if m > mi + dim(X), we see that A m £g> V is 0-regular with respect to A, in the sense of Castelnuovo-Mumford regularity, hence it is globally generated. Therefore it is enough to take mo = mi + dim(X). We now apply the above assertion with A being a suitable power of Li that is very ample, and deduce that there is a positive integer j such that L\®P % is globally generated for every i. In particular, we see that for every positive integer m we have an inclusion H\X,L™)^H\X,L™ +3 ) A FROBENIUS VARIANT OF SESHADRI CONSTANTS 9 induced by a section of L™ 3 <S> L\ m ~ L{ £g> P m +-? that does not vanish at x. This implies s j p(L™ + - 7 ; x) > Sir(L^; x) for every m > 0, and therefore pS F (L 2 J ;x) _ -y s F (Lf';a;) /?) >^ !._:_. (2.10) m + j m m + j By Proposition 2.6 iii), we can find a sequence of m's going to infinity such that the right- hand side of (2.10) converges to Sf(Li,x), and we conclude that £f(L 2 ;x) > ep(Li;x). The reverse inequality follows by symmetry D Remark 2.15. Suppose that L is an ample line bundle on the projective variety A. It is standard to see that given any m and e, the set of points x in the smooth locus A sm of A such that L m separates p e -Frobenius jets at x is open in A sm , hence in X. Given any a > 0, we see that the set {x G X \ ef(L; x) > a} is open: indeed, by Lemma 2.5, this set is the union of the sets of points x G X sra for which L m separates p e -Frobenius jets at x, the union being over those e and m such that £_=_ > a. A formal consequence of this fact is that if the ground field is uncountable, then there is a maximum among all Ef(L;x), for x G A sm , and this is achieved on the complement of a countable union of closed subsets of X. We will make use of the next proposition in the following section, in order to obtain a criterion for point separation in the case of adjoint line bundles. Proposition 2.16. Let Li,...,L r be ample line bundles on a projective n- dimensional variety X , and Z = {x%, . . . , x r } a finite set of smooth points on X . If £p(Li\ Xj) > a for every i, then Ef{L\ ® . . . <g> L r ; Z) > a. Proof. Let mo be such that for all m > mo the following holds: L™ is globally generated, and Hi(X, L™) = for all % and all j > 1. Given a' < a, for every % we may find e« and rrii such that L™* separates p ei -Frobenius jets, and p ' -1 > a'. Note that using Lemma 2.5 we may always replace e* by a multiple tej if we simultaneously replace m« by \ % J[ '- . Therefore we may assume that e^ = e for all i, and also that the m.j > nmo- If m = maxj m^, then £™+ m ° se p ara tes p e -Frobenius jets at Xj (note that £! n + m °~ m * is globally generated). Arguing as in Remark 2.10, we may assume that m^ ® £™ + m ° is 0-regular with respect to L™ , hence it is globally generated. We can thus find sections Si G H°(X,m [ xP <g> Lf +2mo ) that do not vanish at any of the points in Z \ {xj. Since jjm+2mo se p ara t es p e -Frobenius jets at x,, the restriction maps H°(X, L™ +2mo ) -)• H°(X, L™ +2mo ® O x /mP) (2.11) are surjective. Let L — L\ (g) . . . <g) L r and denote by Iz the ideal of Z. We deduce that the composition map r ~ i H°(X, Lf +2m °) ->• if (A, L m+2m ») ->• #°(A, L m+2mo ® Ox//? 1 ) (2.12) 8=1 is surjective. Indeed, H (X,L m+2m ° ® Ox/^ 1 ) = ©Li#°(^ L m+2m ° <g) C x /m^ ] ), and the surjectivity of (2.11) implies that the 2 th component of this direct sum is the image of 10 M. MUSTATA AND K. SCHWEDE si®.. .®H°(X, L™ +2m °)(g). ..®s r . Since the map in (2.12) factors through the restriction map H°{X,L m+2m °) -> H°(X,L m+2mo g> Ox//? 1 ), it follows that £ m + 2m o separates p e -Frobenius jets at Z, hence ep(L; Z) > p ~ . Fur- thermore, the same inequality will hold if we replace e by te and m by e _~ , that is, ^i^) ^^ . (2.13) When t goes to infinity, the right-hand side of (2.13) converges to 2^, which is > a'. We thus obtain ep(L; Z) > a' for every a' < a, which gives the statement of the proposition. □ 3. Frobenius-Seshadri constants and adjoint bundles In this section we show how lower bounds on Frobenius-Seshadri constants imply global generation or very ampleness of adjoint line bundles. As in the previous section, all our varieties are defined over an algebraically closed field of characteristic p > 0. Theorem 3.1. Let L be an ample line bundle on a smooth projective variety X . i) If Z = {xi, . . . , x r } is a finite subset of X such that ep{L; Xj) > r for every i, then the restriction map H°{X, uj x ®L)^ H°{X, uj x ®L® O z ) is surjective. ii) If Ef{L; x) > 1 for every x G X , then Ux ® L is globally generated. iii) If Ep{L', x) > 2 for some x G X, then ux <8> L defines a rational map that is birational onto its image. iv) If Ef(L] x) > 2 for every x G X , then Ux <S> L is very ample. In particular, we obtain the following criterion for global generation and very am- pleness of adjoint line bundles in terms of the usual Seshadri constants. Note that the bounds are the same ones as in characteristic zero, when the assertions are proved via vanishing theorems (see [Laz, Proposition 5.1.19]). Corollary 3.2. Let L be an ample line bundle on a smooth n-dimensional projective variety X . i) If e(L; x) > n, then ux ® L is globally generated at x. In particular, if e(L; x) > n for every x G X , then Ux ® L is globally generated. ii) If e(L; x) > 2n for some x G X , then ux <8> L defines a rational map that is birational onto its image. iii) If e(L; x) > 2n for every x G X, then ux <8> L is very ample. Proof. For every x G X, we have ep(L;x) > £ ^ — by Proposition 2.12. Therefore all assertions follow from Theorem 3.1. □ A FROBENIUS VARIANT OF SESHADRI CONSTANTS 11 In the proof of Theorem 3.1 we will need the following lemma in order to separate tangent vectors. Lemma 3.3. Let L be an ample line bundle on the projective variety X , and x G X a smooth point. If sp(L; x) > a for some a G R>o ; then we can find r and e with p -^— > | and such that the restriction map $: H°(X,L r ) -^H°(X,L r ®O x /{ml)b e] ) (3.1) is surjective. Furthermore, we may assume that p e — 1 — ~r is arbitrarily large. Proof. Let mo be such that for all m > m , the following hold: L m is globally generated and H l (X, L m ) = for all i > 1. It follows from definition that we can find e and m such that p -^— > a and the restriction map H°{X, L m ) -± H°{X, L m (8) Cx/mf ] ) (3.2) is surjective. By Lemma 2.5, we may replace e by se and m by m e _~ for every s > 1. In particular, we may assume that m is arbitrarily large. We may also assume that 2 m+m > |. Indeed, we have pse _ ^ p e — 1 a lim — , „ ., = > — • ,_>«, 2 !lfcl) + mo 2m 2 Furthermore, arguing as in Remark 2.10, we may assume that xxix <S> L m+m ° is 0-regular with respect to L m ° , hence it is globally generated. In order to prove the first assertion in the lemma, it is enough to show that if we take r = 2m + mo, then (3.1) is surjective. Claim. It is enough to show that W := H°(X,L 2m+m ° <g> (m [ f ] / (m 2 x ) [pe] )) is contained in the image o/$. Proof of claim. Indeed, it follows from the surjectivity of (3.2) and the fact that L m+mo is globally generated that the restriction map (p: H°{X,L 2m+mo ) -> H°{X,L 2m+mo ® O x /mf ] ) is surjective as well. We deduce that if u e H (X,L 2m+mo ® C x /(m2)[p e ]) restricts to u e H°(X,L 2m+mo ® O x /m [ f ] ), then there is s e H°(X,L 2m+mo ) such that <p(s) = u. Therefore u — $(s) G W 7 , and if u — 3>(s) lies in the image of $, then so does u. D Let us choose a trivialization of L around x, which induces an isomorphism L x ~ O x ,x that we henceforth tacitly use. Since m^ £g> L m+m ° is globally generated, it fol- lows that given any w G m l f ] / (m 2 x ) lp£ \ there are t x , . . . ,t d G if°(X,mi pel <g> L m+m °) and /i,...,/d6 Cx,! such that w = X/i=i U,xfi,x mod (m^)' pe ] (here tj ja; denotes the restriction of tj to the stalk at x). Using now the surjectivity of (3.2), we can find f\ G H°(X,L m ) such that f itX = f i)X modm l f ] . This implies that if t = ^ti U ® /i e H°(X, L 2m+m °), then w = t x mod (m^)' pe l This completes the proof of the fact that W is contained in the image of $. 12 M. MUSTATA AND K. SCHWEDE The last assertion in the lemma follows from the fact that a [2m(p se — 1) \ . „ . / m \ mna P S£ ~ 1 " 2 ( pf-i + mo J = ^ - ^ (/ - "^37 J " "f- ~» °° when s ~> °°- D The key ingredient in the proof of Theorem 3.1 is the canonical surjective map T: F^ux — > u>x- This can be defined as the trace map for relative duality for the finite flat morphism F (see for example [BK, Chapter 1]). It can also be explicitly described as follows. Recall that X is smooth, and suppose that U C X is an affine open subset and j/i, ... , y n G 0(U) are such that dyi, . . . , dy n give a trivialization of Qx- After possibly replacing [/ by a suitable open cover, we may assume that 0(U) is free over 0(U) P with basis {y'i ■ ■ -Vn I < k < V ~ 1 for all£}. If we put ofa/ = dy\ A ... A dy n , then T: 0{U)dy —¥ 0(U)dy is uniquely characterized by the following properties: 1) T(f p r]) = f ■ T{rj) for every / e 0(U) and every r] e 0{U)dy. 2) T(y^ • • ■ y l ™dy) = y 1 v ■ ■ ■ y n p dy for every ii, . . . ,i n e Z> (with the conven- tion that the expression on the right-hand side is zero, unless all exponents are integers) . We will also consider the e-iterate of T, namely T e : F£(u>x) —> wx- Note that if J is an ideal of Ox, then T e (F:( J^ ■ ux)) = J ■ T e (F:(u x )) = J-uj x . We can now prove the main result of this section. Proof of Theorem 3.1. We first prove i). Let Iz denote the ideal of Z. Note that the hypothesis, together with Proposition 2.16 implies that €p(L; Z) = £F ' - ' ' > 1. Therefore we can find m and e such that m < p e — 1 and L m separates p e -Frobenius jets at Z. Furthermore, by Lemma 2.5 we may replace e by se and m by e _~ for every s > 1. Since m(p se — 1) (p e — 1 — m)(p se — 1) p — 1 — = — ^^ -> oo when s -)• oo p e — 1 p e — 1 it follows that we may assume that p e — 1 — m is as large as we want. In particular, we may assume that ux®F pe ~ m is globally generated, in which case we deduce that ujx®L p " separates p e -Frobenius jets at Z. We now make use of the surjective map T e : F£(u>x) — > wx- As we have seen, this induces a surjective map F^(I^ ux) — >■ Iz^x- Tensoring this by L and using the projection formula gives a surjective map F^(I^ uJx ® £ pC ) — > Iz^x ® ^- Since F^ e is an exact functor, we obtain a commutative diagram with exact rows and surjective vertical maps A FROBENIUS VARIANT OF SESHADRI CONSTANTS 13 + WJTW LP e I z ux ® L T e ®L -> uj x ® L ->■ F:{u x <g> ZX (g) X /J -> ux®I® 0x//z -> By taking global sections we obtain a commutative diagram H°(X, co x ® ^ pe ) — ^ #°(X, Ul ® IT <g) Ox//f G] ) H°(X,u x ®L) H°{X,u x ®L®O x /I z ) in which p is surjective. Since u x <E> L pe separates p e -Frobenius jets at Z, we have that <p is surjective, and we thus obtain that ip is surjective, which completes the proof of i). The assertion in ii) follows from i), by considering Z = {x}, for a point x G X. Under the assumption in iii), it follows from Remark 2.15 that there is an open subset U C X such that ep(L; y) > 2 for every y G U. In order to prove iii), it is enough to show that ou x (g) L separates points and tangent vectors on U. By taking Z = {2/1,2/2} f° r Vi and 2/2 distinct points in U, it follows from i) that uo x <8> L separates points in U . Suppose now that x G U. The hypothesis together with Lemma 3.3 implies that we can find e and m such that m < p e — 1 and such that the restriction map H°{X, L m ) -> #°(X, L m g) C x /(m2) [pe] ) is surjective. Furthermore, since we may assume that p e — 1 — m is large enough, we may assume that uo x ® ]j> e - m j s globally generated, hence also the restriction map #°(X, Wx ® L pe ) ^> F°(X, Ul ® L pe <g) Ox/M^ 1 ) is surjective. Arguing as in the proof of i), we obtain a commutative diagram #°(X, w x <g> ZX) -^ #°(X, w x <g> If <g> Ojr/MM) H°(X } lo x ®L) V-' # (X,w x ®£®e>x/m with // surjective. Since <// is surjective, we deduce that ■0' is surjective, that is, uo x Cg> L separates tangent vectors at x. This completes the proof of iii). The assertion in iv) now follows from the above argument, by taking U — X. □ 4. Frobenius-Seshadri constants on toric varieties It is well-known that the Seshadri constants at the torus-fixed points of a smooth toric variety can be explicitly described in terms of polyhedral geometry (see [DRo] and [B+]). Our goal in this section is to give a similar description for the Frobenius-Seshadri constant. We freely use basic facts and notation on toric varieties from [Ful]. 14 M. MUSTATA AND K. SCHWEDE Let N ~ Z n be a lattice and M = Honiz(iV, Z) the dual lattice. We consider a smooth projective toric variety X corresponding to a fan A in Nn = Af^R. We assume that X is defined over an algebraically closed field k of characteristic p > 0. Let L be an ample line bundle on X and x G X a torus-fixed point. This point corresponds to an n-dimensional cone a G A. We assume that X is smooth and so a is a non-singular cone. We can thus choose a basis e 1; . . . , e n of N such that a is the convex cone generated by these vectors. If e* . . . , e* G M give the dual basis and we put U = x e S then 0{U a ) is a polynomial fc-algebra in t\, . . . , t n , and x G U„ is defined by the ideal (ti, . . . , t n ). There is a torus-invariant divisor D such that L ~ Ox{D). We choose the unique such .D which is effective and whose restriction to the open affine subset U a corresponding to a is zero. Let Pd Q Mr ~ R" denote the lattice polytope corresponding to D. Recall that Pd is the convex hull of those u G M such that div(x") + D > 0. In particular, we have Pd Q R+. Furthermore, since L is ample, it follows that A is the normal fan of Pd- By our normalization of D, this implies that the vertex of Pd corresponding to the cone a G A is the origin G R n , and there are precisely n facets of Pd containing 0, namely P D n {u = (u 1 , ...,u n )eR n \ui = 0}, for 1 < % < n. For every m > 1, we have a basis of H°(X, Ox{mD)) given by {x u \ u G itlPd DM}. Furthermore, our choice of D implies that we have a trivialization Ox{D)\u a — ®u a such that if u = (ui, . . . ,u n ) G raP fl M, then the section x u G H°(X, Ox{rnD)) restricts to Iir=i ^T e ®{U(r)- This proves the following Lemma 4.1. With the above notation, Ox{mD) separates p e -Frobenius jets at x if and only if for every u = (ui, . . . , u n ) G Z> with Ui < p e — 1 for all i, we have u G tuPd- Theorem 4.2. With the above notation, the Frobenius-Seshadri constant of L ~ Ox{D) at x is given by Ef(L; x) = max{r G R>o | r • C n C Pd}, w/iere C„ is t/ie c«6e [0, l] n C R n . Proof. Let M := max{r G R>o | r • C n C P}. Lemma 4.1 gives s F (L m ; x) = max{e G Z> | (p e - 1) • C n C mP D }. In particular, for every m>lwe have v s F (L m ;x) _ ^ < M, hence ep{L]x) < M. Moreover, in order to show that we have equality, it is enough to show that for every 5 G (0, M), we can find positive integers m and e such that M-5<^-^-<M. (4.1) m Indeed, in this case SF(L m ; x) > e and therefore Ef(L; x) > ^-^- > M — 5. It is thus enough to find e > 1 such that there is an integer m with p e — 1 p e — 1 Hw-- m< M3T A FROBENIUS VARIANT OF SESHADRI CONSTANTS This is clearly possible if p e - 1 p e - 1 S(p e - I) 15 M-S M M(M - 5) > 1. This holds for e ^> 0, hence we can find e and m such that (4.1) holds. We thus have e F {L-x) = M. □ Remark 4.3. It is interesting to compare the formula in Theorem 4.2 with the formula for the usual Seshadri constant of L at x (see [B— , Corollary 4.2.2]). This says that if Q n denotes the simplex {u [Ml, , Mr G R> I U\ + . . . + u n < 1}, then (4.2) e(L; x) = max{r G R>o | r ■ Q n C Pq}. Of course, one can also rewrite the right-hand side of (4.2) as r := max{r G R>o | re* G Pp for 1 < i < n}. Note that r is an integer since Pd is a lattice polytope. One can prove (4.2) arguing as above. Indeed, it follows from the discussion preceding Lemma 4.1 that L m separates £-jets at x if and only if for every u = (ui, . . . , u n ) G Z>o with Yui=i u i — ^i we have u G tjiPd- Therefore s(L m ; x) = max{s G Z> | s ■ Q n C mP D } for every m > 1, hence e(P; x) = r . mr Example 4.4. Let M = Z 2 and consider the convex hull P of the following set of points in R 2 {(0,0), (1,0), (2,1), (2, 2), (1,2), (0,1)}. The corresponding toric variety X is the blow-up of P 2 at the three torus-fixed points, and the line bundle L corresponding to P is the anti-canonical line bundle o;^ 1 - If x G X is the torus-fixed point corresponding to G P, then we have ep(L; x) = e(L; x) = 1. Remark 4.5. The recent interesting preprint [Ito] gives estimates for the Seshadri con- stant at a general point on a toric variety (in characteristic zero). It would be interesting to investigate whether one can obtain similar estimates for the Frobenius-Seshadri constant when working over a field of positive characteristic. 16 m. mustata and k. schwede References [B+] T. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. Knutsen, W. Syzdek, and T. Szemberg, A primer on Seshadri constants, in Interactions of classical and numerical algebraic geometry, 3370, Contemp. Math., 496, Amer. Math. Soc., Providence, RI, 2009. 1, 13, 15 [BK] M. Brion and S. Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, 231, Birkhauser Boston, Inc., Boston, MA, 2005. 12 [Dem] J. -P. Demailly, Singular Hcrmitian metrics on positive line bundles, in Complex algebraic varieties (Bayreuth, 1990), 87-104, Springer, Berlin, 1992. 1, 4 [DRo] S. Di Rocco, Generation of k-jets on toric varieties, Math. Z. 231 (1999), 169-188. 13 [EKL] L. Ein, O. Kiichle, and R. Lazarsfeld, Local positivity of ample line bundles, J. Differential Geom. 42 (1995), 193-219. 2 [EL] L. Ein and R. Lazarsfeld, Seshadri constants on smooth surfaces, in Journees de Geometric Algebrique d'Orsay (Orsay, 1992), Asterisque No. 218 (1993), 177186. 2 [Fuj] T. Fujita, Vanishing theorems for semipositive line bundles, in Algebraic geometry (Tokyo/Kyoto, 1982), 519-528, Lecture Notes in Math. 1016, Springer, Berlin, 1983. 8 [Ful] W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, The William H. Rover Lectures in Geometry, Princeton Univ. Press, Princeton, NJ, 1993. 13 [Kun] E. Kunz: Characterizations of regular local rings for characteristic p, Amer. J. Math. 91 (1969), 772-784. 4 [Ito] A. Ito, Seshadri constants via toric degenerations, preprint, 2011, arXiv:1202.6664. 15 [Laz] R. Lazarsfeld, Positivity in algebraic geometry I, Ergebnisse der Mathematik und ihrer Gren- zgebiete, 3. Folge, Vol. 48, Springer- Verlag, Berlin, 2004. 1, 3, 4, 7, 8, 10 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA E-mail address: mmustata@umich.edu Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA E-mail address: schwede@math.psu.edu