SYZYGIES, MULTIGRADED REGULARITY AND TORIC VARIETIES

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1 SYZYGIES, MUTIGRADED REGUARITY AND TORIC VARIETIES MIENA HERING, HA SCHENCK, AND GREGORY G. SMITH Abstract. Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B 1,..., B l on X and m 1,..., m l N, consider the line bundle := B m1 1 B m l l. We give conditions on the m i which guarantee that the ideal of X in P(H 0 (X, ) ) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes. 1. Introduction Understanding the equations defining a projective variety X and the relations among them is a central problem in algebraic geometry. Green [Gre84a] shows that a sufficiently positive line bundle on X gives an embedding X P ( H 0 (X, ) ) such that the first few syzygies are as simple as possible. Explicit conditions certifying that an ample line bundle is suitably positive are given by Green [Gre84a] for curves, Ein and azarsfeld [E93] for smooth varieties, Gallego and Purnaprajna [GP99] for normal surfaces, and Pareschi and Popa [Par00, PP04, PP03] for abelian varieties; see azarsfeld [az04, 1.8.D] for a survey. The primary goal of this paper is to produce similar conditions for toric varieties. et X be a projective variety over a field of characteristic zero and let be a globally generated line bundle on X. The associated morphism is φ : X P ( H 0 (X, ) ) and S := Sym H 0 (X, ) denotes the homogeneous coordinate ring of P ( H 0 (X, ) ). Consider the graded S-module R := j 0 H0 (X, j ) and a minimal free graded resolution E of R. Following Green and azarsfeld [G85], we say that satisfies property (N p ) for p N provided that E 0 = S and Ei = S( i 1) for all 1 i p. Hence, φ (X) is projectively normal if and only if 2000 Mathematics Subject Classification. Primary 14M25; Secondary 13D02, 14C20, 52B20. Key words and phrases. toric variety, syzygy, Castelnuovo-Mumford regularity. The second author was partially supported by NSF Grant DMS

2 2 M. HERING, H. SCHENCK, AND G.G. SMITH satisfies (N 0 ) and φ (X) is normal. If satisfies (N 1 ), then the homogeneous ideal of φ (X) is generated by quadrics and (N 2 ) implies that the relations among the generators are linear. The concepts of normal generation and normal presentation introduced by Mumford [Mum70] correspond to (N 0 ) and (N 1 ) respectively. Typically, the property (N p ) is studied under the additional hypothesis that is ample; our theorems do not require this assumption. To examine this property, we use multigraded Castelnuovo-Mumford regularity. Fix a collection B 1,..., B l of globally generated line bundles on X. For u := (u 1,..., u l ) Z l, set B u := B u 1 1 B u l l and let B be the semigroup {B u : u N l } Pic(X). We say that a line bundle is O X -regular (with respect to B 1,..., B l ) if H i (X, B u ) = 0 for all i > 0 and all u N l with u := u u l = i. Our main result is the following. Theorem 1.1. et w 1, w 2, w 3, w 4,... be a sequence in N l such that B w i l j=1 (B j B) and set m i := w w i for i 1. If B m 1 is O X -regular then the line bundle B mp satisfies (N p ) for p 1. The case l = 1 is in Gallego and Purnaprajna [GP99, Theorem 1.3]. Our proof is a multigraded variant of their arguments. Applying Theorem 1.1 with l = 1 to line bundles on toric varieties yields the following. Corollary 1.2. et be an ample line bundle on an n-dimensional toric variety. If d n 1 + p then the line bundle d satisfies (N p ). The case p = 0, an ingredient in our proof, was established by Ewald and Wessels [EW91]; other proofs appear in iu, Trotter and Ziegler [TZ93], Bruns, Gubeladze and Trung [BGT97], and Ogata and Nakagawa [ON02]. On a toric surface, Koelman [Koe93] proves that a line bundle satisfies (N 1 ) if the associated lattice polytope contains more than three lattice points in its boundary. More generally, Bruns, Gubeladze and Trung [BGT97] show that R is Koszul when d n and this implies that d satisfies (N 1 ) when d n. Assuming n 3, Ogata [Oga03] establishes that n 1 satisfies (N 1 ) and, building on this, Ogata [Oga04] proves that n 2+p satisfies (N p ) when n 3 and p 1. With additional invariants, we can strengthen Corollary 1.2. et h (d) := χ( d ) = n i=0 ( 1)i dim H i (X, d ) be the Hilbert polynomial of and let r() be the number of integer roots of h. Corollary 1.3. et be a globally generated line bundle on a toric variety and let r() be the number of integer roots of its Hilbert polynomial h. If p 1 and d max{deg(h ) r() + p 1, p} then the line bundle d satisfies (N p ).

3 SYZYGIES, MUTIGRADED REGUARITY AND TORIC VARIETIES 3 If X = P n and = O X (1), then we have h (d) = ( ) d+n n and r() = n. In particular, we recover a result by Green [Gre84b, Theorem 2.2] that O P n(d) satisfies (N p ) for p d. Using the dictionary between lattice polytopes and ample line bundles on toric varieties, Corollary 1.3 yields a normality criterion for lattice polytopes. A lattice polytope P is normal if every lattice point in mp is a sum of m lattice points in P. et r(p ) be the largest integer such that r(p ) P does not contain any lattice points in its interior. Corollary 1.4. If P is a lattice polytope of dimension n, then the lattice polytope ( n r(p ) ) P is normal. Theorem 1.1 also applies to syzygies of Segre-Veronese embeddings. Corollary 1.5. If X = l i=1 Pn i then the line bundle O X (d 1,..., d l ) satisfies (N p ) for p min{d i : d i 0}. The Segre embedding O X (1,..., 1) satisfies (N p ) if and only if p 3; see ascoux [as78] or Pragacz and Weyman [PW85] for l = 2, and Rubei [Rub02, Rub04] for l > 2. Eisenbud et al. [EGHP04, 3] provides an overview of results and conjectures about the syzygies of Segre- Veronese embeddings. Inspired by Ein and azarsfeld [E93], we also examine the syzygies of adjoint bundles. Recall that a line bundle on a toric variety is numerically effective (nef) if and only if it is globally generated, and the dualizing sheaf K X is a line bundle if and only if X is Gorenstein. Corollary 1.6. et X be a projective n-dimensional Gorenstein toric variety and let B 1,..., B l be the minimal generators of Nef(X). If w 1, w 2,... is a sequence in N l such that B w i l j=1 (B j B) and m i := w w i for i 1, then for p 1 { K X B m (N n+1+p p+1 ) if X P n, satisfies (N p ) if X = P n. Ein and azarsfeld [E93] prove that for a very ample line bundle and a globally generated line bundle N on a smooth n-dimensional algebraic variety X P n, K X n+p N satisfies (N p ). Corollary 1.6 gives a similar result for ample line bundles on possibly singular Gorenstein toric varieties. Specifically, if is an ample line bundle such that l j=1 (B j B) and N is a nef line bundle on X P n then K X n+p N satisfies (N p ). The proof of Corollary 1.6 combines Theorem 1.1 with Fujita s Freeness conjecture for toric varieties, see Fujino [Fuj03]. Conventions. The nonnegative integers are denoted by N. We work over a field of characteristic zero.

4 4 M. HERING, H. SCHENCK, AND G.G. SMITH Acknowledgments. The first author thanks A. Bayer, W. Fulton, R. azarsfeld, M. Mustaţă and S. Payne for helpful discussions related to this work. Parts of this work were done while the last two authors were visiting and partially supported by the Mathematical Sciences Research Institute in Berkeley and the Mathematisches Forschungsinstitut in Oberwolfach. 2. Multigraded Castelnuovo-Mumford Regularity In this section we review multigraded regularity as introduced by Maclagan and Smith [MS04]. Fix a collection B 1,..., B l of globally generated line bundles on X. For any element u := (u 1,..., u l ) Z l, set B u := B u 1 1 B u l l and let B be the semigroup {B u : u N l } Pic(X). If e 1,..., e l is the standard basis for Z l, then B e j = B j. et F be a coherent O X -module and let be a line bundle on X. We say that F is -regular (with respect to B 1,..., B l ) provided H i (X, F B u ) = 0 for all i > 0 and all u N l satisfying u := u u l = i. As Mumford [Mum66] says, this apparently silly definition reveals itself as follows. Theorem 2.1. If the coherent sheaf F is -regular then for all u N l : (1) F is ( B u )-regular; (2) the natural map H 0 (X, F B u ) H 0 (X, B v ) H 0 (X, F B u+v ) is surjective for all v N l ; (3) F B u is generated by its global sections, provided there exists w N l such that B w is ample. When X is a toric variety, this follows from results in Maclagan and Smith [MS04, 6]. Our proof imitates Mumford [Mum70, Theorem 2] and Kleiman [Kle66, Proposition II.1.1]. Proof. By replacing F with F, we may assume that the coherent sheaf F is O X -regular. We proceed by induction on dim ( Supp(F ) ). The claim is trivial when dim ( Supp(F ) ) 0. As each B j is basepointfree, we may choose a section s j H 0 (X, B j ) such that the induced map F B e j F is injective (see Mumford [Mum70, page 43]). If G j is the cokernel, then we have 0 F B e j F G j 0 and dim ( Supp(G j ) ) < dim ( Supp(F ) ). From this short exact sequence, we obtain the long exact sequence H i (X, F B u e j ) H i (X, F B u ) H i (X, G j B u ) H i+1 (X, F B u e j ).

5 SYZYGIES, MUTIGRADED REGUARITY AND TORIC VARIETIES 5 By taking u = i, we deduce that G j is O X -regular. The induction hypothesis implies that G j is (B j )-regular. Setting u = e j + u with u = i, we see that F is (B j )-regular and (1) follows. For (2), consider the commutative diagram: 0 H 0 (X, F ) H 0 (X, F ) H 0 (X, B j ) H 0 (X, G j ) H 0 (X, B j ) H 0 (X, F B j ) H 0 (X, G j B j ). Since F is O X -regular, the map in the top row is surjective. The induction hypothesis guarantees that the map in the right column is surjective. Thus, the Snake emma implies that the map in the middle column is also surjective. Therefore, (2) follows from the associativity of the tensor product and (1). astly, consider the commutative diagram: H 0 (X, F B u ) H 0 (X, B v ) O X H 0 (X, F B u+v ) O X H 0 (X, F B u ) B v β u id β u+v F B u+v Applying (2), we see that the map in the top row is surjective. By assumption, there is w N l such that B w is ample. If v := kw, then Serre s Vanishing Theorem (azarsfeld [az04, Theorem 1.2.6]) implies that β u+v is surjective for k sufficiently large. Hence, β u is also surjective which proves (3). We end this section with an elementary observation. emma 2.2. et 0 F F F 0 be a short exact sequence of coherent O X -modules. If F is -regular, F is ( B e j )-regular for all 1 j l and H 0 (X, F B e j ) H 0 (X, F B e j ) is surjective for all 1 j l, then F is also -regular. Sketch of Proof. This is similar to the proof of Theorem 2.1.1: tensor the exact sequence 0 F F F 0 with B u and analyze the associated long exact sequence. 3. Proof of Main Theorem The proof of Theorem 1.1 combines the multigraded Castelnuovo- Mumford regularity with a cohomological criterion for (N p ). Given a globally generated line bundle on X, there is a natural surjective map ev : H 0 (X, ) O X and we set M := Ker(ev ). Hence,

6 6 M. HERING, H. SCHENCK, AND G.G. SMITH M is a vector bundle on X which sits in the short exact sequence ( ) 0 M H 0 (X, ) O X 0. The following result shows that M governs the syzygies of φ (X) in P ( H 0 (X, ) ). emma 3.1. et be a line bundle on X that is generated by its global sections and assume that the ground field has characteristic zero. If H 1 (X, M q j ) = 0 for q p + 1 and j 1, then satisfies (N p ). Sketch of Proof. The arguments in azarsfeld [az89, 1.3] show that satisfies (N p ) if and only if H 1 (X, q M j ) = 0 for q p + 1 and j 1. Since k M is a direct summand of tensor product M k in characteristic zero, the claim follows. Proof of Theorem 1.1. Set := B mp and let M be the vector bundle in ( ). We first prove, by induction on q, that M q is (Bmq )-regular for all q 1. Since B m 1 is O X -regular, Theorem implies that H 0 (X, B m1+u ) H 0 (X, B v ) H 0 (X, B m1+u+v ) is surjective for all u, v N l. In particular, the maps H 0 (X, ) H 0 (X, B m 1 e j ) H 0 (X, B m 1 e j ) 1 j l are surjective because B m 1 l j=1 (B j B). Applying Theorem and emma 2.2, we see that M is (B m 1 )-regular. For q > 1, tensor the sequence ( ) with M (q 1) to obtain the exact sequence 0 M q H 0 (X, ) M (q 1) Since the induction hypothesis states that M (q 1) Theorem shows that the maps M (q 1) 0. is (B m q 1 )-regular, H 0 (X, M (q 1) B mq e j ) H 0 (X, ) H 0 (X, M (q 1) B mq e j ) are surjective for 1 j l because B wq l j=1 (B j B). Again by Theorem and emma 2.2, M q is (Bmq )-regular. By emma 3.1, it suffices to prove that H 1 (X, M q j ) = 0 for q p+1 and j 1. Since M q is (Bmq )-regular, Theorem implies that M q is (B mp )-regular for 1 q p; as O X is (B m 1 )-regular, Theorem also implies that O X is (B mp )-regular. It follows that H 1 (X, M q j ) = 0 for q p and j 1. Moreover, Theorem shows that H 0 (X, ) H 0 (X, M p j ) H 0 (X, M p j+1 ) is surjective. Hence, the short exact sequence 0 M (p+1) j H 0 (X, ) M p j M p j+1 0 implies that H 1 (X, M (p+1) j ) = 0 for j 1.

7 SYZYGIES, MUTIGRADED REGUARITY AND TORIC VARIETIES 7 4. Applications to Toric Varieties In this section, we apply our main theorem to line bundles on an n-dimensional projective toric variety X. Consider a globally generated line bundle on X and its associated lattice polytope P. et r() be the number of integer roots of the Hilbert polynomial h (d) := χ( d ) = n i=1 ( 1)i dim H i (X, d ). Since the higher cohomology of d vanishes and the lattice points in the polytope dp = P d form a basis for H 0 (X, d ), it follows that h (d) equals the Ehrhart polynomial of P ; in other words, h (d) is the number of lattice points in dp. If r(p ) is the largest integer such that r(p )P does not contain any interior lattice points, then Ehrhart reciprocity (Stanley [Sta97, Corollary ]) implies that the integer roots of h (d) are { 1,..., r(p )} and r(p ) = r(). emma 4.1. If is a globally generated line bundle on a toric variety X and r() is the number of integer roots of its Hilbert polynomial h, then deg(h ) r() is O X -regular with respect to. Proof. We must establish that H i (X, deg(h ) r() i ) = 0 for all i > 0. If deg(h ) r() i 0, this follows from the vanishing of the higher cohomology of globally generated line bundles on a complete toric variety; see Fulton [Ful93, 3.5]. When deg(h ) r() i < 0, our argument follows Batyrev and Borisov [BB96, Theorem 2.5]. et X be the toric variety corresponding to the normal fan to P. There is a canonical toric map ψ : X X and an ample line bundle A on X such that H i (X, ) = H i (X, A). A toric version of the Kodaira Vanishing Theorem establishes that H j (X, u ) = 0 for u > 0 and j deg(h ) = dim P = dim X (combine Serre duality in Fulton [Ful93, 4.4] with Mustaţă [Mus02, Theorem 3.4]). In particular, we have H i (X, deg(h ) r() i ) = 0 for i deg(h ). When i = deg(h ), we also have 0 = h ( r() ) = χ( r() ) = ( 1) i dim H i (X, deg(h ) r() i ). Proof of Corollary 1.3. By emma 4.1, we can apply Theorem 1.1 with l = 1, B 1 =, w 1 = max{deg(h ) r(), 1} and w i = 1 for i > 1. Proof of Corollary 1.2. The case p = 0 is established in Ewald and Wessels [EW91]. When p 1, the assertion follows from Corollary 1.3. Proof of Corollary 1.4. Given a lattice polytope P, let X be the corresponding toric variety and the associated ample line bundle on X. Since P is normal if and only if satisfies (N 0 ), the result follows from Corollary 1.3 and the fact that r(p ) = r().

8 8 M. HERING, H. SCHENCK, AND G.G. SMITH Proof of Corollary 1.5. et π i : X P n i ( be the projection onto the i-th factor and let B i := π ) i OP n i(1). If I := { i {1,..., l} : d i 0 }, then we have O X (d 1,..., d l ) = i I Bd i i. et d := min{d i 1 : i I} and let B be the semigroup generated by {B i : i I}. Maclagan and Smith [MS04, Proposition 6.10] prove that O X is O X -regular with respect to B 1,..., B l. Thus, Theorem 2.1 shows that i I Bd i d i is O X -regular with respect to {B i : i I} and lies in i I (B i B). Since we have i I B i j I (B j B), Theorem 1.1 applies with w 1 = (d 1 d,..., d l d) and w j = (1,..., 1) for j 2. et B be the semigroup generated by the nef line bundles. To apply our techniques to adjoint bundles, we need to find u with K X B u B. Inspired by Fujita s conjectures, Fujino [Fuj03, Corollary 0.2] provides the necessary criterion. Theorem 4.2 (Fujino). et X be a projective toric variety (not isomorphic to P n ) such that the canonical divisor K X is Q-Cartier. If D is a Q-Cartier divisor such that D C n for all torus invariant curves C, then K X + D is nef. Proof of Corollary 1.6. If X = P n, then K X = O X ( n 1). Either Corollary 1.3 or Corollary 1.5 show that K X B m n+1+p satisfies (N p ). Mustaţă [Mus02, Theorem 3.4] establishes that K X B m n+1 is O X - regular with respect to B 1,..., B l. For any torus invariant curve C, there is a B i such that B i C > 0. Since B mn = Bi n B where B is globally generated, Theorem 4.2 implies that K X B mn B. It follows that K X B m n+1 l j=1 (B j B) and Theorem 1.1 proves the claim. References [1] V. Batyrev and. Borisov, On Calabi-Yau complete intersections in toric varieties, Higher-dimensional complex varieties (Trento, 1994), 39 65, de Gruyter, Berlin, [2] W. Bruns, J. Gubeladze, and N.V. Trung, Normal polytopes, triangulations, and Koszul algebras, J. Reine Angew. Math. 485 (1997), [3]. Ein and R. azarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993), [4] D. Eisenbud, M.. Green, K. Hulek, and S. Popescu, Restricting linear syzygies: algebra and geometry, arxiv:math.ag/ [5] G. Ewald and U. Wessels, On the ampleness of invertible sheaves in complete projective toric varieties, Results Math. 19 (1991), [6] O. Fujino, Notes on toric varieties from Mori theoretic viewpoint, Tohoku Math. J. (2) 55 (2003), [7] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, NJ, 1993.

9 SYZYGIES, MUTIGRADED REGUARITY AND TORIC VARIETIES 9 [8] F.J. Gallego and B.P. Purnaprajna, Projective normality and syzygies of algebraic surfaces, J. Reine Angew. Math. 506 (1999), [9] M.. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), [10], Koszul cohomology and the geometry of projective varieties II, J. Differential Geom. 20 (1984), [11] M.. Green and R. azarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1985), [12] S.. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), [13] R. Koelman, A criterion for the ideal of a projectively embedded toric surface to be generated by quadrics, Beiträge Algebra Geom. 34 (1993), [14] A. ascoux, Syzygies des variétés déterminantales, Adv. in Math. 30 (1978), [15] R. azarsfeld, A sampling of vector bundle techniques in the study of linear series, ectures on Riemann surfaces, World Scientific Publishing Co., Teaneck, NJ, 1989, pp [16], Positivity in algebraic geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge., 48, Springer Verlag, Berlin, [17] J.Y. iu,.e. Trotter, Jr., and G.M. Ziegler, On the height of the minimal Hilbert basis, Results Math. 23 (1993), [18] D. Maclagan and G.G. Smith, Multigraded Castelnuovo-Mumford regularity, J. Reine Angew. Math. 571 (2004), [19] D. Mumford, Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp [20], ectures on curves on an algebraic surface, Annals of Mathematics Studies 59, Princeton University Press, Princeton, N.J., [21] M. Mustaţă, Vanishing theorems on toric varieties, Tohoku Math. J. (2) 54 (2002), [22] S. Ogata, On quadratic generation of ideals defining projective toric varieties, Kodai Math. J. 26 (2003), [23], On higher syzygies of projective toric varieties, preprint, [24] S. Ogata and K. Nakagawa, On generators of ideals defining projective toric varieties, Manuscripta Math. 108 (2002), [25] G. Pareschi, Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), [26] G. Pareschi and M. Popa, Regularity on abelian varieties. I J. Amer. Math. Soc. 16 (2003), [27] Regularity on abelian varieties. II. Basic results on linear series and defining equations, Journal of Algebraic Geometry 13 (2004), [28] P. Pragacz and J. Weyman, Complexes associated with trace and evaluation. Another approach to ascoux s resolution, Adv. in Math. 57 (1985), [29] E. Rubei, On syzygies of Segre embeddings, Proc. Amer. Math. Soc. 130 (2002), [30], Resolutions of Segre embeddings of projective spaces of any dimension, arxiv:math.ag/

10 10 M. HERING, H. SCHENCK, AND G.G. SMITH [31] R.P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, Cambridge, Department of Mathematics, University of Michigan, 525 East University Avenue, Ann Arbor, Michigan 48109, USA address: Mathematics Department, Texas A&M University, College Station, Texas 77843, USA address: Department of Mathematics and Statistics, Queen s University, Kingston, Ontario, K7 3N6, Canada address: ggsmith@mast.queensu.ca