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arXiv:math/0502240v2 [math.AG] 9 Aug 2006
SYZYGIES, MULTIGRADED REGULARITY AND TORIC VARIETIES
MILENA HERING, HAL SCHENCK, AND GREGORY G. SMITH
Abstract. Using multigraded Castelnuovo-Mumford regularity, we study the equations
defining a projective embedding of a variety X. Given g lobally generated line bundles
B
1
, . . . , B
ℓ
on X and m
1
, . . . , m
ℓ
∈ N, consider the line bundle L := B
m
1
1
⊗ · · · ⊗ B
m
ℓ
ℓ
. We
give conditions on the m
i
which guarantee that the ideal of X in P(H
0
(X, L)
∗
) is generated
by qua drics 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 that cut out a projective variety X and the syzygies among
them is a central problem in algebraic geometry. To give precise statement s, consider the
morphism ϕ
L
: X
-
P
H
0
(X, L)
∗
induced by a globally generated line bundle L on X.
Let S =Sym
•
H
0
(X, L) be the homogeneous coordinate ring of P
H
0
(X, L)
∗
, and let E
•
be
a minimal free gra ded resolution of the graded S-module R =
L
j≥0
H
0
(X, L
j
) associated
to L. Following [GL85], we say that L satisfies (N
p
) for p ∈ N provided that E
0
∼
=
S and
E
i
=
L
S(−i − 1) for all 1 ≤ i ≤ p. Thus, ϕ
L
(X) is projectively normal if and only if L
satisfies (N
0
) and ϕ
L
(X) is normal. If L satisfies (N
1
), then the homogeneous ideal of ϕ
L
(X)
is generated by quadrics and (N
2
) implies that the relations among the generators are linear.
In [Mum70], properties (N
0
) and (N
1
) are called normal generation and normal presentation
respectively.
Although [Gr84b] shows that any sufficiently ample line bundle on an arbitrary variety
satisfies (N
p
), it is normally difficult to determine which multiple of a given ample line bundle
suffices. When X is a smooth curve of genus g, [G r 84a] proves that a line bundle L of degree
at least 2g + 1 + p satisfies (N
p
). This is recovered from a n analgous statement for finite sets
in [GL88]. When X is a smoot h variety of dimension n and L is very ample, [EL93] show
that the adjo int line bundles of the form K
X
+ (n + 1 + p)L satisfy (N
p
). Explicit criteria for
(N
p
) are also given in [GP99, GP01] for surfaces and in [Par00, PP04] f or abelian varieties;
we refer to §1.8.D in [Laz04] for a survey. The primary goal of this paper is to produce
similar conditions for toric varieties.
To achieve this, we use multigraded Castelnuovo-Mumford regularity. Fix a list B
1
, . . . , B
ℓ
of glo bally 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 L is
O
X
-regular (with r espect to B
1
, . . . , B
ℓ
) if H
i
(X, L ⊗ B
−u
) = 0 for all i > 0 and all u ∈ N
ℓ
with |u| := u
1
+ · · · + u
ℓ
= i. Our main technical result is the following.
Theorem 1.1. Let w
1
, w
2
, w
3
, . . . be a sequence in N
ℓ
such that B
w
i
⊗B
−1
j
∈ B for 1 ≤ j ≤ ℓ
and set m
i
:= w
1
+ · · · + w
i
for i ≥ 1. If B
m
1
is O
X
-reg ular then B
m
p
satisfies (N
p
) for
p ≥ 1.
2000 Mathematics Subject Classification. Primary 14 M25; Sec ondary 13D02, 14C20, 52B20.
Key words and phrases. toric variety, syzygy, Castelnuovo-Mumford re gularity.
1
2 M. HERING, H. SCHENCK, AND G.G. SMITH
The case ℓ = 1 is Theorem 1.3 in [GP99]. Our proof is a multigraded variant of t heir
arguments. Applying Theorem 1 .1 with ℓ = 1 to line bundles on t oric varieties yields the
following.
Corollary 1.2. Let L be an ample line bundle on an n-dimensional toric variety. If we h ave
d ≥ n − 1 + p then the line bundle L
d
satisfies (N
p
).
The case p = 0, an ingredient in our proof, was established in [EW91]; other proofs appear
in [LTZ93, BGT97, ON02]. On a toric surface, [Koe93] proves that L satisfies (N
1
) if the
associated lattice polygon contains more than three lattice points in its boundary. [GP01]
gives a criteria for (N
p
) on smoo th rational surfaces which, when restricted to to r ic surfaces,
shows that L satisfies (N
p
) if the associated polygon conta ins at least p+3 lattice points in its
boundary. This result extends to all toric surfaces and is sharp; see [Sch04]. Related results
on toric surfaces appear in [Fak02] which studies multiplication maps and in [Har97] which
studies (N
0
) for anticanonical rational surfaces. For an arbitrary toric variety, [BGT97]
shows that R is Koszul when d ≥ n and this implies that L
d
satisfies (N
1
) when d ≥ n.
Assuming n ≥ 3, Ogata establishes in [Oga03] that L
n−1
satisfies (N
1
) and, building on this
in [Oga04], he proves that L
n−2+p
satisfies (N
p
) when n ≥ 3 and p ≥ 1.
We can strengthen Corollary 1.2 by using additional invariants. Let
h
L
(d) := χ(L
d
) =
n
P
i=0
(−1)
i
dim H
i
(X, L
d
)
be the Hilbert polynomial of L and let r(L) be the number of integer roots of h
L
.
Corollary 1.3. Let L be a globally genera ted line bundle on a toric variety and let r(L) be
the number of integer roots of its Hilbert polynomial h
L
. I f p ≥ 1 and
d ≥ max{deg(h
L
) − r(L) + p − 1, p}
then the l i ne bundle L
d
satisfies (N
p
).
If X = P
n
and L = O
X
(1), then we have h
L
(d) =
d+n
n
and r(L) = n. In particular, we
recover Theorem 2.2 in [Gr84b] which states that O
P
n
(d) satisfies (N
p
) for p ≤ d. On the
other hand, Theorem 2.1 in [OP01] shows that, for n ≥ 2 and d ≥ 3, O
P
n
(d) does not satisfy
(N
3d−2
) and it is conjectured that this is sharp.
Using the dictionary between lattice polytopes and line bundles on toric varieties, Corol-
lary 1.3 yields a normality criterion f or lattice polytopes. A lattice polytope P is normal
if every lattice point in mP for m ≥ 1 is a sum of m lattice points in P . Let 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 la ttice polytope of dimension n, then
n − r(P )
P is normal.
Theorem 1 .1 also applies to syzygies of Segre-Veronese embeddings.
Corollary 1.5. If X = P
n
1
× · · · × P
n
ℓ
then the line bundl e O
X
(d
1
, . . . , d
ℓ
) satisfies (N
p
) for
p ≤ min{d
i
: d
i
6= 0}.
The Segre embedding O
X
(1, . . . , 1) satisfies (N
p
) if and only if p ≤ 3; see [Las78, PW85]
for ℓ = 2, and [Rub02, Rub04] for ℓ > 2. An overview of results and conjectures about the
syzygies of Segre-Veronese embeddings appears in §3 of [EGHP05].
Inspired by [EL93], 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.
SYZYGIES, MULTIGRADED REGULARITY AND TORIC VARIETIES 3
Corollary 1.6. Let X be a projective n-dimensional Gorenstein toric va riety and assume
that B
1
, . . . , B
ℓ
are the minimal generators of Nef(X). Suppose w
1
, w
2
, . . . i s a sequence in
N
ℓ
such that B
w
i
⊗ B
−1
j
∈ B for 1 ≤ j ≤ ℓ and m
i
:= w
1
+ · · · + w
i
for i ≥ 1. If X 6= P
n
and p ≥ 1, then the adjoint line bundl e K
X
⊗ B
m
n+p
satisfies (N
p
). If X = P
n
and p ≥ 1,
then K
X
⊗ B
m
n+1+p
satisfies (N
p
).
[EL93] proves that for a very ample line bundle L and a nef line bundle N on a smoo th
n-dimensional algebraic variety X 6= P
n
, K
X
⊗ L
n+p
⊗ N satisfies (N
p
). Corollary 1.6
gives a similar result for ample line bundles on po ssibly singular Gorenstein toric varieties.
Specifically, if L is an ample line bundle such that L ⊗ B
−1
j
∈ B for 1 ≤ j ≤ ℓ and N is a nef
line bundle on X 6= P
n
then K
X
⊗ L
n+p
⊗ N satisfies (N
p
). For an ample line bundle L on
a ruled variety X, §5 in [But94] demonstrates that K
X
⊗ L
n+1+p
need not satisfy (N
p
) for
p = 0 and 1. Hence, the conclusions of Corollary 1.6 are stronger than one can expect for
a general variety. The proof of Corollary 1.6 combines Theorem 1.1 with Fujita’s Freeness
conjecture for toric varieties, see [Fuj03].
Conventions. We work over a field of characteristic zero and N denotes the nonnegative
integers.
Acknowledgements. We thank A. Bayer, W. Fulton, B. Harbourne, R. Lazarsfeld,
M. Mustat¸˘a and S. Payne for helpful discussions. Parts of this work were done while the
last two authors were visiting the Mathematical Sciences Research Institute in Berkeley
and the Mathematisches Forschungsinstitut in Oberwolfach. The second author was
partially supported by NSF Grant DMS 0 3–11142 and the third author was partially
suppo r t ed by NSERC.
2. Multigraded Castelnuovo-Mumford Regularity
This section reviews multigraded regularity as introduced in [MS04]. Fix a list 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). If e
1
, . . . , e
ℓ
is the standard basis for
Z
ℓ
, then B
e
j
= B
j
.
Let F be a coherent O
X
-module and let L be a line bundle on X. We say that F is
L-regular (with respect to B
1
, . . . , B
ℓ
) provided H
i
(X, F ⊗ L ⊗ B
−u
) = 0 fo r all i > 0 and
all u ∈ N
ℓ
satisfying |u| := u
1
+ · · · + u
ℓ
= i. When X = P
n
, this definition specializes to
Mumford’s version of regularity (see [Mum66]) and as Mumford says, “this apparently silly
definition reveals itself as follows.”
Theorem 2.1. If the coherent sheaf F is L-regular then for all u ∈ N
ℓ
:
(1) F is (L ⊗ B
u
)-regular;
(2) the m ap H
0
(X, F ⊗ L ⊗ B
u
) ⊗ H
0
(X, B
v
)
-
H
0
(X, F ⊗ L ⊗ B
u+v
) is surjective for
all v ∈ N
ℓ
;
(3) F ⊗L ⊗B
u
is globally gene rated provided there exists w ∈ N
ℓ
such that B
w
is ample.
When X is a toric variety, this follows from results in §6 of [MS04]. Our approach imitates
the proofs of Theorem 2 in [Mum70] and Proposition II.1.1. in [Kle66].
Proof. By replacing F with F ⊗L, 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 basepoint-free, we may choose a section s
j
∈ H
0
(X, B
j
) such that the induced
4 M. HERING, H. SCHENCK, AND G.G. SMITH
map F ⊗ B
−e
j
-
F is injective (see page 43 in [Mum70]). If G
j
is the cokernel of this map,
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
)
-
.
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) fo llows.
For (2), consider the commutative diagram:
H
0
(X, F ) ⊗ H
0
(X, B
j
)
-
H
0
(X, G
j
) ⊗ H
0
(X, B
j
)
0
-
H
0
(X, F )
-
-
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
guarant ees that the map in the right column is surjective. Thus, the Snake Lemma (e.g.
Prop. 1.2 in [Bou80]) implies that the map in the middle column is also surjective. Therefore,
(2) fo llows from the associativity of the tensor product and (1).
Lastly, 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
-
F ⊗ B
u+v
β
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 (e.g.
Theorem 1.2.6 in [Laz04]) implies that β
u+v
is surjective for k ≫ 0. Hence, β
u
is also
surjective which proves (3).
We end this section with an elementar y observation.
Lemma 2.2. Let 0
-
F
′
-
F
-
F
′′
-
0 be a short ex a ct sequence of coherent
O
X
-modules. If F is L-regular, H
0
(X, F ⊗L⊗B
−e
j
)
-
H
0
(X, F
′′
⊗L⊗B
−e
j
) is surjective
for all 1 ≤ j ≤ ℓ, and F
′′
is (L ⊗ B
−e
j
)-regular for all 1 ≤ j ≤ ℓ, then F
′
is also L-regular.
Sketch of Proof. Tensor the exact sequence 0
-
F
′
-
F
-
F
′′
-
0 with L ⊗ B
u
and analyze the associated long exact sequence. The argument is similar to the proof of
Theorem 2 .1.1.
3. Proof of Main Theorem
The proof of Theorem 1.1 combines multigraded Castelnuovo-Mumford regularity with a
cohomological criterion for (N
p
). Given a globally generated line bundle L on X, there is a
natural surjective map ev
L
: H
0
(X, L) ⊗ O
X
-
L and we set M
L
:= Ker(ev
L
). Hence, M
L
is a vector bundle on X which sits in the short exact sequence
(†) 0
-
M
L
-
H
0
(X, L) ⊗ O
X
-
L
-
0 .
It is well-known that M
L
governs the syzygies of ϕ
L
(X) in P
H
0
(X, L)
∗
. Specifically, L
satisfies (N
p
) if and only if H
1
(X,
V
q
M
L
⊗ L
j
) = 0 for q ≤ p + 1 and j ≥ 1; see Lemma 1.10
in [GL88] or Proposition 1.3.3 in [Laz89]. In characteristic zero,
V
k
M
L
is a direct summand
SYZYGIES, MULTIGRADED REGULARITY AND TORIC VARIETIES 5
of M
⊗k
L
, so it suffices to show that H
1
(X, M
⊗q
L
⊗ L
j
) = 0 for q ≤ p + 1 and j ≥ 1 in our
situation.
Proof of Theorem 1.1. Set L := B
m
p
and let M
L
be the vector bundle in (†). We first prove,
by induction on q, that M
⊗q
L
is (B
m
q
)-regular for a ll q ≥ 1. Since B
m
1
is O
X
-regular,
Theorem 2.1.2 implies that H
0
(X, B
m
1
+u
) ⊗ H
0
(X, B
v
)
-
H
0
(X, B
m
1
+u+v
) is surjective
for all u, v ∈ N
ℓ
. In particular, the maps H
0
(X, L) ⊗ H
0
(X, B
m
1
−e
j
)
-
H
0
(X, L ⊗ B
m
1
−e
j
)
for 1 ≤ j ≤ ℓ are surjective because B
m
1
∈
T
ℓ
j=1
(B
j
⊗ B). Combining Theorem 2.1.1
and Lemma 2.2, we see that M
L
is (B
m
1
)-regular. For q > 1 , tensor the sequence (†) with
M
⊗(q−1)
L
to obtain the exact sequence 0
-
M
⊗q
L
-
H
0
(X, L)⊗M
⊗(q−1)
L
-
M
⊗(q−1)
L
⊗L
-
0.
The induction hypothesis states that M
⊗(q−1)
L
is (B
m
q−1
)-regular. Since B
w
q
⊗ B
−1
j
∈ B for
all 1 ≤ j ≤ ℓ, Theorem 2.1.2 shows that
H
0
(X, M
⊗(q−1)
L
⊗ B
m
q
−e
j
) ⊗ H
0
(X, L)
-
H
0
(X, M
⊗(q−1)
L
⊗ L ⊗ B
m
q
−e
j
)
is surjective for 1 ≤ j ≤ ℓ. Again by Theorem 2.1.1 and Lemma 2.2, M
q
L
is (B
m
q
)-regular.
As observed above, it suffices to prove that H
1
(X, M
⊗q
L
⊗ L
j
) = 0 for q ≤ p + 1 and
j ≥ 1. Since M
⊗q
L
is (B
m
q
)-regular, Theorem 2.1.1 implies that M
⊗q
L
is (B
m
p
)-regular for
1 ≤ q ≤ p; as O
X
is (B
m
1
)-regular, Theorem 2.1.1 also implies that O
X
is (B
m
p
)-regular.
It follows that H
1
(X, M
⊗q
L
⊗ L
j
) = 0 for q ≤ p and j ≥ 1. Moreover, Theorem 2.1.2
shows that H
0
(X, L) ⊗ H
0
(X, M
⊗p
L
⊗ L
j
)
-
H
0
(X, M
⊗p
L
⊗ L
j+1
) is surjective and the exact
sequence 0
-
M
⊗(p+1)
L
⊗ L
j
-
H
0
(X, L) ⊗ M
⊗p
L
⊗ L
j
-
M
⊗p
L
⊗ L
j+1
-
0 implies that
H
1
(X, M
⊗(p+1)
L
⊗ L
j
) = 0 for j ≥ 1.
4. Applications to Toric Varieties
In this section, we apply the main t heorem to line bundles on an n-dimensional projective
toric var iety X. Consider a globally generated line bundle L on X and its associated lattice
polytope P
L
. Let r(L) be the number of integer roots of the Hilbert polynomial h
L
(d) =
χ(L
d
). Since the higher cohomology of L
d
vanishes and the lattice points in the polytope
dP
L
= P
L
d form a basis for H
0
(X, L
d
), it follows that h
L
(d) equals the Ehrhart po lynomial of
P
L
. In other words, h
L
(d) is the number of lattice points in dP . If r(P
L
) is the largest integer
such that r(P
L
)P
L
does not contain any interior lattice points, then Ehrhart reciprocity (e.g.
Corollary 4.6.28 in [Sta97]) implies that the integer roots o f h
L
(d) are {−1, . . . , −r(P
L
)} and
r(P
L
) = r(L).
Lemma 4.1. If L is a globally generated line bundle on a toric variety X and r(L) is the
number of integer roots of its Hilbert polynomial h
L
, then L
deg(h
L
)−r(L)
is O
X
-reg ular with
re spect to L.
Proof. We must prove that H
i
(X, L
deg(h
L
)−r(L)−i
) = 0 for all i > 0. If deg(h
L
) −r(L) −i ≥ 0,
this follows from the vanishing of the higher cohomology of globally generated line bundles
on a complete toric variety; see §3.5 of [Ful93]. When deg(h
L
) − r(L) − i < 0, we follow the
proof of Theorem 2.5 in [BB96]. Let X
′
be the to ric variety corresponding to the normal
fan to P
L
. There is a canonical to r ic map ψ : X
-
X
′
and an ample line bundle A on X
′
such that H
i
(X, L
r
)
∼
=
H
i
(X
′
, A
r
) for all r ∈ Z. A toric version of the Kodaira Vanishing
Theorem establishes that H
j
(X, L
−u
) = 0 for u > 0 and j 6= deg(h
L
) = dim P
L
= dim X
′
(combine Serre duality from §4.4 o f [Ful93] with Theorem 3.4 in [Mus02]). In particular,
6 M. HERING, H. SCHENCK, AND G.G. SMITH
we have H
i
(X, L
deg(h
L
)−r(L)−i
) = 0 for i 6= deg(h
L
). When i = deg(h
L
), we also have
0 = h
L
−r(L)
= χ(L
−r(L)
) = (−1)
i
dim H
i
(X, L
deg(h
L
)−r(L)−i
).
Proof of Corollary 1.3. In light of Lemma 4.1, the claim is the special case of Theorem 1.1
with ℓ = 1, B
1
= L, w
1
= max{deg(h
L
) − r(L), 1} and w
i
= 1 fo r i > 1.
Proof of Corollary 1.2. The case p = 0 is in [EW91]; the case p ≥ 1 f ollows from Corol-
lary 1.3.
The following well-known examples illustrate that Corollary 1.3 is sharp in some cases.
Example 4.2. Let L be the ample line bundle on the toric variety X corresponding to the
polytope conv{(1, 0), (0, 1), (1, 1), (2, 2)} ⊂ R
2
. The ideal of X in P
3
= P
H
0
(X, L)
∗
is
generated by the cubic x
3
2
− x
0
x
1
x
3
which implies that L does not satisfy (N
1
). Calculations
in [M2] show that L
2
satisfies (N
3
) but not (N
4
).
Example 4.3. Let e
1
, . . . , e
n
be the standard basis of R
n
and let L be the ample line bundle
on the toric variety X corresponding to
P = conv{0, e
1
, . . . , e
n−1
, e
1
+ · · · + e
n−1
+ (n − 1)e
n
} ⊂ R
n
.
Since X is n-dimensional and singular, the morphism X
-
P
n
= P
H
0
(X, L)
∗
is obviously
not an embedding. The natural map S
-
R is not surjective and (n − 2 )P is not normal
because (1, . . . , 1) lies in 2(n − 2)P but cannot be written as an integral combination of two
lattice points in (n − 2)P . For n = 3, calculations in [M2] show that L
2
satisfies (N
1
) but
not (N
2
).
Proof of Corollary 1.4. Given a lattice polytope P , let X be the corresponding toric variety
and L the associated ample line bundle on X. Since P is normal if and only if L satisfies
(N
0
), the result follows from Corollary 1.3 and the fact that r(P ) = r(L).
Proof of Corollary 1.5. Let π
i
: X
-
P
n
i
be the i-th projection and set B
i
:= π
∗
i
O
P
n
i
(1)
.
If I :=
i ∈ {1, . . . , ℓ} : d
i
6= 0
, then O
X
(d
1
, . . . , d
ℓ
)
∼
=
N
i∈I
B
d
i
i
. Let B be the semigroup
generated by {B
i
: i ∈ I} and let d := min{d
i
− 1 : i ∈ I}. Proposition 6.10 in [MS04]
proves that O
X
is O
X
-regular with respect to B
1
, . . . , B
ℓ
. Thus, Theorem 2.1 shows that
N
i∈I
B
d
i
−d
i
is O
X
-regular with respect to {B
i
: i ∈ I} and lies in
T
i∈I
(B
i
⊗ B). Since we
have
N
i∈I
B
i
∈
T
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.
Now assume that B
1
, . . . , B
ℓ
are the minimal generators of Nef(X). To apply our tech-
niques to adjoint bundles, we need to find u with K
X
⊗ B
u
∈ B = Nef(X). Inspired by
Fujita’s conjectures, Corollary 0.2 in [Fuj03] provides the fo llowing necessary criterion: Let
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); Corollary 1.3 proves that
K
X
⊗ B
m
n+1+p
satisfies (N
p
). Theorem 3.4 in [Mus02] shows 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
m
n
= B
n
i
⊗ B
′
where B
′
is globally generated, Corollary 0.2 in [Fuj03] implies that
K
X
⊗ B
m
n
∈ B. It follows that K
X
⊗ B
m
n+1
∈
T
ℓ
j=1
(B
j
⊗ B) and Theorem 1.1 proves the
claim.
SYZYGIES, MULTIGRADED REGULARITY AND TORIC VARIETIES 7
The singular cubic surface in Example 4.2 also demonstrates that Corollar y 1.6 can be
sharp; see [GP99] for more examples of this type.
Example 4.4. Let (X, L) be the normal cubic surface and ample line bundle defined in Ex-
ample 4.2. It follows that K
−1
X
= L and L is the minimal generator of Nef(X). Example 4.2
shows that K
X
⊗ L
2
= L does not satisfy (N
1
). Hence, Corollary 1.6 provides the smallest
m ∈ N (namely m = 3) such that K
X
⊗ L
m
satisfies (N
1
).
For toric surfaces, it follows from [Sch04] that all of our corollaries are not optimal for
p ≥ 2. Specifically, given an ample line bundle L 6
∼
=
O
P
2
(1) on a Gorenstein toric surface X,
K
X
⊗ L
m
satisfies N
3(m−2)
for m ≥ 2 and m 6= 4, and K
X
⊗ L
4
satisfies N
5
.
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E-mail address: mhering@umich.edu
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA
E-mail address: schenck@math.tamu.edu
Mathematics Department, Texas A&M University, College Station, Texas 77843, USA
E-mail address: ggsmith@mast.queensu.ca
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L
3N6, Canada
