Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points
We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each -trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the 3 case, we extend recent constructions and results of Bini, Bo...
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| Цитувати: | Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points. Dragos Oprea. SIGMA 18 (2022), 061, 21 pages |
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| author | Oprea, Dragos |
| author_facet | Oprea, Dragos |
| citation_txt | Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points. Dragos Oprea. SIGMA 18 (2022), 061, 21 pages |
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| description | We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each -trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the 3 case, we extend recent constructions and results of Bini, Boissière, and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the -trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of 3s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 061, 21 pages
Big and Nef Tautological Vector Bundles
over the Hilbert Scheme of Points
Dragos OPREA
Department of Mathematics, University of California San Diego,
9500 Gilman Drive, La Jolla, CA, USA
E-mail: doprea@math.ucsd.edu
URL: http://math.ucsd.edu/~doprea/
Received January 31, 2022, in final form July 31, 2022; Published online August 12, 2022
https://doi.org/10.3842/SIGMA.2022.061
Abstract. We study tautological vector bundles over the Hilbert scheme of points on
surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the
tautological bundles are big and nef, and illustrate it by examples. In the K3 case, we
extend recent constructions and results of Bini, Boissière and Flamini from the Hilbert
scheme of 2 and 3 points to an arbitrary number of points. Among the K-trivial surfaces,
the case of Enriques surfaces is the most involved. Our techniques apply to other smooth
projective surfaces, including blowups of K3s and minimal surfaces of general type, as well
as to the punctual Quot schemes of curves.
Key words: Hilbert scheme; Quot scheme; tautological bundles
2020 Mathematics Subject Classification: 14C05; 14D20; 14C17
1 Introduction
Let X be a smooth projective surface, and let X [k] denote the Hilbert scheme of k points on X.
Each vector bundle F → X of rank r yields a tautological vector bundle F [k] → X [k] of rank rk
given by
F [k] = p⋆(q
⋆F ⊗OZ).
Here, p, q are the natural projections from X [k] ×X, and Z ⊂ X [k] ×X denotes the universal
subscheme.
The literature surrounding the geometry of the tautological bundles is vast. Likewise, many
notions of positivity for vector bundles have been studied in algebraic and complex differential
geometry. Merging these two themes, it is natural to investigate the positivity properties of the
tautological bundles.
In this note, we address the question whether the bundles F [k] → X [k] are big and nef. To our
knowledge, for K3s, this has been considered for the first time in the recent article [7], alongside
the stability and bigness of twists of the tangent bundle of X [k]. Specifically, if X a K3 surface
of Picard rank 1, and the number of points is k = 2, 3, it is shown in [7] that F [k] is big and nef
when F is either
(a) a positive line bundle,
(b) a twist of a Lazarsfeld–Mukai bundle (for suitable numerics),
(c) a twist of an Ulrich bundle.
This paper is a contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants
in honor of Lothar Göttsche on the occasion of his 60th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Gottsche.html
mailto:doprea@math.ucsd.edu
http://math.ucsd.edu/~doprea/
https://doi.org/10.3842/SIGMA.2022.061
https://www.emis.de/journals/SIGMA/Gottsche.html
2 D. Oprea
Recall that a vector bundle V over a scheme Y is said to be big and nef if the line bundle
OP(V )(1) → P(V ) is big and nef, where P(V ) denotes the projective bundle of one dimensional
quotients. A discussion of big and nef vector bundles can be found in [21, Chapters 6 and 7].
A useful well-known characterization occurs when V → Y is globally generated. In this case, if
the top Segre class
(−1)dimY
∫
Y
s(V ) > 0 (1.1)
it follows that V → Y is big and nef.
1.1 Results
The original motivation for our work was provided by the recent results of [7], which we extend
in several directions.
(i) ForK3s, we allow for arbitrary number of points k, and derive a condition that ensures F [k]
is big and nef, see Theorem 2.1. We apply this theorem to obtain analogues of examples
(a)–(c) above for any k.
(ii) We allow for arbitraryK-trivial surfaces. The case of Enriques surfaces is the most difficult,
and we only have results in odd rank, see Theorem 3.6, as well as a conjectural bound in
general.
(iii) We consider other smooth projective surfaces, including blowups of K3s in Theorem 4.3,
and minimal surfaces of general type, in rank 1, in Theorem 4.4. The latter theorem is
the most involved result we prove here, requiring a more detailed analysis than for other
geometries.
(iv) We show how the same ideas yield similar results over the punctual Quot schemes of curves,
see Theorem 5.2.
Compared to [7], the new ingredient is the closed form calculation of the Segre integrals in
[25, 26, 27, 33]. The formulas are explicit, and the goal here is to show how to apply them to
derive geometric positivity results. This is not always immediate, and the arguments require
several different ideas. We thus believe it is worthwhile to record the outcome. We also illustrate
our calculations by a few geometric examples.
1.2 Applications
By [40, Proposition 1.4], taking determinants of big and nef vector bundles yields big and nef
divisors. There are several results in the literature concerning the positivity of the determinants
detF [k], see for instance [5, 10] with regards to very ampleness when F has rank 1, for arbitrary
surfaces. In general, nef divisors over the Hilbert scheme of K3s were studied in [4, Section 10].
Over other surfaces, related results can be found in [3, 6, 9, 19, 23, 24, 30, 34, 39], among
others. The nef cones of divisors of the punctual Quot schemes of curves of genus 0 and 1 were
determined in [15, 36].
Whenever F [k] → X [k] is a big and nef bundle, for every m1, . . . ,mh ≥ 0, Demailly vanishing
gives1
H i
(
X [k], ωX[k] ⊗ Symm1F [k] ⊗ · · · ⊗ SymmhF [k] ⊗
(
detF [k]
)⊗h)
= 0, i > 0.
1For an ample vector bundle V → Y , Demailly’s vanishing theorem [13] states
Hi(Y, ωY ⊗ Symm1V ⊗ · · · ⊗ SymmhV ⊗ (detV )⊗h) = 0, m1, . . . ,mh ≥ 0, h > 0, i > 0.
This is derived in [21, Theorem 7.3.14] from Griffiths vanishing. The same argument applies to big and nef vector
bundles; [21, Example 7.3.3] notes that Griffiths vanishing holds in this context.
Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points 3
For instance, using Theorem 2.1 or Corollary 2.4, if L → X is an ample line bundle on a K3
surface X of Picard rank 1, with χ(L) ≥ 3k, we have
H i
(
X [k], Symm1L[k] ⊗ · · · ⊗ SymmhL[k] ⊗
(
detL[k]
)⊗h)
= 0, i > 0.
Analogous statements hold in all geometric situations covered by items (i)–(iv) above. Coho-
mology with values in the tautological bundles and their representations was studied for instance
in [2, 12, 14, 35, 41], but the vanishings results above are new.
To further understand the cohomology, the next step would be to compute the holomorphic
Euler characteristics, that is to find the series
Zm1,...,mh
X,F =
∞∑
k=0
qkχ
(
X [k], ωX[k] ⊗ Symm1F [k] ⊗ · · · ⊗ SymmhF [k] ⊗
(
detF [k]
)⊗h)
.
This is a difficult but interesting question. We expect that the answer is given by algebraic
functions. The simplest case m1 = · · · = mh = 0 corresponds to the Verlinde series determined
in [14, Theorem 5.3] for K-trivial surfaces, and conjectured for small h for all surfaces in [27,
Section 1.6]. (After the writing was completed, we learned about the recent announcement [16]
regarding expressions for the Verlinde series for all surfaces and all values of h.)
Similar vanishing statements can be made over the punctual Quot schemes of curves.
1.3 Plan of the paper
The case of K3 surfaces is the simplest and is discussed first, see Section 2. To illustrate the
results of Section 2, in Section 2.1 we extend the constructions in [7] to arbitrary number of
points. Other K-trivial surfaces, and in particular Enriques surfaces, are considered in Section 3.
Other geometries, specifically K3 blowups and minimal surfaces of general type, are studied in
Section 4. Sections 3 and 4 are the most involved. Finally, Section 5 concerns the punctual
Quot scheme of curves.
2 K3 surfaces
Let X be a smooth projective surface. The bundle F [k] → X [k] is globally generated, and
therefore nef, provided F → X is (k − 1)-very ample. By definition, (k − 1)-very ampleness is
the requirement that the natural map
H0(X,F ) → H0(X,F ⊗Oζ)
is surjective for all zero-dimensional subschemes ζ of X of length k. Thus, via (1.1), if F is
(k − 1)-very ample and∫
X[k]
s
(
F [k]
)
> 0,
then F [k] → X [k] is big and nef. This is explained for instance in [7, Propositions 2.4 and 4.5].
Let (X,H) be a polarized K3 surface, and let r = rankF . Central for our argument is the
following structural expression for the Segre integrals established in [27]:
∞∑
k=0
zk
∫
X[k]
s
(
F [k]
)
= A0(z)
c2(F )A1(z)
c21(F )A2(z). (2.1)
4 D. Oprea
The series A0, A1, and A2 are given by explicit algebraic functions
A0(z) = (1 + (1 + r)t)−r−1(1 + (2 + r)t)r,
A1(z) = (1 + (1 + r)t)
r
2 (1 + (2 + r)t)−
r−1
2 ,
A2(z) = (1 + (1 + r)t)r
2+2r(1 + (2 + r)t)−r2+1(1 + (1 + r)(2 + r)t)−1, (2.2)
for the change of variables
z = t(1 + (1 + r)t)1+r.
We point out that in [27], r stands for rank(F ) + 1, while for us r = rank(F ); the above
expressions account for the different notational conventions. Related formulas over the moduli
space of higher rank sheaves were proposed in [17] and were recently proven in [31].
For each rank r vector bundle F → X, we write v = v(F ) = ch(F )
√
Td(X) for its Mukai
vector, and we set
χ = χ(F ), δ = 1 +
1
2
⟨v, v⟩.
Here ⟨ , ⟩ is the Mukai pairing given by
⟨v, v⟩ =
∫
X
v22 − 2v0v4 for vectors v = (v0, v2, v4) ∈ H0(X)⊕H2(X)⊕H4(X).
Moreover, δ equals the expected dimension of the moduli space of sheaves of type v. We show
Theorem 2.1. Assume χ ≥ (r + 2)k and δ ≥ 0. If F is (k − 1)-very ample, then F [k] is a big
and nef vector bundle over X [k].
Proof. By the first paragraph of this section, it suffices to show that the integral of the Segre
class s2k
(
F [k]
)
is positive. To this end, we use an equivalent form of equation (2.1), which can
be found in [27, p. 11]. Specifically, via a residue calculation, it was established there that∫
X[k]
s2k
(
F [k]
)
= Coefftk
[
(1 + (2 + r)t)δ(1 + (1 + r)t)χ−δ−(r+1)k
]
. (2.3)
Expanding via the binomial theorem, equation (2.3) shows that the Segre integral is positive
when the expression between brackets is a polynomial of degree at least k, which is the case for
δ ≥ 0, χ− δ − (r + 1)k ≥ 0, χ− (r + 1)k ≥ k.
However, since δ could be large, we seek better bounds. To this end, we change variables, setting
t =
u
1− (1 + r)u
.
We rewrite (2.3) as∫
X[k]
s2k
(
F [k]
)
= Rest=0(1 + (2 + r)t)δ(1 + (1 + r)t)χ−δ−(r+1)k dt
tk+1
= Resu=0(1 + u)δ(1− (1 + r)u)−χ+(r+2)k−1 du
uk+1
= Coeffuk(1 + u)δ(1− (1 + r)u)−χ+(r+2)k−1.
Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points 5
Letting ai denote the coefficients of the term (1 + u)δ, we have ai > 0 for 0 ≤ i ≤ δ. Similarly,
the coefficients of the second term are
bj = (1 + r)j(−1)j
(
−χ+ (r + 2)k − 1
j
)
> 0,
since −χ+ (r + 2)k − 1 < 0. Here, we use the standard definition of binomial numbers(
x
j
)
=
x(x− 1) · · · (x− j + 1)
j!
for arbitrary x. Thus, the Segre integral equals
∑
aibj , the sum ranging over i+j = k, 0 ≤ i ≤ δ,
j ≥ 0. The integral is positive since each term aibj > 0, and the sum is non-empty (it contains
the term a0bk). The proof is complete. ■
Remark 2.2. The theorem is certainly not optimal in all cases, but it suffices for our purposes.
To illustrate it, when δ = 0, equation (2.3) yields the following result originally noted in [27,
Proposition 2.1]:∫
X[k]
s2k
(
F [k]
)
= (r + 1)k
(
χ− (r + 1)k
k
)
. (2.4)
In this case, the positivity of the Segre integral is guaranteed when χ ≥ (r+2)k, but also when
χ < (r + 1)k and k even.
The vanishing of the Segre integrals (2.4) for exceptional bundles with (r+1)k ≤ χ < (r+2)k
played an important role in the proof of (2.2) in [27]. The point of Theorem 2.1 is that we can
furthermore pin down the sign of the Segre integral for χ to the right of the above interval.
Remark 2.3. In rank 1, (k − 1)-very ampleness of nef line bundles over K3 surfaces can be
effectively studied using [5, Theorem 2.1]. Specifically, if L is nef and L2 > 4k, then either L
is (k − 1)-very ample or else there exists an effective divisor D such that L− 2D is Q-effective,
with
L.D − k ≤ D2 < L.D/2 < k. (2.5)
Furthermore, D contains a subscheme ζ of length less or equal to k such that
H0(L) → H0(L|ζ) is not surjective.
Over arbitrary smooth projective surfaces, a similar result ensures the (k − 1)-very ampleness
of the adjoint bundles KX + L.
To our knowledge, an analogous criterion in higher rank is missing. We point out two con-
structions yielding (k − 1)-very ample bundles over K3 surfaces:
(i) If (X,H) satisfies Pic(X) = Z⟨H⟩ and F is a µH -stable vector bundle with detF = H and
χ ≥ (r + 1)k + δ then F is (k − 1)-very ample. This assertion follows by the proof of [27,
Proposition 2.2].
(ii) By [7, Proposition 4.5], over any surface, twisting a globally generated vector bundle by
a (k−1)-very ample line bundle yields a (k−1)-very ample bundle (with large determinant).
For abelian surfaces, other constructions are possible via isogenies or extensions, see Section 3.1.
6 D. Oprea
2.1 Big and nef tautological bundles over the Hilbert scheme of K3s
Theorem 2.1 applies to the three examples considered in Theorems 5.3, 5.5 and 5.7 and Corol-
laries 5.4, 5.6 and 5.8 of [7], and mentioned in items (a)–(c) of the Introduction: line bundles
and twists of Lazarsfeld–Mukai or Ulrich bundles. The goal here is to show how to extend the
results in [7] from k ≤ 3 to any number of points.
Throughout this section, we assume (X,H) is a K3 surface of Picard rank 1, and Pic(X) =
Z⟨H⟩. Let H2 = 2g − 2.
Corollary 2.4. Let Ln = H⊗n for n ≥ 1. Assume g ≥ 3k − 1. Then (Ln)
[k] is big and nef
over X [k].
Proof. Global generation, and thus nefness, is explained in [7, Theorem 5.3]. To prove bigness,
as also noted in [7], it suffices to establish the positivity of the Segre integral∫
X[k]
s2k
(
(Ln)
[k]
)
> 0.
By formula (2.4), we have∫
X[k]
s2k
(
(Ln)
[k]
)
= 2k
(
χ(Ln)− 2k
k
)
,
which is positive provided
χ(Ln) ≥ 3k ⇐⇒ 2 + n2(g − 1) ≥ 3k.
The latter inequality is clear under our hypothesis. ■
We next consider Lazarsfeld–Mukai bundles and their induced tautological bundles over X [k].
This geometric situation corresponds to Theorem 5.5 and Corollary 5.6 in [7]. There are several
ways of formulating the result, but in keeping with [7], we prefer bounds which do not depend
on the rank.
Recall that the Lazarsfeld–Mukai bundles are obtained as duals E = K∨
C,L to kernels
0 → KC,L → H0(X,L)⊗OX → ι⋆L → 0.
Here L → C is a line bundle over a nonsingular curve C ∈ |H|, of degree d and with r ≥ 2
sections, such that L and ωC⊗L∨ are globally generated. It follows that E is globally generated,
with
rkE = r ≥ 2, c1(E) = H, c2(E) = d =⇒ v(E) = (r,H, g − 1− d+ r).
We let ρ = g − r(r − 1 + g − d) denote the Brill–Noether number.
Example 2.5. Let E be a globally generated Lazarsfeld–Mukai bundle as above. Assume that
ρ ≥ 0, g > 2k − 2 > 0, g >
2
5
(d+ 1).
Then (E ⊗H)[k] is big and nef.
Proof. Let F = E⊗H. The assumption g > 2k− 2 was used in [7, Theorem 5.5] to prove that
F = E ⊗H is (k − 1)-very ample; this is based on the result cited in Remark 2.3(ii). As noted
Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points 7
in [7], bigness follows once we verify that
∫
X[k] s2k
(
F [k]
)
> 0. To this end, we check that the
assumptions of Theorem 2.1 hold true. A simple calculation yields
χ(F ) = g(r + 3)− d+ r − 3, δ(F ) = 1 +
1
2
⟨v(F ), v(F )⟩ = ρ.
The inequality χ(F ) ≥ (r + 2)k is satisfied. Indeed, by hypothesis g ≥ 2k − 1 and g ≥ 2d+3
5 ,
hence averaging we have g ≥ k + d−1
5 . Then
χ(F )− (r + 2)k = g(r + 3)− d+ r − 3− (r + 2)k
≥
(
k +
d− 1
5
)
(r + 3)− d+ r − 3− (r + 2)k
=
d+ 4
5
(r − 2) + (k − 2) ≥ 0,
since r, k ≥ 2. ■
Example 2.6. The (untwisted) Lazarsfeld–Mukai bundles also yield big and nef vector bundles
over X [k], under more restrictive assumptions. Take r = 2 for simplicity. Assume
2d− 2 ≥ g > 2k − 3 +
3
2
d, which implies χ(E) ≥ 4k + ρ, ρ ≥ 0.
This is a bit stronger than what is needed, but it ensures χ ≥ 4k and χ ≥ 3k+ρ simultaneously.
It is well known that E is µH -stable when Pic(X) = Z⟨H⟩. (Reason: any destabilizing quotient
has rank 1, slope ≤ 0, and is globally generated since E is. Hence the quotient is trivial, and
thus c2(E) = d = 0, a contradiction.) Since c1(E) = H, it follows E is (k − 1)-very ample by
Remark 2.3(i). By Theorem 2.1, we have that E[k] is big and nef.
Finally, we turn to Ulrich bundles considered in Theorem 5.7 and Corollary 5.8 in [7].
We write H2 = 2h, so that g = h + 1. Recall that a bundle E over (X,H) is said to be
Ulrich if
H⋆(X,E(−H)) = H⋆(X,E(−2H)) = 0.
Such bundles always exist for K3 surfaces of Picard rank 1 by [1, Theorem 1.5], and they have
numerics
rkE = 2a, c1(E) = 3aH, c2(E) = 9a2h− 4a(h− 1).
Example 2.7. Assume h > 2k−3 > 0. Consider an Ulrich bundle E as above. Then (E⊗H)[k]
is big and nef on the Hilbert scheme X [k].
Proof. Letting F = E ⊗H, we compute
χ(F ) = 12ah, δ(F ) = 1 + a2h+ 4a2.
As noted in [7, Theorem 5.6], the bundle F = E ⊗H is (k − 1)-very ample if h > 2k − 3; this
uses the statement cited in Remark 3(ii). To prove bigness, it remains to verify the positivity
of the top Segre integral. By Theorem 2.1, we check that
χ(F ) ≥ (r + 2)k ⇐⇒ 12ah ≥ (2a+ 2)k.
This is clear since h > 2k − 3 > 0. ■
Example 2.8. We can extend the result to Ulrich bundles E over the polarized K3 surface
(X,mH) for all m ≥ 1. By [11, Proposition 4.4], the Mukai vectors of such Ulrich bundles are
of the form
v =
(
r,
3rm
2
H,h
(
2m2r
)
− r
)
.
By the same arguments, (E ⊗H)[k] is big and nef for h > 2k − 3 > 0.
8 D. Oprea
3 Other K-trivial surfaces
3.1 Abelian and bielliptic surfaces
The formulas in [27] can be used to treat the case of abelian or bielliptic surfaces. For vector
bundles F → X, we set
r = rankF, χ = χ(F ), v = ch(F ), δ =
1
2
⟨v, v⟩.
We show
Theorem 3.1. For a (k − 1)-very ample vector bundle F → X, the tautological bundle F [k] →
X [k] is big and nef provided that χ ≥ (r + 2)k and δ ≥ 0.
Proof. We follow the same steps as for K3 surfaces, but a few numerical changes are necessary.
First, it was noted on [27, p. 19] that the analogue of equation (2.3) for abelian or bielliptic
surfaces takes the form∫
X[k]
s2k
(
F [k]
)
= Coefftk
[
(1 + (2 + r)t)δ(1 + (1 + r)t)χ−δ−(r+1)k−1(1 + (1 + r)(2 + r)t)
]
.
Next, the change of variables
t =
u
1− (1 + r)u
turns the above expression into∫
X[k]
s2k
(
F [k]
)
= Coeffuk
[
(1 + u)δ(1− (1 + r)u)−χ+(r+2)k−1
(
1 + (1 + r)2u
)]
.
The proof is completed by the same argument as in Theorem 2.1, this time letting ai be the coef-
ficients of (1+u)δ
(
1+(1+r)2u
)
and letting bj be the coefficients of (1−(1+r)u)−χ+(r+2)k−1. ■
Corollary 3.2. If L is (k − 1)-very ample line bundle over an abelian or bielliptic surface and
L2 ≥ 6k, then L[k] is big and nef.
In the following examples, we assume X is an abelian surface with Picard rank 1, with
Néron–Severi ample generator H.
Example 3.3. Assume H2 ≥ 6k. For H2 > 4k, the line bundle Ln = H⊗n is (k − 1)-very
ample for all n ≥ 1. This is an immediate consequence of [5, Theorem 2.1]. Indeed, using that
the Picard rank is 1, the inequality (2.5) is impossible. For H2 ≥ 6k, we also have χ(Ln) ≥ 3k.
It follows from Corollary 3.2 that (Ln)
[k] is big and nef.
In higher rank, just as for K3s, the reader can consider twists of Ulrich and Lazarfeld–
Mukai bundles. Here, taking advantage of the abelian surface geometry, we discuss simple
semihomogeneous bundles, and twists of unipotent and homogeneous bundles.
Example 3.4. Assume that H is a principal polarization. Let (a, b) = 1 be coprime positive
integers with
b > a2k.
By [29, Remark 7.13], there exist simple semihomogeneous vector bundles W → X with
rkW = a2, µ(W ) =
bH
a
∈ NS(X)⊗Q, χ(W ) = b2.
We claim W [k] is big and nef.
Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points 9
We note first that W is (k − 1)-very ample. Indeed, it is shown in [29, Theorem 5.8] that
W = f⋆L
for some line bundle L → Y , where f : Y → X is an isogeny of degree a2. It was remarked in
[32, Section 2.4] that W has no higher cohomology for b > 0. Consequently, the same is true
about L. Since
h0(L) = h0(W ) = χ(W ) = b2,
we conclude that L is effective and L2 = 2b2.
We claim L is
(
a2k − 1
)
-very ample for b > a2k. This follows again by [5, Theorem 2.1].
Indeed, the Picard rank is invariant under isogenies [8, Proposition 3.2], hence Y has Picard
rank 1 since X does. If M is the ample Néron–Severi generator, write L ≡num M ℓ, for ℓ > 0.
We have
2b2 = L2 = ℓ2M2 ≥ 2ℓ2 =⇒ 0 < ℓ ≤ b.
For any effective divisor D ̸= 0, we have
L.D = ℓM.D ≥ ℓM.M =
L2
ℓ
=
2b2
ℓ
≥ 2b ≥ 2a2k.
This violates (2.5). Since L is nef and L2 = 2b2 > 4a2k, it follows that L is
(
a2k − 1
)
-very
ample.
To check (k − 1)-very ampleness for W , note that if ζ is a subscheme of X of length k, then
H0(W ) → H0(W ⊗Oζ) surjective ⇐⇒ H0(L) → H0(L⊗Of⋆ζ) surjective.
The latter is true since L is
(
a2k − 1
)
-very ample and f⋆ζ has length a2k.
Finally, the inequality
χ(W ) ≥ (r + 2)k ⇐⇒ b2 ≥
(
a2 + 2
)
k
is certainly true when b > a2k.
Example 3.5. Assume H2 > 4k. Let E be a unipotent bundle of rank r ≥ 2, that is, E admits
a filtration whose successive quotients are the trivial line bundle [29, Definition 4.5]. Let F =
E ⊗ H. If H2 > 4k then H is (k − 1)-very ample by [5, Theorem 2.1], see (2.5). It follows
that F is also (k− 1)-very ample. Indeed, F is obtained as an iterated extension of H. At each
step, we inductively show that the extension is (k − 1)-very ample without higher cohomology,
by examining the relevant short exact sequences. The filtration of F also gives
χ(F ) = rχ(H) > 2rk ≥ (r + 2)k.
Thus (E ⊗H)[k] is big and nef.
Let E be homogeneous bundle of rank r ≥ 2, that is, E is invariant by translations on X.
By [29, Theorem 4.17], we can write
E =
⊕
i
Ui ⊗ Pi,
with Ui unipotent, and Pi a line bundle of degree 0. Repeating the argument above for each
summand, we show first that E ⊗H is (k − 1)-very ample, and then (E ⊗H)[k] is big and nef.
For bielliptic surfaces, we leave specific examples to the reader, mentioning only that (k−1)-
very ampleness of line bundles is studied in [28].
10 D. Oprea
3.2 Enriques surfaces
By contrast, the case of Enriques surfaces requires more care, due to the shape of the algebraic
functions giving the Segre integrals. As before, we write v(F ) = ch(F )
√
Td(X) for the Mukai
vector, and set
δ =
1
2
+
1
2
⟨v, v⟩ =⇒ δ = rc2 −
r − 1
2
c21 −
r2 − 1
2
.
Note that δ is an integer if and only if the rank r is odd. In this case, we prove:
Theorem 3.6. Let F be a (k−1)-very ample bundle of odd rank r on an Enriques surface, with
χ ≥ 2k(r + 1), δ ≥ 0.
Then F [k] is big and nef.
Computer experiments show that the bound χ ≥ (r + 2)k imposed for the other K-trivial
surfaces is insufficient here. Nonetheless, we have the following
Conjecture 3.7. For all ranks, odd or even, Theorem 3.6 holds under the weaker assumption
χ ≥
(
5r
4
+ 2
)
k, δ ≥ 0.
Proof of Theorem 3.6. Since F is (k − 1)-very ample, F [k] is globally generated, hence nef.
To establish bigness, it remains to show that the degree of the top Segre class s2k
(
F [k]
)
> 0.
By [27, Theorem 1], the Enriques analogue of (2.1) takes the form
∞∑
k=0
zk
∫
X[k]
s
(
F [k]
)
= A0(z)
c2(F )A1(z)
c1(F )2A2(z)
1
2 (3.1)
for exactly the same universal functions A0, A1, A2 which appear for K3 surfaces. Using (3.1)
and following the reasoning in [27, p. 11], with the modified numerics, we obtain an expression
for the top Segre integral∫
X[k]
s2k
(
F [k]
)
= Coefftk
[
(1 + (1 + r)t)χ−δ−k(r+1)− 1
2 (1 + (2 + r)t)δ(1 + (1 + r)(2 + r)t)
1
2
]
. (3.2)
This is the Enriques analogue of equation (2.3), but the half integer exponents complicate our
analysis.
To determine the sign of the Segre integral, we carry out the usual change of variables
t =
u
1− (1 + r)u
.
Then we rewrite (3.2) as∫
X[k]
s2k
(
F [k]
)
= Rest=0(1+ (1+ r)t)χ−δ−k(r+1)− 1
2 (1+ (2+ r)t)δ(1+ (1+ r)(2+ r)t)
1
2
dt
tk+1
= Resu=0(1− (1 + r)u)−χ+k(r+2)−1(1 + u)δ
(
1 + (1 + r)2u
) 1
2
du
uk+1
= Coeffuk(1− (1 + r)u)−α(1 + u)δ
(
1 + (1 + r)2u
) 1
2 ,
where α = χ− k(r + 2) + 1 ≥ kr + 1.
Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points 11
We note that when r is odd, δ ≥ 0 is an integer. Thus, the middle term (1 + u)δ has
nonnegative coefficients and constant term equal to 1. We claim that
(1− (1 + r)u)−α
(
1 + (1 + r)2u
) 1
2
has positive coefficients up to order k. The same will therefore be true after multiplying by
(1 + u)δ, showing that the top Segre class is positive.
A different idea is needed to establish the above claim. We change variables u 7→ u/(1 + r)
and consider instead the series
W = (1− u)−α(1 + (1 + r)u)
1
2 .
For each 0 ≤ m ≤ k, the coefficient of um in W equals
∑
i+j=m
(
−α
i
)(1
2
j
)
(−1)i(1 + r)j =
∑
i+j=m
(
α+ i− 1
i
)(1
2
j
)
(1 + r)j . (3.3)
The term corresponding to i = m, j = 0 is clearly positive since α ≥ 1. We will ignore this
term for the analysis. The next term i = m− 1, j = 1 is positive as well. The remaining terms
however have alternating signs because of the fractional binomials. We show nonetheless that
the sum of the consecutive (i, j) and (i− 1, j + 1) terms is positive, for j odd:(
α+ i− 1
i
)(1
2
j
)
(1 + r)j +
(
α+ i− 2
i− 1
)( 1
2
j + 1
)
(1 + r)j+1 > 0. (3.4)
This proves that the alternating sum (3.3) is positive as well. To justify (3.4), we note that(
α+ i− 1
i
)
=
(
α+ i− 2
i− 1
)
α+ i− 1
i
> 0,
( 1
2
j + 1
)
=
(1
2
j
) 1
2 − j
j + 1
.
For j odd, we have
( 1
2
j
)
> 0. After cancellation, the inequality to establish becomes
α+ i− 1
i
+ (1 + r)
1
2 − j
j + 1
> 0.
Writing i = m− j, and using α− 1 = χ− (r + 2)k ≥ rk ≥ rm, it suffices to establish
rm+m− j
m− j
+ (1 + r)
1
2 − j
j + 1
> 0 ⇐⇒ j2r +
r + 3
2
(m− j) +mr > 0,
which is clearly true. ■
Corollary 3.8. If L is a (k − 1)-very ample line bundle over an Enriques surface X, k ≥ 2,
then L[k] is big and nef.
In particular, if H is an ample line bundle over an Enriques surface, and Ln = H⊗n then
(Ln)
[k] is big and nef for all n ≥ k + 1.
Proof. The second half of the corollary follows from the first. Indeed, it is noted in [37,
Proposition 2.5] that if H is ample then Ln = H⊗n is (k − 1)-very ample for all n ≥ k + 1.
We prove the first statement. If χ := χ(L) ≥ 4k, the assertion follows from Theorem 3.6
with r = 1. Now, over Enriques surfaces, (k− 1)-very ampleness imposes numerical restrictions
on L which are stronger than for the other K-trivial surfaces. Indeed, it was noted in [37,
12 D. Oprea
Theorem 2.4] that if L is (k − 1)-very ample, then L2 ≥ (k + 1)2. When k ≥ 6, this is sufficient
to guarantee that
χ(L) = 1 +
L2
2
≥ 1 +
(k + 1)2
2
> 4k.
When 2 ≤ k ≤ 5, the bound χ ≥ 4k required by Theorem 3.6 may fail. However, the finitely
many cases
4k > χ ≥ 1 +
(1 + k)2
2
, 2 ≤ k ≤ 5,
can be checked by hand using equation (3.2). ■
4 Other geometries
The Segre integrals for arbitrary surfaces are established only when F has rank 1 or 2, see
[26, 27]. In rank 1, the answers were conjectured by Lehn [22]. The formulation below can be
found in [27]:
∞∑
n=0
zk
∫
X[k]
s
(
L[k]
)
= A1(z)
L2
A2(z)
χ(OX)A3(z)
L.KXA4(z)
K2
X . (4.1)
Here, for z = t(1 + 2t)2, we have
A1(z) = (1 + 2t)
1
2 ,
A2(z) = (1 + 2t)
3
2 (1 + 6t)−
1
2 ,
A3(z) =
1
2
(1 + 2t)−1
(√
1 + 2t+
√
1 + 6t
)
,
A4(z) = 4(1 + 2t)
1
2 (1 + 6t)
1
2
(√
1 + 2t+
√
1 + 6t
)−2
.
4.1 A positivity result
It is more difficult to determine the sign of the top Segre integral from the above formulas.
The following lemma plays an important role in the analysis. We will apply it in the next
section to geometric situations.
Lemma 4.1. For all integers m, n, p with m ≥ 0, p ≥ 0 such that m+n+ p is even, the series
f(t) =
(√
1 + 2t+
√
1 + 6t
)m
(1 + 2t)
n−1
2 (1 + 6t)
p−1
2
has positive coefficients up to order less or equal than min
(
1
2(m+ n+ p)− 1,m− 1
)
.
Taking the minimum is necessary. Indeed, for (m,n, p) = (2, 19, 1) the term of order 1
2(m+
n+p)−1 has negative coefficient, while for (m,n, p) = (4, 0, 0), the term of order m−1 has zero
coefficient. The lemma also holds for m + n + p odd, but this case never occurs geometrically.
The hypothesis m, p ≥ 0 can fail in geometric examples, but our proof requires it.
Proof. The argument is not straightforward due to the alternating signs in the expansions
√
1 + 2t = 1 + t− t2
2
+
t3
2
− 5t4
8
+ · · · ,
√
1 + 6t = 1 + 3t− 9t2
2
+
27t3
2
− 405t4
8
+ · · · .
Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points 13
However, a residue calculation and suitable changes of variables will render the answer manifestly
positive.
We begin by considering the case n ≥ 0. In this situation, assume first m, n, p are even,
and set
F (t) =
(√
1 + 2t+
√
1 + 6t
)m
(1 + 2t)−
1
2 (1 + 6t)−
1
2 .
Claim 4.2. The series F has positive coefficients up to order less or equal than m
2 − 1, and no
terms of order between m
2 and m− 1.
We assume this for now. Note that
f(t) = F (t)P (t), P (t) = (1 + 2t)
n
2 (1 + 6t)
p
2 .
Clearly, P is a polynomial of degree n+p
2 with positive coefficients. This observation together
with Claim 4.2 gives the argument. Indeed, let k ≤ min
(
1
2(m+ n+ p)− 1,m− 1
)
. Writing Fi
and Pj for the coefficients of F and P , we have
Coefftk f(t) =
∑
FiPj , the sum ranging over i+ j = k, i ≥ 0, 0 ≤ j ≤ n+ p
2
.
The sum is nonnegative since i ≤ k ≤ m− 1, so Fi ≥ 0 by the Claim 4.2, and Pj > 0. The sum
is in fact strictly positive. Indeed, for k ≥ n+p
2 , it contains the term
Fk−n+p
2
Pn+p
2
> 0.
The last statement is also a consequence of Claim 4.2, using that k − n+p
2 ≤ m
2 − 1. When
k < n+p
2 , the sum contains the term F0Pk = 2mPk > 0.
Proof of Claim 4.2. We seek to show that for all k ≤ m
2 − 1, the residue
Rest=0
F (t)
tk+1
dt > 0.
The peculiar change of variables
t =
s(s+ 1)
2
√
3s+ (2 +
√
3)
will simplify the calculation. For convenience, set
a =
2−
√
3
2
√
3
> 0 so that t =
s(s+ 1)
2
√
3(s+ a+ 1)
.
We note the following identities
√
1 + 2t =
s+
√
3+1
2
3
1
4 (s+ a+ 1)
1
2
,
√
1 + 6t =
3
1
4
(
s+
√
3+1
2
√
3
)
(s+ a+ 1)
1
2
,
√
1 + 2t+
√
1 + 6t =
(
√
3 + 1)(s+ 1)
3
1
4 (s+ a+ 1)
1
2
, dt =
(
s+
√
3+1
2
)(
s+
√
3+1
2
√
3
)
2
√
3(s+ a+ 1)2
ds.
From here, we obtain
Rest=0
F (t)
tk+1
dt = Ress=0
G(s)
sk+1
ds,
14 D. Oprea
where
G(s) = α(s+ 1)m−k−1(s+ a+ 1)k−
m
2 , α = 2k
(√
3 + 1
)m√
3
k−m
2 > 0.
A second change of variables will be needed next (this change of variables could have been carried
out simultaneously with the first, but this is a price worth paying for readability). We set
s =
(a+ 1)2
a
u
1− a+1
a u
.
The reader can verify that
s+ 1 =
1 + (a+ 1)u
1− a+1
a u
, s+ a+ 1 =
a+ 1
1− a+1
a u
, ds =
(a+ 1)2
a
1(
1− a+1
a u
)2 du.
By direct calculation, we find
Ress=0
G(s)
sk+1
ds = Resu=0
H(u)
uk+1
du
for
H(u) = β(1 + (a+ 1)u)m−k−1
(
1− a+ 1
a
u
)k−m
2
, β = αak(a+ 1)−k−m
2 > 0.
The first term is a polynomial with positive coefficients ai given by binomial numbers, for
0 ≤ i ≤ m− k − 1. The second term also has positive coefficients
bj =
(
−a+ 1
a
)j(k − m
2
j
)
> 0
since k − m
2 < 0. Thus
Coeffuk H(u) =
∑
aibj , the sum ranging over i+ j = k, 0 ≤ i ≤ m− k − 1, j ≥ 0.
This coefficient is positive since ai > 0, bj > 0 and the sum is non-empty (the term a0bk = bk > 0
appears in the sum).
When m is even, and m
2 ≤ k ≤ m − 1, the expression H(u) is a polynomial in u of degree
m
2 − 1 < k. Hence, the coefficient of uk in H(u) vanishes. This proves the claim (and completes
the argument when m, n, p are even and n ≥ 0). ■
When n ≥ 0, there are three other cases to consider, which require different choices for F
and P :
(i) when m even, n, p odd, we set
F (t) =
(√
1 + 2t+
√
1 + 6t
)m
, P (t) = (1 + 2t)
n−1
2 (1 + 6t)
p−1
2 ,
(ii) when m odd, n even, p odd, we set
F (t) =
(√
1 + 2t+
√
1 + 6t
)m
(1 + 2t)−
1
2 , P (t) = (1 + 2t)
n
2 (1 + 6t)
p−1
2 ,
(iii) when m odd, n odd, p even, we set
F (t) =
(√
1 + 2t+
√
1 + 6t
)m
(1 + 6t)−
1
2 , P (t) = (1 + 2t)
n−1
2 (1 + 6t)
p
2 .
Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points 15
In case (i), F has positive coefficients up to order m
2 , and no terms of order between m
2 + 1 and
m− 1. In cases (ii) and (iii), F has positive coefficients up to order m−1
2 , and no terms of order
between m+1
2 and m− 1. The arguments are similar, and we leave the details to the reader.
When n < 0, the reasoning used to prove Claim 4.2 also works to deal directly with the
function
f(t) =
(√
1 + 2t+
√
1 + 6t
)m
(1 + 2t)
n−1
2 (1 + 6t)
p−1
2 .
Following exactly the same steps, first changing from t to s and then from s to u, we obtain
H(u) = γ(1 + (a+ 1)u)m−k−1
(
1− a+ 1
a
u
)k−m+n+p
2
(
1− (
√
3 + 1)3
4
√
3
u
)n
×
(
1 +
(√
3 + 1
)3
4
u
)p
,
for γ > 0. By the same arguments, this has positive coefficients when the exponents
m− k − 1 ≥ 0, k − m+ n+ p
2
< 0, n < 0, p ≥ 0,
which we assumed. ■
4.2 K3 blowups
The top Segre classes computed by formula (4.1) are always coefficients in series of the form f(t)
as in Lemma 4.1, barring the condition m, p ≥ 0. When this condition is satisfied, we easily
obtain big and nef criteria for the tautological bundles L[k] → X [k].
There are several specific examples where our techniques apply. We illustrate them first when
π : X → S
is the blowup of a K3 surface S at a point p ∈ S. We assume S has Picard rank 1, with ample
Picard generator H. Let H2 = 2h.
Theorem 4.3. Let E be the exceptional divisor on X, and set L = H − ℓE. Assume ℓ ≥ k − 1
and
2h > max
(
(ℓ+ 2)2 − 6, (ℓ+ 1)2 + 4k, ℓ(ℓ+ 1) + 6k − 6
)
.
Then L[k] is big and nef on X [k].
Proof. We first show H − (ℓ+ 1)E nef. Recall the Sheshadri constant
ϵ(S, p) = max{t ∈ R≥0 : H − tE is nef},
see for instance [20, Definition 5.1.1]. Note that since H is in the nef cone of X, if H − tE is
nef, then H − t′E is also nef for all 0 ≤ t′ ≤ t. Thus it suffices to explain that
ϵ(S, p) ≥ ℓ+ 1.
The Sheshadri constants of K3 surfaces of Picard rank 1 have been studied in [18] and shown
to satisfy
ϵ(S, p) ≥
⌊√
H2
⌋
,
16 D. Oprea
with two possible exceptions
H2 = α2 + α− 2, ϵ(S, p) ≥ α− 2
α+ 1
and
H2 = α2 +
α− 1
2
, ϵ(S, p) ≥ α− 1
2α+ 1
, α ∈ Z>0.
Since the inequality 2h > ℓ(ℓ+ 3) is implied by our hypothesis, it follows that ϵ(S, p) ≥ ℓ+ 1 in
all cases (using α ≥ ℓ+ 2 in the two exceptional cases).
We next show that L is (k − 1)-very ample. Note that KX = E, and write
L = KX +M, M = H − (ℓ+ 1)E.
Observe that M is nef by the first paragraph of the proof, and
M2 = 2h− (ℓ+ 1)2 > 4k.
By [5, Theorem 2.1], if L is not (k − 1)-very ample, there exists an effective divisor D ̸= 0 such
that
D.M − k ≤ D2 <
D.M
2
< k.
Furthermore, M − 2D is Q-effective and D contains a subscheme ζ of length at most equal to k
such that
H0(L) → H0(L⊗Oζ)
is not surjective. Write
D = aH + bE,
and note that D effective implies a ≥ 0. Similarly,
M − 2D = (1− 2a)H + (−ℓ− 1− 2b)E
is Q-effective, so 1− 2a ≥ 0. Thus a = 0, D = bE with b > 0. We have
D.M
2
< k =⇒ b(ℓ+ 1) < 2k =⇒ b <
2k
ℓ+ 1
≤ 2
since k ≤ ℓ+ 1. Hence b = 1 and D = E. For subschemes ζ of E, the map
H0(L) → H0(L⊗Oζ)
can be written as composition
H0(L) → H0(L|E), H0(L|E) → H0(L⊗Oζ).
Since L|E = OE(ℓ) and ζ has length less or equal to k ≤ ℓ + 1, the second map is clearly
surjective. The first map is also surjective since
H1(L(−E)) = 0.
Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points 17
This is a consequence of [38, Proposition 4.1] and requires the bound 2h > (ℓ + 2)2 − 6, which
we assumed. We conclude L[k] is globally generated and thus nef.
It remains to explain that the top Segre class of L[k] is positive. We use (4.1) and we change
variables
z = t(1 + 2t)2 so that dz = (1 + 2t)(1 + 6t) dt.
Then ∫
X[k]
s
(
L[k]
)
= Resz=0A
2h−ℓ2
1 A2
2A
ℓ
3A
−1
4
dz
zk+1
= Rest=0 2
−ℓ−2(1 + 2t)h−
ℓ2
2
−2k−ℓ+ 3
2 (1 + 6t)−
1
2
(√
1 + 2t+
√
1 + 6t
)ℓ+2 dt
tk+1
= 2−ℓ−2Coefftk(1+ 2t)h−
ℓ2
2
−2k−ℓ+ 3
2 (1+ 6t)−
1
2
(√
1 + 2t+
√
1 + 6t
)ℓ+2
. (4.2)
We are now in the situation considered in Lemma 4.1. Thus, the coefficient above is positive
provided k ≤ ℓ+ 1 and
k ≤ 1
2
(
(ℓ+ 2) + 2
(
h− ℓ2
2
− 2k − ℓ+ 2
))
− 1 ⇐⇒ 2h > ℓ(ℓ+ 1) + 6k − 6.
This completes the proof. ■
4.3 Surfaces of general type
It is natural to wonder how far these techniques take us. We show
Theorem 4.4. Assume X is a smooth projective minimal surface of general type. Let L be
a (k − 1)-very ample line bundle such that
χ(L) ≥ 3k, L.KX ≥ 2K2
X + k + 1.
Then L[k] is big and nef over X [k].
Proof. Arguing as in the proof of Theorem 4.3, in particular equation (4.2), we express the
Segre integral as the tk-coefficient in the series
f(t) = 2−m
(√
1 + 2t+
√
1 + 6t
)m
(1 + 2t)
n−1
2 (1 + 6t)
p−1
2 ,
where
m = L.K − 2K2, n = (L−K)2 + 3χ− 4k − 1, p = K2 − χ+ 3,
with K = KX and χ = χ(OX). Note that
m+ n+ p = 2χ(L)− 4k + 2
is even. Furthermore, p ≥ 0. Indeed, let pg, q denote the genus and irregularity of X. By Noet-
her’s inequality K2 ≥ 2pg − 4, and K2 > 0 since X is minimal of general type. Averaging, we
obtain K2 > pg − 2, and thus
p = K2 − χ+ 3 > (pg − 2)− (1− q + pg) + 3 = q ≥ 0.
By Lemma 4.1, the coefficients of f(t) up to order min
(
1
2(m + n + p) − 1,m− 1
)
are positive.
In particular, if
k ≤ min
(
1
2
(m+ n+ p)− 1,m− 1
)
,
then the Segre integral is positive. The latter inequality is exactly our hypothesis, and thus L[k]
is big and nef. ■
18 D. Oprea
5 Curves
Let C be a smooth projective curve of genus g, and let V → C be a vector bundle with
χ = χ(V ). It is natural to ask whether the previous results also apply to the tautological bundles
V [k] → C [k]. In fact, by similar considerations, we establish an analogue of Theorem 2.1:
Proposition 5.1. Assume V → C is a (k−1)-very ample rank r vector bundle with χ ≥ (r+1)k.
Then V [k] → C [k] is big and nef over C [k].
To illustrate, if V is stable of degree d > r(2g − 2 + k), then V is (k − 1)-very ample. This
holds because for any divisor Z ⊂ C of degree k, we have
H1(V (−Z)) = H0
(
V ∨ ⊗KC(Z)
)
= 0
by Serre duality, stability and the assumption µ(V ) > 2g − 2 + k.
Proof. Since V is (k − 1)-very ample, it follows that V [k] is globally generated, hence nef.
To prove V [k] is big, it suffices to verify
(−1)k
∫
C[k]
s
(
V [k]
)
> 0.
The latter integrals are computed in [26, Theorem 2]; they can be viewed as higher rank ana-
logues of the classical k-secant integrals. The answer bears analogies with equation (2.1):
∞∑
k=0
zk
∫
C[k]
sk
(
V [k]
)
= Ad
1A
1−g
2 ,
where
A1(z) = 1 + t, A2(z) =
(1 + t)r+1
1 + (1 + r)t
, z = −t(1 + t)r.
We can express the Segre integrals as residues
(−1)k
∫
C[k]
sk
(
V [k]
)
= (−1)k Resz=0A
d
1A
1−g
2
dz
zk+1
= Rest=0(1 + t)χ−kr−g(1 + (1 + r)t)g
dt
tk+1
.
Just as in Theorem 2.1, we further change variables t = u
1−u , so that
(−1)k
∫
C[k]
sk
(
V [k]
)
= Resu=0(1 + ru)g(1− u)−χ+k(r+1)−1 du
uk+1
= Coeffuk(1 + ru)g(1− u)−χ+k(r+1)−1.
By the same reasoning as in Theorem 2.1, we see that this coefficient is positive when −χ +
k(r + 1)− 1 < 0 ⇐⇒ χ ≥ k(r + 1). This completes the argument. ■
To go further, we consider the punctual Quot schemes QuotC
(
CN , k
)
parametrizing quotients
0 → S → CN ⊗OC → Q → 0, rkQ = 0, length Q = k.
These are smooth projective varieties of dimensionNk, and carry tautological vector bundles V [k]
for each vector bundle V → C:
V [k] = p⋆(Q⊗ q⋆V ).
Here Q is the universal quotient and p, q are the two projections over QuotC
(
CN , k
)
× C. The
associated Segre integrals were studied in [33]. We extend the above results to the punctual
Quot scheme, in rank 1. The higher rank case appears more involved.
Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points 19
Theorem 5.2. Let L be a line bundle with χ(L) ≥ k + g and χ(L) ≥ k
(
1 + 1
N
)
. Then, the
vector bundle L[k] is big and nef over QuotC
(
CN , k
)
.
Proof. We show that L[k] is globally generated, hence nef. The universal sequence over
QuotC
(
CN , k
)
× C
0 → S → CN ⊗O → Q → 0
induces, via tensorization by L followed by pushforward, a morphism
H0
(
C,L⊕N
)
⊗OQuot → L[k].
To prove that the morphism is surjective, we establish that
H0
(
C,L⊕N
)
→ H0(C,L⊗Q) (5.1)
is surjective for all length k punctual quotients Q of the rank N trivial bundle.
The argument only requires L to be (k−1)-very ample, which is certainly true for us. In fact,
L is (k− 1)-very ample whenever degL ≥ 2g− 1+ k ⇐⇒ χ(L) ≥ g+ k, as we remarked before
the proof of Proposition 5.1. Surjectivity of (5.1) for N = 1 is a rephrasing of (k − 1)-very
ampleness.
For the general case, we induct on N . We pick a splitting CN = C ⊕ CN−1, and form the
diagram with exact rows and columns:
0
��
0
��
0
��
0 // S′ //
��
S //
��
S′′ //
��
0
0 // OC
//
��
CN ⊗OC
//
��
CN−1 ⊗OC
//
��
0
0 // Q′ //
��
Q //
��
Q′′ //
��
0
0 0 0
Tensoring with L and taking global sections yields
0 // H0(C,L) //
��
H0
(
C,L⊕N
)
//
��
H0
(
C,L⊕(N−1)
)
//
����
0
0 // H0(C,L⊗Q′) //
��
H0(C,L⊗Q) // H0(C,L⊗Q′′) //
��
0
0 0
Both Q′, Q′′ have lengths less or equal to k, the length of Q. Since L is (k − 1)-very ample,
it is also (ℓ− 1)-very ample for all 1 ≤ ℓ ≤ k, in particular when ℓ equals the length of Q′ or Q′′.
Thus, the first and last vertical arrows are surjective by induction. This implies that the middle
vertical arrow is surjective as well.
20 D. Oprea
It remains to show that L[k] is big. By (1.1), it suffices to determine the sign of the top Segre
class of L[k]. No additional calculation is needed in this case. Indeed, the Segre integrals were
noted in [33, Corollary 10] to satisfy the symmetry
(−1)Nk
∫
QuotC(CN ,k)
s
(
L[k]
)
= (−1)k
∫
C[k]
s
((
L⊕N
)[k])
.
In the proof of Proposition 5.1, the latter integral was shown to be positive if Nχ(L) ≥ (N+1)k,
which is true by hypothesis. ■
Acknowledgements
We are grateful to G. Bini, S. Boissière, F. Flamini for correspondence related to [7]; their paper
served as motivation for this work. We thank A. Marian and R. Pandharipande for collaboration
that led to [25, 26, 27, 33]. We thank the referees for their careful reading of the manuscript
and for their comments. The author is supported by NSF grant DMS1802228.
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1 Introduction
1.1 Results
1.2 Applications
1.3 Plan of the paper
2 K3 surfaces
2.1 Big and nef tautological bundles over the Hilbert scheme of K3s
3 Other K-trivial surfaces
3.1 Abelian and bielliptic surfaces
3.2 Enriques surfaces
4 Other geometries
4.1 A positivity result
4.2 K3 blowups
4.3 Surfaces of general type
5 Curves
References
|
| id | nasplib_isofts_kiev_ua-123456789-211726 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T04:20:49Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Oprea, Dragos 2026-01-09T12:53:12Z 2022 Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points. Dragos Oprea. SIGMA 18 (2022), 061, 21 pages 1815-0659 2020 Mathematics Subject Classification: 14C05; 14D20; 14C17 arXiv:2208.06599 https://nasplib.isofts.kiev.ua/handle/123456789/211726 https://doi.org/10.3842/SIGMA.2022.061 We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each -trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the 3 case, we extend recent constructions and results of Bini, Boissière, and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the -trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of 3s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves. We are grateful to G. Bini, S. Boissiere, and F. Flamini for correspondence related to [7]; their paper served as motivation for this work. We thank A. Marian and R. Pandharipande for collaboration that led to [25, 26, 27, 33]. We thank the referees for their careful reading of the manuscript and for their comments. The author is supported by NSF grant DMS1802228. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points Article published earlier |
| spellingShingle | Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points Oprea, Dragos |
| title | Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points |
| title_full | Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points |
| title_fullStr | Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points |
| title_full_unstemmed | Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points |
| title_short | Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points |
| title_sort | big and nef tautological vector bundles over the hilbert scheme of points |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211726 |
| work_keys_str_mv | AT opreadragos bigandneftautologicalvectorbundlesoverthehilbertschemeofpoints |