COMPLETE FAMILIES OF SMOOTH SPACE CURVES AND
STRONG SEMISTABILITY
OLIVIER BENOIST
Abstract. We study complete families of non-degenerate smooth space curves,
that is proper subvarieties of the Hilbert scheme of non-degenerate smooth
curves in P3 . On the one hand, we construct the first examples of such families. Our examples parametrize curves of genus 2 and degree 5. On the other
hand, we obtain necessary conditions for a complete family of smooth polarized
curves to induce a complete family of non-degenerate smooth space curves.
These restrictions imply for example that the base of such a family cannot be
a rational curve. Both results rely on the study of the strong semistability of
certain vector bundles.
Introduction
0.1. Complete families of smooth space curves. Let k be an algebraically
closed field. By a curve, we will always mean a projective connected one-dimensional
variety over k.
If B is an integral variety over k, a family of smooth space curves over B is a
closed subvariety C ,→ P3B := P3 × B such that C → B is a smooth family of curves.
Equivalently, it is a morphism from B to the Hilbert scheme of smooth curves in P3 .
Such a family will be said to be trivial if all its fibers are isomorphic as subvarieties
of P3 , and isotrivial if all its fibers are isomorphic as abstract curves. We will be
interested in complete families: those whose base B is proper.
The family of lines parametrized by the grassmannian is an example of a nontrivial complete family of smooth space curves. It is also easy to construct (necessarily isotrivial) non-trivial complete families whose members are plane curves
([2] Proposition 2.1). For this reason, we will restrict our attention to families
parametrizing non-degenerate space curves, that is curves whose linear span is P3 .
Non-trivial complete families of non-degenerate smooth space curves have been
studied by Chang and Ran in [5] and [6]. First, they showed that the curves
parametrized by the family can be neither rational nor elliptic curves ([6] Theorem
3). Moreover, they proved that every such family comes by base-change from a
family over a one-dimensional base, so that it is possible to restrict the study to the
case where B is a curve ([6] Theorem 1). However, they do not provide examples
of such families. The existence of non-trivial complete families of non-degenerate
smooth space curves is also stated as an open question in [14] p. 57.
One of our goals is to construct examples, and our first main theorem is:
Theorem 0.1. (i) There exist non-trivial complete families of non-degenerate
smooth space curves. The base of such a family may be chosen to be an
elliptic curve.
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2
OLIVIER BENOIST
(ii) Moreover, if k is of positive characteristic p, and if p ≡ ±1[8], there exists
such a family over a smooth curve of genus ≥ 2 that does not come by basechange from a family over a curve of genus ≤ 1.
The curves parametrized by our families are curves of genus 2 and degree 5. As
the moduli space of smooth curves of genus 2 is affine, such families are necessarily
isotrivial. It is the degree 5 line bundle providing the embedding that varies in the
family. In view of Chang and Ran’s results, those examples are minimal: they have
both smallest genus and smallest degree possible.
Theorem 0.1 shows the existence of elliptic curves (and, when char(k) ≡ ±1[8], of
a curve of genus ≥ 2) in the Hilbert scheme of non-degenerate smooth space curves:
it fits into the classical theme of constructing complete subvarieties of moduli spaces
initiated by Oort [30].
0.2. Embedding abstract families. Our second main theorem gives necessary
conditions for an abstract complete family of smooth polarized curves to induce a
complete family of non-degenerate smooth space curves.
We start with a smooth projective family of curves over a smooth projective
curve π : C → B, endowed with a relatively ample line bundle L. We want to
decide whether this abstract polarized family gives rise to a complete family of
non-degenerate smooth space curves. For this, we need to choose the sections that
will embed the fibers of π in P3 , that is a subbundle of rank 4 of E := π∗ L.
It turns out that, surprisingly, there is no choice at this point: this subbundle
has to be the first graded piece of the strong Harder-Narasimhan filtration of E (the
definition is recalled in Paragraph 1.1):
Theorem 0.2. Let π : C → B, φ : C ,→ P3B be a non-trivial complete family of
non-degenerate smooth space curves over a smooth curve B, and let E := π∗ φ∗ O(1).
Then H 0 (P3 , O(1)) ⊂ E is the first graded piece of the strong Harder-Narasimhan
filtration of E.
This theorem gives strong restrictions on the polarized family (π : C → B, L)
that need to be satisfied if one wishes to construct a non-trivial family of nondegenerate smooth space curves out of it: the first graded piece of the strong
Harder-Narasimhan filtration of E := π∗ L has to be of rank 4, and the corresponding
sections have to induce embeddings of the fibers of π in P3 .
We will deduce the following corollary, showing that in the examples of Theorem
0.1 (i), the genus of the base curve is minimal.
Corollary 0.3. There are no non-trivial complete families of non-degenerate smooth
space curves over P1 .
Finally, the necessary conditions discussed above turn out to be sufficient in
certain situations, up to replacing the base B by a finite surjective cover:
Proposition 0.4. Let π : C → B be a smooth projective family of curves over a
smooth curve. Let L be a line bundle on C and let E := π∗ L. Let F ⊂ E be a
subbundle of rank 4 such that for every b ∈ B, F|b ⊂ H 0 (Cb , Lb ) embeds Cb in P3 .
Suppose that one of the following conditions is satisfied:
(i) F is stable and B is an elliptic curve,
(ii) F is strongly semistable and k is the algebraic closure of a finite field.
COMPLETE FAMILIES OF SMOOTH SPACE CURVES
3
Then there exists a finite morphism from a smooth curve f : B 0 → B such that,
denoting by (π 0 : C 0 → B 0 , L0 ) the base-change, there exists an embedding φ0 : C 0 ,→
P3B 0 that is a complete family of non-degenerate smooth space curves satisfying
L0 |Cb0 ' φ0∗ O(1)|Cb0 for every b ∈ B 0 .
Moreover, in case (i), f may be chosen to be an isogeny.
Remark 0.5. In Proposition 0.4, the families constructed will be non-trivial if the
(Cb , Lb ) are not all isomorphic as polarized curves. In this case, Theorem 0.2 shows
that F has to be the first graded piece of the strong Harder-Narasimhan filtration
of E.
The first section of this paper is devoted, after recalling generalities about strong
semistability, to the proofs of Theorem 0.2, Corollary 0.3 and Proposition 0.4.
0.3. Curves of genus 2 and degree 5. Let us now explain how we will use
Proposition 0.4 to prove Theorem 0.1.
Let C be a smooth curve of genus 2. If L is a line bundle of degree 5 on C,
a straightforward application of Riemann-Roch theorem shows that h1 (C, L) = 0,
h0 (C, L) = 4, and that these four sections embed C in P3 . Let us denote A :=
Pic5 (C) and let P be a Poincaré bundle on C × A. The vanishing of H 1 implies
by base-change theorems for cohomology that E := p2∗ P is a rank 4 vector bundle
whose formation commutes with base-change.
Now, let B be a smooth curve and i : B → A be a non-constant morphism. We
consider the constant family π : C := C × B → B polarized by L := (Id, i)∗ P. By
base-change, π∗ L = i∗ E. We want to apply Proposition 0.4 (i) and (ii) to polarized
families of the form (C → B, L), with F := i∗ E and we need to choose C and B
carefully so that the stability hypotheses are satisfied.
Proof of Theorem 0.1. By Proposition 2.2, there exists a curve C of genus 2, an
elliptic curve B and a non-constant morphism i : B → A such that i∗ E is stable.
By Proposition 0.4 (i) and the above discussion, this proves Theorem 0.1 (i).
By Proposition 3.1, when k is of characteristic p ≡ ±1[8], there exists a curve
C of genus 2 and an immersion i : C → A both defined over F̄p , such that i∗ E is
strongly semistable. By Proposition 0.4 (ii) and the above discussion, this proves
Theorem 0.1 (ii).
Of course, in both cases, Remark 0.5 ensures that the families constructed are
non-trivial.
The second and third section of this paper are respectively devoted to the proofs
of Propositions 2.2 and 3.1. The short proof of Proposition 2.2 is based on a classical
construction of curves of genus 2 whose jacobians are not simple.
Let us be more precise about Proposition 3.1 and its proof. We set k = F̄p , we
choose our curve C to be defined by the equation Z 2 = X 6 + Y 6 , and consider
⊗3
the immersion i : C → A given by i(P ) = ωC
(−P ). Then, we prove that i∗ E is
strongly semistable exactly when p ≡ ±1[8].
The reason for our choice of i is that it turns out that i∗ E is a syzygy bundle
on C (Lemma 3.3). Strong semistability of syzygy bundles on curves have been
related by Brenner ([3] Theorem 1) and Trivedi ([32] Theorem 4.12) to HilbertKunz multiplicities of associated section rings (see Definition 3.14). In the case
of Fermat curves (our situation of a curve with equation Z 2 = X 6 + Y 6 is close),
Hilbert-Kunz multiplicities have been studied and computed by Han and Monsky
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OLIVIER BENOIST
([11], [12], [28]). Moreover, in some cases where syzygy bundles over Fermat curves
are strongly semistable, Brenner and Kaid [4] have been able to obtain stronger
results, namely explicit Frobenius periodicity up to a twist for those syzygy bundles.
Here, we do not know how to compute the relevant Hilbert-Kunz multiplicities
beforehand, so that we proceed the other way around. We show directly that i∗ E is
Frobenius periodic up to a twist when p ≡ ±1[8], which implies its strong semistability (Corollary 3.12). To do so, we follow very closely the arguments of Brenner
and Kaid [4]. However, at several places, Brenner and Kaid use the semistability of
their syzygy bundles, that is available to them from the Hilbert-Kunz multiplicity
computations due to Han and Monsky. We need to replace these arguments by
different ones: instead, we perform explicit syzygy computations using the strong
Lefschetz property of appropriate homogeneous ideals (in Paragraph 3.2).
A benefit of our method is that it also allows us to compute the strong HarderNarasimhan filtration of i∗ E when i∗ E is not strongly semistable, that is when
p ≡ ±3[8] (Proposition 3.13). In Paragraph 3.5, applying the aforementioned results
of Brenner and Trivedi, we are able to deduce the Hilbert-Kunz multiplicity of
⊗3
(C, ωC
), giving a new example of how this invariant varies with the characteristic
of the base field:
Theorem 0.6. Let C be the curve of genus 2 with equation Z 2 = X 6 + Y 6 . Then:
⊗3
(i) eHK (C, ωC
)=
⊗3
(ii) eHK (C, ωC
)=
15
4
15
4
if p ≡ ±1[8],
+ 4p12 if p ≡ ±3[8].
0.4. Further examples. Let us explain the reason why we were specifically interested in producing an example over a curve that is not an elliptic curve: since
there do not exist non-isotrivial complete families of smooth curves over rational or
elliptic curves ([31] Théorème 4), non-isotrivial complete families of smooth space
curves must have a base of genus ≥ 2. For this reason, one may view the second
part of Theorem 0.1 as a step forward towards the construction of non-isotrivial
families.
In the last section of this paper, we explain in greater detail the difficulties
encountered when trying to construct families beyond our examples, including nonisotrivial ones. We discuss specific examples and open problems.
Acknowledgements. I would like to thank Chungsim Han for having kindly made
available to me a copy of her thesis [11].
1. Embedding abstract families
The goal of this section is to prove Theorem 0.2, Corollary 0.3 and Proposition
0.4. We start by recalling general facts about strong semistability of vector bundles
on smooth curves.
1.1. Strong semistability. If k is of positive characteristic, and X is a variety
over k, we denote by F : X → X the absolute Frobenius morphism.
Definition 1.1. A vector bundle E on a smooth curve B is said to be strongly
semistable if k is of characteristic 0 and E is semistable, or if k is of positive
characteristic and for every k ≥ 0, F k∗ E is semistable.
COMPLETE FAMILIES OF SMOOTH SPACE CURVES
5
Compared to semistability, strong semistability has the advantage of being preserved by finite base-change, tensor products and symmetric powers (see [23] 2.2.2,
2.2.3).
The following important theorem is due to Langer ([22] Theorem 2.7):
Theorem 1.2. Let E be a vector bundle on a smooth curve B. Then there exists
a finite morphism from a smooth curve f : B 0 → B such that the graded pieces of
the Harder-Narasimhan filtration of f ∗ E are strongly semistable.
Such a filtration will be called a strong Harder-Narasimhan filtration. In characteristic 0, the Harder-Narasimhan filtration is always strong.
Over elliptic curves, the situation is very simple:
Proposition 1.3. Let E be an indecomposable vector bundle over an elliptic curve.
Then:
(i) E is strongly semistable.
(ii) E is stable if and only if its degree is prime to its rank.
Proof. In the first statement, the semistability of E is proved in [16] Lemma 1. The
strong semistability then follows from the more general Theorem 2.1 of [26].
It is easy to see that a semistable vector bundle whose rank and degree are prime
to each other is stable. Conversely, when the degree and the rank of E are not prime
to each other, Oda has proved ([29] Corollary 2.5) that E is not simple, hence not
stable.
Finally, to prove Proposition 0.4, we will need conditions ensuring that a vector
bundle becomes isomorphic to a direct sum of isomorphic line bundles after an
appropriate base-change. This is the goal of the two following propositions. The
first one might be well-known, but I do not know a reference for it. The second one
is Lange-Stuhler theorem.
Proposition 1.4. Let E be a stable vector bundle over an elliptic curve E. Then
there exists an isogeny f : E 0 → E such that f ∗ E is isomorphic to a direct sum of
isomorphic line bundles.
Proof. By Proposition 1.3 (i), we will be able to use in our proof the fact the
pull-back of E by any isogeny is semistable.
Let us first show that there exists an isogeny f : E 0 → E such that f ∗ E is
isomorphic to a direct sum of line bundles. Let r be the rank of E and let f :
E 0 → E be an isogeny whose degree is divisible by r. Write f ∗ E as a direct
sum of indecomposable bundles. If k is of characteristic 0, those indecomposable
bundles must be stable of the same slope by [17] Lemma 3.2.3, and by Proposition
1.3 (ii), the only possibility is that they are all line bundles. On the other hand,
if k is of positive characteristic p, Proposition 1.3 (ii) shows that f ∗ E cannot be
stable. Consider a Jordan-Hölder filtration for E. Using induction on the rank
of E, it is possible to suppose that all the graded pieces of this filtration have
rank 1. Now, extensions between line bundles of the same degree are trivial if
the line bundles are not isomorphic, and parametrized by H 1 (E 0 , O) otherwise.
Since [p]∗ : H 1 (E 0 , O) → H 1 (E 0 , O) is zero in characteristic p, base-changing by an
appropriate power of the isogeny [p] concludes.
It remains to prove that up to base-changing
L again by an isogeny, all these line
bundles are isomorphic. Let us write f ∗ E ' i Fi , where the Fi are the isotypical
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OLIVIER BENOIST
factors: each Fi is the direct sum of isomorphic line bundles. Write f = g ◦h, where
h : E 0 → F is separable of Galois group G and g : F → E is purely inseparable.
If the Galois group G did not act transitively on the isotypical factors, a nontrivial direct sum G of some of them would descend to F by Galois descent. Since,
Hom(G, f ∗ E/G ⊗ Ω1E 0 ) = Hom(G, f ∗ E/G) = 0, inseparable descent ([18] Theorem
5.1) shows that this sheaf descends even to E, contradicting the stability of E.
Hence G permutes the isotypical components. But since G acts on E 0 as a finite
subgroup of translations, it follows that the line bundles appearing in E differ from
each other by torsion bundles. Hence all these line bundles will become isomorphic
after further pull-back by a well-chosen isogeny.
Proposition 1.5. Let E be a vector bundle on a smooth curve B over the algebraic
closure of a finite field. Then the following conditions are equivalent:
(i) E is strongly semistable.
(ii) There exists a finite morphism from a smooth curve f : B 0 → B such that
f ∗ E is isomorphic to a direct sum of isomorphic line bundles.
Proof. The implication (ii) =⇒ (i) is trivial. Let us explain the other one: it is due
to Lange and Stuhler [21]. First, it is easy to find a finite morphism from a smooth
curve g : B 00 → B and a line bundle N on B 00 such that g ∗ E ⊗ N has degree 0.
Moreover, it is still strongly semistable. Applying [21] Satz 1.9 shows that g ∗ E ⊗ N
is trivialized by a finite surjective morphism h : B 0 → B 00 . Setting f = g ◦ h, one
sees that f ∗ E is a direct sum of line bundles isomorphic to h∗ N −1 .
1.2. The strong Harder-Narasimhan filtration. Let B be a smooth curve,
π : C → B be a smooth projective family of curves over B and L be a line bundle
on C. Let E := π∗ L. In this situation, the following holds:
Lemma 1.6. The sheaf E is locally free. Its formation commutes with base-change
by any finite map from a smooth curve B 0 → B.
Moreover, for every b ∈ B, the natural map E|b → H 0 (Cb , Lb ) is injective.
Proof. From its definition, one sees that E has no torsion, and a torsion-free coherent
sheaf over a smooth curve is necessarily locally free. The second statement is a
consequence of flat base-change [15] III Proposition 9.3. As for the third statement,
consider the exact sequence 0 → O(−b) → O → O|b → 0. Pull it back to C,
tensor with L and push it forward to B to get an exact sequence 0 → E(−b) →
E → Im(E → H 0 (Cb , Lb )) → 0. Restricting to b using right-exactness of tensor
product, and noticing that the morphism E(−b)|b → E|b vanishes, one sees that
E|b → H 0 (Cb , Lb ) is indeed injective.
Let us now prove Theorem 0.2:
Theorem 1.7 (Theorem 0.2). Let π : C → B, φ : C ,→ P3B be a non-trivial
complete family of non-degenerate smooth space curves over a smooth curve B, and
let E := π∗ φ∗ O(1). Then the constant subbundle H 0 (P3 , O(1)) ' O⊕4 ⊂ E is the
first graded piece of the strong Harder-Narasimhan filtration of E.
Proof. Let us suppose for contradiction that the proposition does not hold and
that there exists a counter-example. Note that at any point in the proof, it will be
possible to replace B by a finite cover by a smooth curve B 0 . Indeed, the formation
of E commutes with this base-change (see Lemma 1.6).
COMPLETE FAMILIES OF SMOOTH SPACE CURVES
7
Let Q be the quotient of E by O⊕4 . Using Theorem 1.2, perform a base-change
to ensure that the strong Harder-Narasimhan filtration of Q is defined over B.
Our hypothesis that O⊕4 is not the first graded piece of the Harder-Narasimhan
filtration of E implies that the first graded piece of the Harder-Narasimhan filtration
of Q has non-negative degree.
Our counter-example may be defined over a field of finite type: spreading out
and specializing to a general closed point, we get data defined over a finite field. It is
still a counter-example to the proposition, as after such a general specialization, Q
still has a subbundle of non-negative degree. Hence, from now on, we may suppose
that k is the algebraic closure of a finite field. As before, we may assume that the
strong Harder-Narasimhan filtration of Q is defined over B and that its first graded
piece has non-negative degree.
Base-changing again using Proposition 1.5, we may assume that this subbundle
is a direct sum of line bundles of non-negative degree: in particular, Q contains
a subbundle of rank 1 of non-negative degree M. Consequently, there exists a
subbundle F of E that is an extension of a line bundle of non-negative degree M
by O⊕4 . Base-changing again, it is possible to suppose that the strong HarderNarasimhan filtration of F is defined over B: we will let G1 ⊂ F be the first graded
piece of this filtration.
Now, let us use F to embed C in a relative projective bundle over B: we get
an immersion ψ : C ,→ PB F. Moreover, one recovers the original embedding φ by
projecting away from PB M. Note that, as F|b → H 0 (Cb , Lb ) is injective by Lemma
1.6, ψ embeds all fibers of π in a non-degenerate way in P4 . Let us introduce the
relative secant variety: it is the subvariety S ,→ PB F that is the union of the secant
varieties of the embedded curves Cb ,→ PFb . It is a hypersurface of PB F because
secant varieties of non-degenerate curves in P4 are of dimension 3. It does not
meet PB M because, for every b ∈ B, the linear system H 0 (P3 , O(1)) induced an
embedding of Cb .
Let q : PB F → B be the projection and Oq (1) be the relative tautological
bundle. By description of the Picard group of a projective bundle, there exists
a line bundle A on B and an integer l such that S is the zero-locus of a section
in H 0 (PB F, Oq (l) ⊗ q ∗ A) = H 0 (B, Syml F ⊗ A). The hypothesis that S does
not meet PB M means exactly that this section induces a non-vanishing section in
H 0 (B, M⊗l ⊗ A): in particular, A ' M⊗−l .
We will distinguish three cases to get a contradiction. Let us suppose first that
µ(G1 ) < µ(M), so that all the graded pieces Gi of the strong Harder-Narasimhan
filtration of F have slope < µ(M). This filtration induces a filtration of Syml F
whose graded pieces are tensor products of symmetric powers of the Gi : these are
strongly semistable of slope < µ(M⊗l ). Consequently, H 0 (B, Syml F ⊗M⊗−l ) = 0,
which is a contradiction.
Next, suppose that µ(G1 ) ≥ µ(M) > 0. The morphism G1 → M cannot be zero
as there are no non-zero morphisms G1 → O⊕4 by semistability of G1 . Again by
semistability of G1 , this morphism has to be surjective. Then G1 is an extension of
M by a subbundle of O⊕4 , and the inequality µ(G1 ) ≥ µ(M) implies that G1 → M
is an isomorphism. Hence F splits as a direct sum O⊕4 ⊕ M. Now the space
H 0 (B, Syml F ⊗ M⊗−l ) is one-dimensional because µ(M) > 0, and the zero locus
of one of its sections on a fiber of q is a hyperplane with multiplicity l. This
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OLIVIER BENOIST
contradicts the fact that, the curve Cb being embedded in a non-degenerate way in
P4 , its secant variety is also non-degenerate.
Finally, let us consider the case where µ(M) = 0. Then, F is strongly semistable
as an extension of strongly semistable bundles of the same degree. Applying Proposition 1.5, it is possible to perform a base-change so that F becomes a direct sum
of isomorphic line bundles. In particular, PB F ' P4B . Then, the relative secant
variety S is a hypersurface of P4B that avoids a constant section. It follows that S
is a product hypersurface: it is isomorphic to S × B, where S is a hypersurface of
P4 . Consequently, S is the secant variety of all curves Cb ,→ P4 . Recall that, by a
theorem of Chang and Ran ([6] Theorem 3), the curves Cb have genus at least 2.
Hence, by Lemma 1.8 below, there is only finitely many possibilities for the curves
Cb , so that the subvariety ψ : C ,→ P4B has to be a product itself. Since the original
family φ is obtained by projecting away from a constant section, it follows that our
original family was itself trivial: this is a contradiction.
We used the following lemma:
Lemma 1.8. Let C ⊂ P4 be a smooth non-degenerate curve of genus at least 2,
and let S be its secant variety. Then there is a unique family of lines that covers
S, namely the 2-dimensional family of secants of C. Moreover, C is an irreducible
component of the set of points included in infinitely many of these lines.
Proof. The secant variety S is naturally the image of a P1 -bundle over the two-fold
symmetric product C (2) of C. By results of Dale [8], this natural morphism is
birational. As it is not explicitly stated in [8], let us quickly explain how to deduce
this claim from this reference. In order to fit with Dale’s notations, we temporarily
set X := C and Sec(X) := S. Let us consider the diagram p. 331, and let us
introduce T (X) := pr
¯ 23 (M (X)). There is a natural morphism X 2 → T (X) defined
by (x, y) 7→ (x, hx, yi). Using Theorem 1.8, this morphism is seen to be birational.
Consequently, M (X) is birational to the natural P1 -bundle over X 2 . The map pr
¯ 13
is separable of degree 2 by Theorem 1.8 and Lemma 3.5. The map β is birational
by Theorem 4.1 and Theorem 1.10. It follows that the composite M (X) → Sec(X)
is separable of degree 2. Since it is induced by our natural map from the P1 -bundle
over C (2) to S, the latter is birational.
The genus of C being at least 2, the Abel-Jacobi map shows that C (2) is not
covered by rational curves. Hence, the only family of rational curves that covers
S is the one induced by the fibers of the P1 -bundle structure, that is the family of
secants of C.
Finally, a dimension count shows that the subset of S consisting of points included in infinitely many of these secants is at most 1-dimensional. Since C is
obviously contained in it, C has to be an irreducible component of this locus. The proof of Corollary 0.3 follows:
Corollary 1.9 (Corollary 0.3). There are no non-trivial complete families of nondegenerate smooth space curves over P1 .
Proof. Let π : C → P1 , φ : C ,→ P3P1 be a complete family of non-degenerate smooth
space curves over P1 . This family has to be isotrivial by [31] Théorème 4: all the
fibers of π are isomorphic to a fixed curve C. Moreover, by a theorem of Chang
and Ran ([6] Theorem 3), C is of genus ≥ 2. Since the automorphism group of C
is finite and P1 is simply connected, the family has to be a product: C ' C × P1 .
COMPLETE FAMILIES OF SMOOTH SPACE CURVES
9
Since the Picard scheme Pic(C) does not contain non-trivial rational curves, all
the fibers are even isomorphic as polarized curves and φ∗ O(1) ' p∗1 M ⊗ p∗2 N for
some line bundles M (resp. N ) on C (resp. P1 ). Consequently, E := π∗ φ∗ OP3 (1) is
isomorphic to a direct sum of isomorphic line bundles, hence is strongly semistable.
It follows from Theorem 0.2 that the subbundle of E used to construct the embedding φ is E itself, so that our family is trivial.
1.3. Constructing embeddings. The converse results of Proposition 0.4 are consequences of Propositions 1.4 and 1.5:
Proof of Proposition 0.4. Under the hypotheses of Proposition 0.4, Propositions 1.4
and 1.5 ensure the existence of a finite morphism from a smooth curve f : B 0 → B
such that f ∗ F is isomorphic to a direct sum of isomorphic line bundles. Moreover,
in case (i), the morphism f may be chosen to be an isogeny. Using Lemma 1.6, one
sees that f ∗ F is a subbundle of π∗0 L0 , and that for every b ∈ B, (f ∗ F)|b ⊂ H 0 (Cb0 , L0b )
embeds Cb0 in a non-degenerate way in P3 . Consequently f ∗ F induces an embedding
φ0 : C 0 ,→ PB 0 (f ∗ F) that is non-degenerate over every b ∈ B 0 . Since this projective
bundle is trivial by our choice of f , we are done.
Remark 1.10. In Proposition 0.4 (ii), the genus of the curve B 0 is not explicit as
the construction of B 0 relies on Lange-Stuhler theorem (Proposition 1.5). However,
it follows from the proof of this theorem (see [21] Satz 1.4 b)) that we can give
bounds for the genus of B 0 if we now an explicit Frobenius periodicity property for
the strongly semistable vector bundle F (that is a relation of the form F r∗ F '
F s∗ F ⊗ N for some r 6= s and some line bundle N ).
Fortunately, in our applications to Theorem 0.1 (ii), we prove the strong semistability of the relevant vector bundle precisely by exhibiting such a relation (see the
proof of Corollary 3.12). Consequently, Lange-Stuhler’s proof provides bounds for
the genus of the base of the families constructed in Theorem 0.1 (ii).
2. Families over elliptic curves
The goal of this section is to prove Proposition 2.2, thus completing the proof of
Theorem 0.1 (i), as explained in Paragraph 0.3.
Let us start by describing a construction of curves of genus 2 whose jacobian is
not simple, that is very well explained in the first section of [10].
Let E and F be two non-isogenous elliptic curves. Let n be an odd integer that is
invertible in k. Choose an isomorphism α : E[n] → F [n] whose graph Γ is isotropic
with respect to the Weil pairings on E[n] and F [n]. Let A := (E×F )/Γ. Notice that
the quotient of A by the image F of {0} × F is (E × F )/h{0} × F, Γi ' E/E[n] ' E
so that we have an exact sequence of abelian varieties 0 → F → A → E → 0. By
[10] Propositions 1.1 and 1.4, A is isomorphic to the jacobian of a smooth curve C
of genus 2. Moreover, the theta divisor of A has degree n on F .
Choose an isomorphism A ' Pic5 (C), and recall that we had defined E := p2∗ P,
where P is a Poincaré bundle on C × A.
Lemma 2.1. The degree on F of any line bundle N ∈ Pic(A) is a multiple of n.
The degree of E on F is −n.
Proof. Consider the projection u : E × F → A. The line bundle u∗ N is obviously
Γ-invariant. Since E and F are not isogenous, Pic(E × F ) ' Pic(E) ⊕ Pic(F ).
Consequently, the action of Γ on Pic(E × F ) is easy to describe, and one sees that
10
OLIVIER BENOIST
Pic(E × F )Γ consists of line bundles of the form p∗1 M ⊗ p∗2 M0 , where M (resp.
M0 ) have degree divisible by n on E (resp. F ). Hence N · F = u∗ N · ({0} × F ) is
a multiple of n.
Finally, by [1] VII.4, the determinant of E on A is numerically equivalent to
minus the theta divisor. Hence, its degree on F is −n.
We may now prove:
Proposition 2.2. The restriction of E to a general fiber of q : A → E is stable.
Proof. Suppose for contradiction that it is not the case. Then the restriction of E
to a general fiber of q is a fortiori not semistable, as its rank 4 is prime with its
degree −n.
The existence of a relative Harder-Narasimhan filtration with respect to q implies
that there exists a saturated subsheaf F ⊂ E whose restriction to a general fiber of
q destabilizes E. Outside of a finite number of points of A, F is a vector bundle,
and det(F) is well-defined as a line bundle. It extends to a line bundle still denoted
by det(F) on all of A because A is smooth. By construction, det(F) has degree
> −n on the fibers of F. By Lemma 2.1, this degree is ≥ 0.
Hence, the restriction of F to a general fiber of q is of non-negative degree.
Equivalently, the restriction of F ∨ to a general fiber of q is of non-positive degree.
Consequently, the vector bundle E ∨ is not ample. This contradicts [1] VII.2.2. 3. Families over higher genus curves
In all this section, we suppose that k is of positive characteristic p ≥ 5.
Let C be a smooth curve of genus 2, c ∈ C a point and M a line bundle of degree
6 on C. Let A := Pic5 (C), let P be the Poincaré bundle on C × A normalized so
that P|{c}×A ' OA , and let E := p2∗ P. We consider the morphism i : C → A
defined by i(P ) := M(−P ).
The main goal of this section is to prove the following proposition, thus completing the proof of Theorem 0.1 (ii), as explained in Paragraph 0.3. More precisely,
Proposition 3.1 follows from Lemma 3.3, Corollary 3.12 and Proposition 3.13.
Proposition 3.1. Suppose that C is given by the equation Z 2 = X 6 + Y 6 , and that
⊗3
M is the tricanonical line bundle ωC
. Then i∗ E is strongly semistable if and only
if p ≡ ±1[8].
The restrictive assumptions on the curve C and the line bundle M in the hypotheses of this proposition will be made explicitely later, when they become useful.
As already explained in the introduction, the general strategy of the proof is
borrowed from [4].
3.1. Syzygy bundles. Let us first recall what a syzygy bundle is.
Definition 3.2. Let X be a variety, let (Li )1≤i≤n be line bundles on it and let
σi ∈ H 0 (X, Li ) be sections with no common zero. The syzygy bundle associated
to these sections is the vector bundle of rank n − 1 on X defined by the following
exact sequence:
M
·σi
0 → SyzX (σi ) →
L−1
−→
OX → 0.
i
i
COMPLETE FAMILIES OF SMOOTH SPACE CURVES
11
If L is a base-point free line bundle on X and the σi form a base of H 0 (X, L), we
will denote the corresponding syzygy bundle by SyzX (L). If N is a line bundle on
X, left-exactness of H 0 makes it easy to compute
H 0 (X, SyzX (σi ) ⊗ N ): it consists
P
−1
0
of sections τi ∈ H (X, Li ⊗ N ) such that i τi σi = 0.
Let S := SyzC (M). The relevance of syzygy bundles to our problem is shown
by the following:
Lemma 3.3. There is an isomorphism i∗ E ' S ⊗ M(c).
Proof. Consider the pull-back (Id, i)∗ P of the Poincaré bundle on C ×C. Its restriction to {c} × C is trivial and its restriction to C × {P } is isomorphic to M(−P ).
It follows that (Id, i)∗ P ' p∗1 M ⊗ p∗2 O(c)(−∆), where ∆ is the diagonal. As a
consequence, there is a short exact sequence on C × C:
0 → (Id, i)∗ P → p∗1 M ⊗ p∗2 O(c) → (p∗1 M ⊗ p∗2 O(c))|∆ → 0.
Pushing it forward by p2 and using vanishing of the appropriate H 1 , one gets:
0 → i∗ E → H 0 (C, M) ⊗ O(c) → M(c) → 0,
where the arrow H 0 (C, M) ⊗ O(c) → M(c) is the evaluation. One recognizes the
definition of a syzygy bundle, up to a twist.
From now on, we will restrict to the case where the line bundle M is the tricano⊗3
nical line bundle ωC
. Since ωC ' f ∗ O(1), where f : C → P1 is the hyperelliptic
double cover, this will allow us to compare F ∗ S with bundles on P1 , that are easier
to describe.
Let X, Y be coordinates on P1 and let P (X, Y ) be the homogeneous degree 6
polynomial defining the ramification locus
C is the hyperelliptic
L of f : the curve
⊗i
curve Z 2 = P (X, Y ). The canonical ring i≥0 H 0 (C, ωC
) of C is then isomorphic
to k[X, Y, Z]/hZ 2 − P (X, Y )i, where the generators X, Y are of degree 1 and Z is of
degree 3. In particular, it is isomorphic to k[X, Y ]⊕ k[X, Y ]·Z as a k[X, Y ]-module.
Let us introduce the two following syzygy bundles on P1 :
(
p+1
S+ := SyzP1 (X 3p , X 2p Y p , X p Y 2p , Y 3p , P (X, Y ) 2 ),
p−1
S− := SyzP1 (X 3p , X 2p Y p , X p Y 2p , Y 3p , P (X, Y ) 2 ).
Lemma 3.4. There is an exact sequence:
⊗−3
⊗−3
0 → F ∗ S(ωC
) → f ∗ S+ ⊕ f ∗ S− (ωC
) → F ∗ S → 0.
⊗m
Moreover, if m ≥ 0, the complex obtained by tensoring by ωC
and taking global
sections is exact.
Proof. From the definition of a syzygy bundle, one sees that:
F ∗ S ' SyzC (X 3p , X 2p Y p , X p Y 2p , Y 3p , Z p ).
It is easy to describe the morphisms in the exact sequence at the level of local
sections. The morphism f ∗ S+ → F ∗ S is (A, B, C, D, E) 7→ (A, B, C, D, ZE), the
⊗−3
morphism f ∗ S− (ωC
) → F ∗ S is (A, B, C, D, E) 7→ (ZA, ZB, ZC, ZD, E). In a
⊗−3
∗
similar manner, F S(ωC
) → f ∗ S+ is (A, B, C, D, E) 7→ (ZA, ZB, ZC, ZD, E)
⊗−3
⊗−3
∗
∗
and F S(ωC ) → f S− (ωC ) is (A, B, C, D, E) 7→ −(A, B, C, D, ZE).
To prove the exactness of this complex, it suffices to prove the second statement.
This is easy using the description of the canonical ring as k[X, Y ] ⊕ k[X, Y ] · Z. 12
OLIVIER BENOIST
3.2. The strong Lefschetz property and syzygy computations. In this paragraph, we want to compute the syzygy bundles S+ and S− on P1 . To be able to
perform those computations, we need to restrict the situation again, by choosing
carefully the polynomial P . We will take P (X, Y ) = X 6 + Y 6 , so that C is the
curve of equation Z 2 = X 6 + Y 6 .
Our main tool will be the strong Lefschetz property for homogeneous ideals. Let
us recall the definition:
Definition 3.5. Let R := k[x1 , . . . , xn ] graded by the degree. An homogeneous
artinian ideal I ⊂ R is said to satisfy the strong Lefschetz property if there exists a
·ld
linear form l ∈ R1 such that for every r, d ≥ 0, the multiplication map (R/I)r −→
(R/I)r+d is of maximal rank.
Lemma 3.6. Let I ⊂ k[x, y] be an homogeneous artinian ideal. Moreover, if k is
of positive characteristic p, suppose that (R/I)r = 0 for r ≥ p. Then I satisfies the
strong Lefschetz property.
Proof. When k is of characteristic 0, this is [13] Proposition 4.4. The way this
hypothesis is used in the proof is via the explicit description of Borel-fixed ideals
in characteristic 0, applied to the generic initial ideal of I. The description of
Borel-fixed ideals in positive characteristic p is more complicated in general (see [9]
Theorem 15.23), but one sees from this description that it coincides with the simple
one in characteristic 0 when the condition that (R/I)r = 0 for r ≥ p is satisfied.
Consequently, under this hypothesis, the proof goes through.
It is now possible to prove:
Proposition 3.7.
) ⊕ O( −15p−1
)⊕3 and S− ' O( −15p+3
)⊕4 .
(i) If p ≡ 1[8], S+ ' O( −15p−9
4
4
4
−15p−3 ⊕4
−15p+1 ⊕3
−15p+9
(ii) If p ≡ −1[8], S+ ' O( 4 ) and S− ' O( 4 ) ⊕ O( 4 ).
)⊕2 ⊕ O( −15p+1
)⊕2 and S− ' O( −15p−3
)⊕
(iii) If p ≡ 3[8], S+ ' O( −15p−7
4
4
4
−15p+5 ⊕3
O( 4 ) .
(iv) If p ≡ −3[8], S+ ' O( −15p−5
)⊕3 ⊕ O( −15p+3
) and S− ' O( −15p−1
)⊕2 ⊕
4
4
4
−15p+7 ⊕2
O( 4 ) .
Proof. We know the degrees of S+ and S− from their definition. Moreover, by
Grothendieck’s theorem, a vector bundle on P1 splits as a direct sum of line bundles.
As a consequence, to prove the proposition, it will be enough to compute the global
sections of some twists of S+ and S− . For instance, to prove that S+ ' O( −15p−9
)⊕
4
15p−3
⊕3
0
1
O( −15p−1
)
if
p
≡
1[8],
it
is
sufficient
to
show
that
h
(P
,
S
(
))
=
0
and
+
4
4
that h0 (P1 , S+ ( 15p+1
4 )) = 3.
Even if the result depends only on p modulo 8, it will be necessary in the proof
to distinguish between different values of p modulo 24. As all the global section
computations needed are similar, we will only carry out one: we will show that if
p ≡ 1[24], h0 (P1 , S+ ( 15p+1
4 )) = 3. The other cases are absolutely analogous, and
left to the reader.
Let us assume that p ≡ 1[24]. Applying global sections to an appropriate twist of
the exact sequence defining S+ , we see that H 0 (P1 , S+ ( 15p+1
4 )) is the vector space
of solutions of the equation:
(3.1)
AX 3p + BX 2p Y p + CX p Y 2p + DY 3p + E(X 6 + Y 6 )
p+1
2
= 0,
COMPLETE FAMILIES OF SMOOTH SPACE CURVES
13
where the unknowns A, B, C, D, E are homogeneous polynomials in X and Y , the
first four being of degree 3p+1
and E being of degree 3p−11
4
4 . This is a linear system
in the coefficients of A, B, C, D, E.
The matrix of this linear system in the monomial bases is made of six rectangular blocks, by separating the monomials according to the value modulo 6 of the
exponent of X. As a consequence, the solution space of (3.1) is the direct sum of
the solution spaces of six smaller systems, that we may solve independently.
Let us look at the first one, obtained by conserving in the equation (3.1) only
monomials such that the exponent of X is a multiple of 6. Then, setting x := X 6
and y := Y 6 , it is possible to write A = X 3 Y 4 a(x, y), B = X 4 Y 3 b(x, y), C =
X 5 Y 2 c(X, Y ), D = Y d(x, y) and E = Y 4 e(x, y). Dividing by Y 4 , we get the new
equation:
ax
(3.2)
p+1
2
+ bx
p+2
3
y
p−1
6
+ cx
p+5
6
y
p−1
3
+ dy
p−1
2
+ e(x + y)
p+1
2
= 0,
where the unknowns a, b, c, d, e are homogeneous polynomials in x and y of respecp−9 p−9 p−1
p−9
5p+3
unknowns
tive degrees p−9
8 , 8 , 8 , 8 and 8 : it is a linear system in
8
and as many equations.
p+1
p+2
p−1
p+5
p−1
p−1
Now introduce the ideal I := hx 2 , x 3 y 6 , x 6 y 3 , y 2 i of R := k[x, y].
p+1
The linear system (3.2) has maximal rank exactly when ·(x + y) 2 : (R/I) p−9 →
8
(R/I) 5p−5 has maximal rank. If α, β ∈ k∗ , this rank is equal to the rank of
8
p+1
multiplication by (αx + βy) 2 , as one sees by performing the change of variables
x0 = αx, y 0 = βy, hence of the multiplication by a power of a general linear
form. But I satisfies the strong Lefschetz property by Lemma 3.6, so that such
multiplication maps are of maximal rank. It follows that the linear system (3.2) is
of maximal rank. Since it has as many unknowns as equations, it has no nontrivial
solution.
The same argument using the strong Lefschetz property shows that the five
other sub-linear systems have maximal rank. Three of them have exactly one
more unknown than equations (those corresponding to exponents of X congruent
to 1,2 and 3 modulo 6). Another has as many unknowns as equations (the one
corresponding to exponents of X congruent to 4 modulo 6), and the last one has
more equations than unknowns. Consequently, only three have non-trivial solutions, and moreover a one-dimensional solution space. It follows, as wanted, that
h0 (P1 , S+ ( 15p+1
4 )) = 3.
Remark 3.8. The matrices of the linear systems in the proof of Proposition 3.7 are
complicated matrices of binomial coefficients, very similar to those appearing in
Han’s thesis [11]. It seems difficult to check directly that they are of maximal rank.
Remark 3.9. Proposition 3.7 and Lemma 3.4 show at once that F ∗ S is unstable
when p ≡ ±3[8]. We will obtain more precise information in Paragraph 3.4.
3.3. Frobenius periodicity. We are ready to prove the strong semistability of S
when p ≡ ±1[8]. We will denote by R the ramification locus of f : it consists of the
points Pi = [ζi : 1], where the ζi are the sixth roots of −1. We will view R either
as a subset of P1 or as a subset of C. We will use the fact that these ramification
points are transitively permuted by the natural action of the group µ6 of sixth roots
of unity on P1 .
14
OLIVIER BENOIST
Proposition 3.10. There exist exact sequences:
⊗ −15p+3
⊕5
4
0 → F ∗ S → (ωC
(3.3)
⊗ −15p−15
4
0 → ωC
(3.4)
)
⊗ −15p+15
4
→ ωC
⊗ −15p−3
⊕5
4
→ (ωC
)
→ 0, if p ≡ 1[8],
→ F ∗ S → 0, if p ≡ −1[8].
Proof. We start with the first statement. By Proposition 3.7, the injective mor⊗ −15p+11
⊕3
4
phism in the exact sequence of Lemma 3.4 writes: F ∗ S → ((ωC
⊗ −15p+3
⊗ −15p+3
ωC 4 ) ⊕ (ωC 4 )⊕4 .
⊗ −15p+3
(ωC 4 )⊕5 is injective at
)
⊕
∗
We will prove that the induced morphism F S →
the level of fibers at every P ∈ C. This will allow to
⊗ −15p+15
conclude, because the quotient will be a line bundle, isomorphic to ωC 4
for
degree reasons.
From its description, it is easy to see that the morphism F ∗ S → f ∗ S− '
⊗ −15p+3
(ωC 4 )⊕4 is an isomorphism on the fibers outside R, and that if P ∈ R,
the kernel of F ∗ S|P → f ∗ S− |P consists of syzygies (A, B, C, D, E) such that
A, B, C, D vanish at P . It remains to see that this kernel is not killed by the
⊗ −15p+3
4
⊗3
composition F ∗ S|P → f ∗ S+ (ωC
)|P → ωC
∗
|P or
⊗ −15p+11
⊕3
4
) |P .
(ωC
⊗3
S+ (ωC
)|P
equivalently that its image
in f
does not belong to
Suppose that it is not the case for P = P1 : then there exists a non-zero
⊗ 15p+1
section (A, B, C, D, E) ∈ H 0 (C, f ∗ S+ (ωC 4 )) = H 0 (P1 , S+ ( 15p+1
4 )) such that
A, B, C, D vanish at P1 . Writing A = (X − ζ1 Y )Ã1 , B = (X − ζ1 Y )B̃1 , C =
Q6
(X − ζ1 Y )C̃1 , D = (X − ζ1 Y )D̃1 , Ẽ1 = i=2 (X − ζi Y )E, one gets a section
σ1 = (Ã1 , B̃1 , C̃1 , D̃1 , Ẽ1 ) ∈ H 0 (P1 , S− ( 15p−3
4 )) such that Ẽ1 vanishes in P2 , . . . , P6 .
For symmetry reasons, using the µ6 -action, there exists, for every 1 ≤ i ≤ 6 a nonzero section σi = (Ãi , B̃i , C̃i , D̃i , Ẽi ) ∈ H 0 (P1 , S− ( 15p−3
4 )) such that Ẽi vanishes at
Pj for j 6= i. Since H 0 (P1 , S− ( 15p−3
))
is
4-dimensional
by Proposition 3.7, these six
4
sections cannot be linearly independent: for instance, σ1 ∈ hσ2 , . . . , σ6 i. It follows
that Ẽ1 vanishes at all Pi . Write F := Ẽ1 /(X 6 + Y 6 ). Then (Ã1 , B̃1 , C̃1 , D̃1 , F ) ∈
H 0 (P1 , S+ ( 15p−3
4 )) is non-zero, contradicting Proposition 3.7.
Let us quickly explain how to prove the second statement using an analogous ar⊗ −15p−3
gument. By Lemma 3.4 and Proposition 3.7, there is a morphism (ωC 4 )⊕5 →
F ∗ S, and it suffices to prove its surjectivity. Using only the four factors coming
from S+ , one gets surjectivity at points P ∈
/ R, and the fact that, if P ∈ R,
all (A, B, C, D, E) ∈ F ∗ S|P such that E = 0 are in the image. Hence, it suf⊗ 15p−9
fices to prove that the unique section (A, B, C, D, E) ∈ H 0 (C, f ∗ S− (ωC 4 )) =
H 0 (P1 , S− ( 15p−9
4 )) satisfies E(P ) 6= 0. But if it didn’t, E would vanish on all of
R by symmetry, and (A, B, C, D, E/(X 6 + Y 6 )) ∈ H 0 (P1 , S+ ( 15p−9
4 )) would be a
non-zero section contradicting Proposition 3.7.
Proposition 3.11. There are isomorphisms:
⊗ 15−15p
4
(i) F ∗ S ' S(ωC
∗
(ii) F S ' S
∨
), if p ≡ 1[8],
⊗ −15−15p
(ωC 4 ),
if p ≡ −1[8].
⊗3
Proof. Let us start with the first statement. We denote by τi ∈ H 0 (C, ωC
) the
sections appearing in the last arrow of the exact sequence (3.3). Tensoring this
COMPLETE FAMILIES OF SMOOTH SPACE CURVES
⊗ 15p−3
4
exact sequence by ωC
15
and taking cohomology, one gets:
⊗ 15p−3
4
0 → H 0 (C, F ∗ S(ωC
i
⊗3
H 0 (C, ωC
)) → k⊕5 −→
).
τ
⊗ 15p−3
But H 0 (C, F ∗ S(ωC 4 )) = 0 by the second part of Lemma 3.4 applied with
m = 15p−3
4 , and Proposition 3.7, so that the τi are linearly independant. This
shows that, up to a twist, (3.3) is the exact sequence defining the syzygy bundle S.
We use a similar argument to prove the second statement. We denote by τi ∈
⊗3
H 0 (C, ωC
) the sections appearing in the first arrow of the exact sequence (3.4).
⊗ 15p+7
4
Tensoring it by ωC
and taking cohomology, one gets:
⊗ 15p+7
4
0 → H 0 (C, ωC )⊕5 → H 0 (C, F ∗ S(ωC
τ
i
⊗−2
)) → H 1 (C, ωC
H 1 (C, ωC )⊕5 .
)) −→
The first vector space in this exact sequence is 10-dimensional. Using the second
part of Lemma 3.4 and Proposition 3.7, one sees that the dimension of the second
⊗−2
one is ≤ 10. It follows that H 1 (C, ωC
)) → H 1 (C, ωC )⊕5 is injective. This map
⊗3
⊗3
⊕5 τi
0
being Serre-dual to k −→ H (C, ωC ), the τi generate H 0 (C, ωC
). This shows
that, up to a twist, the dual of (3.4) is the exact sequence defining the syzygy
bundle S.
Corollary 3.12. If p ≡ ±1[8], S is strongly semistable.
⊗ 15p
2 −15
Proof. Proposition 3.11 shows that when p ≡ ±1[8], F 2∗ S ' S(ωC 4 ): S is
Frobenius periodic up to a twist. It is classical that such bundles are strongly
semistable, let us recall the argument. If S were not semistable, consider the first
2 +15
⊗ −15p
4
graded piece F ⊂ S of its Harder-Narasimhan filtration. Then F 2∗ F(ωC
)⊂
S would have greater slope than F contradicting the choice of F. Hence S is
semistable. By the periodicity property, so are all its Frobenius pull-backs.
3.4. Unstability. Let us now describe what happens when p ≡ ±3[8].
Proposition 3.13. If p ≡ ±3[8], then F ∗ S is not semistable and its HarderNarasimhan filtration is strong. This filtration is of the form:
⊗ −15p−3
4
(i) 0 → T → F ∗ S → ωC
(ii) 0 →
⊗ −15p+3
ωC 4
→ 0 if p ≡ 3[8],
→ F ∗ S → T → 0 if p ≡ −3[8].
Proof. We will only prove the first statement, as the second is similar. From
⊗ −15p−3
Lemma 3.4 and Proposition 3.7, we get a morphism F ∗ S → f ∗ S− → ωC 4 .
Let us prove that it is surjective. Since F ∗ S → f ∗ S− is surjective at all points
P ∈
/ R, and since if P ∈ R, the image of F ∗ S|P → f ∗ S− |P consists of syzygies (A, B, C, D, E) such that E(P ) = 0, we need to show that not all sections
⊗ 15p−5
(A, B, C, D, E) ∈ H 0 (C, f ∗ S− (ωC 4 )) = H 0 (P1 , S− ( 15p−5
4 )) satisfy E(P ) = 0.
Suppose it is not the case: then, by symmetry using the µ6 -action, for all sections
6
6
(A, B, C, D, E) ∈ H 0 (P1 , S− ( 15p−5
4 )), E would vanish on R. Dividing E by X +Y ,
15p−5
0
1
we would get a non-sero section in H (P , S+ ( 4 )), contradicting Proposition
3.7. Hence our morphism was surjective, we denote its kernel by T .
⊗ −15p+1
From Lemma 3.4 and Proposition 3.7 again, we get a morphism (ωC 4 )⊕2 →
∗
f S+ → F ∗ S. Let us prove that it is injective on every fiber. Since f ∗ S+ → F ∗ S is
injective on the fibers at P ∈ R, and since, if P ∈
/ R, the kernel of f ∗ S+ |P → F ∗ S|P
16
OLIVIER BENOIST
consist of syzygies (A, B, C, D, E) such that A, B, C, D all vanish at P , it suffices
⊗ 15p−1
to rule out the existence of a section (A, B, C, D, E) ∈ H 0 (C, f ∗ S+ (ωC 4 )) =
H 0 (P1 , S+ ( 15p−1
4 )) such that A, B, C, D all vanish at P . We proceed by contradiction. Then, for symmetry reasons, there exist for 1 ≤ i ≤ 6 a section
Di all vanish at Pi .
(Ai , Bi , Ci , Di , Ei ) ∈ H 0 (P1 , S+ ( 15p−1
4 )) such that Ai , Bi , Ci , Q
Dividing Ai , Bi , Ci , Di by X − ζi Y and multiplying E by j6=i (X − ζj Y ), we
get non-zero sections σi = (Ãi , B̃i , C̃i , D̃i , Ẽi ) ∈ H 0 (P1 , S− ( 15p−5
4 )) such that Ẽi
))
is 3-dimensional,
vanishes at Pj for j 6= i. By Proposition 3.7, H 0 (P1 , S− ( 15p−5
4
hence the σi cannot be linearly independent, say σ1 ∈ hσ2 , . . . , σ6 i. Then Ẽ1 vanishes at all the Pi and (Ã1 , B̃1 , C̃1 , D̃1 , Ẽ1 /(X 6 + Y 6 )) ∈ H 0 (P1 , S+ ( 15p−5
4 )) is a
non-zero section contradicting Proposition 3.7.
⊗ −15p+1
4
Since there are obviously no non-zero morphisms ωC
⊗ −15p+1
(ωC 4 )⊕2
⊗ −15p−3
4
→ ωC
, the
subbundle
factors through T , and a degree computation shows that
this realizes T as an extension:
(3.5)
⊗ −15p+1
⊕2
4
0 → (ωC
)
⊗ −15p+1
4
→ T → ωC
→ 0.
Now T is strongly semistable as an extension of strongly semistable bundles of the
⊗ −15p−3
4
same slope, and writing F ∗ S as an extension of ωC
Harder-Narasimhan filtration of F ∗ S.
by T indeed realizes the
3.5. Hilbert-Kunz multiplicities. We will apply here our results to the computation of Hilbert-Kunz multiplicities. Let us first recall the definition.
Definition 3.14. Let A be a noetherian n-dimensional ring of characteristic p and
m be a maximal ideal of A. We denote by m[e] the ideal of A generated by pe -th
powers of elements of m. Then the Hilbert-Kunz multiplicity of (A, m) is defined
to be:
l(A/m[e] )
eHK (A, m) := lim
.
e→∞
pne
This invariant was first considered by Kunz [20], and the limit was shown to
exist and to be finite by Monsky [27]. It is difficult to compute in general.
We will be interested in the following geometric case:
Definition 3.15. Let C be a smooth curve endowed with a line bundle L whose
sections
embed C as a projectively normal curve.
the section ring A :=
L
L Consider
0
⊗l
0
⊗l
H
(C,
L
)
with
its
maximal
ideal
m
:=
H
(C,
L
). Then define:
l≥0
l>0
eHK (C, L) := eHK (A, m).
In this particular case, Brenner ([3] Theorem 1) and Trivedi ([32] Theorem 4.12)
have related the Hilbert-Kunz multiplicity to the behaviour of a syzygy bundle with
respect to the Frobenius:
Theorem 3.16. Let C be a smooth curve endowed with a degree d line bundle L
whose sections embed C in Pk−1 as a projectively normal curve. Using Theorem
1.2, choose a finite morphism of degree e from a smooth curve f : C 0 → C such that
f ∗ SyzC (L) has a strong Harder-Narasimhan filtration. Let ri and eδi be the ranks
and degrees of the graded pieces of this filtration (so that ri and δi are independent
COMPLETE FAMILIES OF SMOOTH SPACE CURVES
17
of f ). Then:
eHK (C, L) =
1 X δi2
kd
−
.
2d i ri
2
Applying this theorem using Corollary 3.12 and Proposition 3.13, we get:
Theorem 3.17 (Theorem 0.6). Let C be the curve of genus 2 with equation Z 2 =
X 6 + Y 6 . Then:
⊗3
(i) eHK (C, ωC
) = 15
4 if p ≡ ±1[8],
⊗3
15
(ii) eHK (C, ωC ) = 4 + 4p12 if p ≡ ±3[8].
4. Open questions
In this section, we discuss the difficulties encountered when trying to give other
examples of complete families of non-degenerate smooth space curves.
4.1. Curves of genus 2 and degree 5. Let us start by considering families of
curves of genus 2 and degree 5: we may ask whether it is possible to apply the
strategy of Section 3 to other curves than {Z 2 = X 6 + Y 6 }, in order to get more
examples of families over higher genus curves. On the one hand, it is very likely
that our method extends to other particular curves, on the other hand, it seems
more difficult to obtain examples in arbitrary positive characteristic. An even more
optimistic question is:
Question 4.1. Let C be a curve of genus 2 defined over the algebraically closure
of a finite field. Does there exist a non-trivial complete family of degree 5 smooth
space curves all of whose members are isomorphic to C ?
Keeping the notations of Paragraph 0.3, and applying Lange-Stuhler theorem
(Proposition 1.5), we see that it is equivalent to find a non-constant curve B → A
defined over a finite field such that the restriction of E to B is strongly semistable. It
is not difficult to check that E is semistable when restricted to a general translate of
the theta divisor, so that E itself is semistable with respect to the natural principal
polarization Θ, hence strongly Θ-semistable by [26] Theorem 2.1. Consequently,
we are exactly in need of a restriction theorem for strong semistability. The best
known result, due to Langer [24] Theorem 3.1, shows that the restriction of E to a
very general member of |nΘ|, n 0, is strongly semistable. It is not sufficient to
answer Question 4.1, as we need a curve defined over a finite field.
4.2. Other isotrivial families. In view of Theorem 0.2, it would be interesting
to answer the following question:
Question 4.2. Do there exist non-trivial isotrivial complete families of non-linearly
normal smooth space curves ?
Indeed, such an example would come from a complete family of smooth polarized
curves whose section bundle E is not strongly semistable, the first graded piece of
its strong Harder-Narasimhan filtration providing the embedding. The first case of
Question 4.2 is:
Question 4.3. Do there exist non-trivial complete families of smooth space curves
of genus 2 and degree 6 ?
A question in the same spirit as Question 4.2, but with a different flavor is:
18
OLIVIER BENOIST
Question 4.4. Do there exist non-trivial complete families of smooth space curves
whose general member is linearly normal, but such that some particular member is
not ?
Let us point out and discuss an interesting particular case of Question 4.4:
Question 4.5. Do there exist non-trivial isotrivial complete families of smooth
space curves of genus 5 and degree 8 ?
By Riemann-Roch, if C is a smooth curve of genus 5, the locus in Pic8 (C)
consisting of line bundles having 4 sections embedding C in P3 is the complement
of a divisor ∆ := {ωC (P + Q − R − S)|P, Q, R, S ∈ C}. The line bundles in ∆ all
have 4 sections that do not induce an embedding of C, except for ωC itself that has
5 sections embedding C in P4 . One checks easily that degree 8 line bundles on C
together with a 4-dimensional space of sections are parametrized by the blow-up Z
of Pic8 (C) at ωC , and that the locus where the 4 sections provide an embedding of
¯ of ∆. A first step to answer Question
C is the complement of the strict transform ∆
¯ (note that, since ∆ ∈ |6Θ| is ample,
4.5 is to construct a curve in Z that avoids ∆
8
no curve in Pic (C) avoids ∆). I do not know if such curves exist.
Consequently, when trying to answer Question 4.5 using Proposition 0.4, there
is a difficulty in constructing an appropriate complete family of polarized curves,
even before checking the relevant strong semistability condition.
4.3. Non-isotrivial families. Let us finally discuss non-isotrivial families. The
general question is:
Question 4.6. Do there exist non-isotrivial complete families of smooth space
curves ?
Note that, since there are no non-isotrivial families of smooth curves over an
elliptic curve ([31] Théorème 4), it will not be possible to construct examples in
characteristic 0 using Proposition 0.4.
A first idea, close to the use of curves of genus 2 and degree 5 in the isotrivial
case, is to use curves of genus g and degree g + 3 for small values of g, to ensure
the existence of exactly four sections. This strategy fails:
Proposition 4.7. There do not exist complete families of smooth space curves of
genus 3 and degree 6, nor of genus 4 and degree 7.
Proof. Let us explain the genus 3 and degree 6 case, the other one being similar.
By Riemann-Roch, every line bundle of degree 6 on a curve C of genus 3 has 4
sections, and those that do not induce an embedding of C are those in the theta
divisor {ωC (P + Q)|P, Q ∈ C}. Since this theta divisor is ample, this rules out
the existence of isotrivial families. To show that non-isotrivial families do not exist
either, apply [7] Corollary 4.5.
A more promising particular case is given by the following question:
Question 4.8. Do there exist non-trivial complete families of canonical genus 4
space curves ?
The four sections of the canonical bundle of a curve of genus 4 embed this curve
in P3 if and only if it is not hyperelliptic. Hence, in order to answer Question 4.8
COMPLETE FAMILIES OF SMOOTH SPACE CURVES
19
using Proposition 0.4, we need to construct complete families of non-hyperelliptic
genus 4 curves, and then to study the Hodge bundle on the base of the family.
There are two classical ways to construct complete families of smooth curves. In
the first one, dating back to Kodaira [19], all the curves are coverings of a fixed curve
with moving ramification points. Here, the curves in the family could be coverings
of degree 2 of a fixed curve of genus 2 ramified over two points. Unfortunately, such
families will never allow to answer Question 4.8: one checks that the Hodge bundle
on the base of such families is unstable by splitting it using the covering group.
The second one is attributed to Mumford by Oort [30]. The base of the family is
constructed as a general complete intersection in a compactification of the moduli
space of smooth curves whose boundary has codimension 2. Such families are not
explicit. In characteristic p, when trying to prove that the Hodge bundle on the base
of such a family is strongly semistable, one would want to use restriction theorems
for strong semistability as above. Another possibility would be to construct a family
in characteristic 0 for which the Hodge bundle is semistable, and then to reduce it
to characteristic p. At this point, we would like to use the fact that a semistable
bundle in characteristic 0 specializes modulo p to a strongly semistable bundle for
infinitely many p. This question is completely open (see for instance the discussion
in [25], paragraph 5).
It might also be possible to use different arguments to answer Question 4.8.
Canonical genus 4 curves in P3 are exactly smooth complete intersections of a
quadric and a cubic. Consequently, their Hilbert scheme is very easy to describe.
A strategy to answer Question 4.8 in arbitrary characteristic, based on the study
of the birational geometry of this Hilbert scheme has been proposed in [2].
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Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg
et CNRS, 7 rue René Descartes, 67000 Strasbourg, FRANCE
E-mail address: olivier.benoist@unistra.fr