On stable rank two bundles with canonical determinant and many sections

A. BERTRAM^{1} Department of Mathematics, University of Utah, Salt Lake City, UT 84112, email: bertram@math.utah.edu

B. FEINBERG email:bfeinberg@state.ma.us

1 INTRODUCTION

Let C be a smooth projective curve of genus g ≥ 2, and consider the moduli space M_{C}^{s}(2, 2g − 2) of stable rank two bundles on C of degree 2g − 2. This is a smooth, quasiprojective variety of dimension (3g − 3) + g, and via the “determinant map”

Δ : M_{C}^{s}(2, 2g − 2) → P ic^{2g−2}(C)

sending E to ∧^{2}E, is a fiber bundle over the Jacobian, with fibers of dimension 3g − 3. Let M_{C}^{s}(2, K) := Δ^{−1}(K_{C}) be the moduli space of stable rank two bundles with canonical determinant.

The moduli space M_{C}^{s}(2, 2g − 2) naturally compactifies to M_{C}(2, 2g − 2), the moduli space of s-equivalence classes of semistable bundles (see [Se]). For our purposes, the potential advantages of the fact that M_{C}(2, 2g − 2) is projective are outweighed by the disadvantages: (1) that M_{C}(2, 2g − 2) is singular (for g ≥ 3) away from M_{C}^{s}(2, 2g − 2) and (2) that M_{C}(2, 2g − 2) does not parametrize bundles away from M_{C}^{s}(2, 2g − 2), but merely equivalence classes of bundles. As a result, we will work with M_{C}^{s}(2, 2g − 2) in this paper.

Because χ(E) = 0 for any rank two bundle of degree 2g − 2, it follows that the subscheme Θ ⊂ M_{C}^{s}(2, 2g − 2) parametrizing stable bundles E with H^{0}(C, E) ≠ 0 is a divisor, and following the line bundle precedent, this is called the Riemann theta divisor for rank two bundles. More generally, using Brill-Noether notation, we denote by W_{2,2g−2}^{r} the locus (with a natural scheme structure) of stable bundles E ∈ M_{C}^{s}(2, 2g − 2) such that h^{0}(C, E) ≥ r + 1.

The central theme of this paper may be stated as follows:

The Brill-Noether loci are not fiber bundles over the Jacobian.

Indeed, if we define W_{2,K}^{r} as the scheme-theoretic intersection of W_{2,2g−2}^{r} and M_{C}^{s}(2, K), then we will see that the W_{2,K}^{r} are distinguished among the intersections of W_{2,2g−2}^{r} with fibers of Δ. We hope to convince the reader that these loci are sufficiently interesting to be worth further study.

Reasoning by analogy with the well-established Brill-Noether theory for line bundles (see e.g. [ACGH]), several authors postulated expected dimensions for the Brill-Noether loci W_{2,d}^{r} ⊂ M_{C}^{s}(2, d) (cf. [T] and [Su]). In our case, this gives an expected dimension for W_{2,2g−2}^{r} of

ρ_{2g−2}(r) = 4g − 3 − (r + 1)^{2}.

1Research of the first author was in part supported by NSF grant DMS-9218215

Thus following the lead of Brill-Noether for line bundles, one might be tempted to make the following predictions (listed in descending order of likelihood) about the W_{2,2g−2}^{r} on a general curve C of genus g:

(a) Non-existence: If ρ_{2g−2}(r) < 0, then W_{2,2g−2}^{r} = ∅.

(b) Dimension: If ρ_{2g−2}(r) ≥ 0, then dim(W_{2,2g−2}^{r}) = ρ_{2g−2}(r).

(c) Smoothness: W_{2,2g−2}^{r} is smooth of dimension ρ(r) away from W_{2,2g−2}^{r+1}.

However these predictions are too naive. We give counterexamples in genus 5 and 6, which we believe are systematic, and are due to the fact that for genus 5 and above and large r, the loci W_{2,K}^{r} can contribute whole components of the “wrong” dimension to W_{2,2g−2}^{r}. We will present our analysis of W_{2,K}^{r} for small genus and r in §3 and §4.

The source of our counterexamples in genus 5 and 6 is the fact that for large r, the dimension of W_{2,K}^{r} itself is greater than ρ_{2g−2}(r). If W_{2,2g−2}^{r} were a fiber bundle over the Jacobian, then the expected dimension of W_{2,K}^{r} ought to be ρ_{2g−2}(r) − g. In fact, we will give an asymptotically far larger expected dimension for W_{2,K}^{r} of

ρ_{K}(r) = 3g − 3 − \binom{r + 2}{2}

in §2. In addition, bolstered by our analysis in §3 and §4, we anticipate that this expected dimension is the “correct” one for W_{2,K}^{r} and make the following (we hope less naive) conjecture, which is also conjectured by Mukai in [Mu2]:

Conjecture: If C is a general curve of genus g, then:

(a) Non-existence: If ρ_{K}(r) < 0 then W_{2,K}^{r} is empty.

(b) Dimension: If ρ_{K}(r) ≥ 0 then dim(W_{2,K}^{r}) = ρ_{K}(r).

(c) Smoothness: W_{2,K}^{r} is smooth away from W_{2,K}^{r+1}.

Note that part (b) of the conjecture logically splits into two parts, namely existence of W_{2,K}^{r} when ρ_{K}(r) ≥ 0 and the dimension of W_{2,K}^{r} once it is shown to be nonempty. In the line bundle case, it can be shown that for all curves, the Brill-Noether loci exist when their expected dimension is non-negative. However, all proofs of this fact rely on the projectivity of P ic^{d}(C). In our case, as we already mentioned, the varieties M_{C}^{s}(2, K) are not projective, and we suspect that for special curves dim(W_{2,K}^{r}) may be empty even though ρ_{K}(r) ≥ 0. We will prove existence for general curves of genus g ≤ 12 in §3. The rest of part (b) and (c), as in the line bundle case, is equivalent to the injectivity of a “Petri” map. We will introduce this map in §2 and investigate it in §4 for general curves of genus ≤ 7. In §3, we will also establish (a) for general curves of small genus.

2 EXPECTED DIMENSIONS

In §1, we introduced the expected dimension ρ_{K}(r) = 3g − 3 − \binom{r+2}{2} for W_{2,K}^{r}. We will justify this in three ways:

(1) W_{2,K}^{r} is a skew-isotropic degeneracy locus.

We use the skew-symmetric form ∧^{2}E ≅ K just as Mumford uses a symmetric form in [M1] and [M2] to construct Brill-Noether loci in Prym varieties as symmetric-isotropic degeneracy loci.

Suppose X is a quasi-projective scheme and E is a family of stable rank two bundles with canonical determinant on C parametrized by X. That is, E is a rank two bundle on C ×X such that for each closed point x ∈ X, E_{x} := E|_{C×{x}} is stable with canonical determinant. Let π_{C} and π_{X} denote the projections from C × X to C and X respectively. Let D be an effective divisor on C of degree d >> 0. Let E(D) := E ⊗ π_{C}^{∗}O_{C}(D).

Then exactly as in [M1], there is a nondegenerate skew-symmetric form on the rank 4d bundle π_{X}_{∗}(E(D)/E(−D)) such that the rank 2d subbundles π_{X}_{∗}(E(D)) and π_{X}_{∗}(E/E(−D)) are isotropic everywhere. Furthermore, for each closed point x ∈ X, we have an identity on the intersection of the fibers of the subbundles: H^{0}(E_{x}) = H^{0}(C, E_{x}(D)) ∩ H^{0}(C, E_{x}/E_{x}(−D)) ⊂ H^{0}(C, E_{x}(D)/E_{x}(−D)).

Suppose V is a fixed vector space of dimension 2k with a non-degenerate skew form B : ∧^{2}V → C. Let Σ ⊂ Gr(k, V) be the subvariety of the Grassmannian consisting of the k-dimensional isotropic subspaces of V. Then it is an easy check (as in [H]) that for fixed W_{0} ∈ Σ, the set

{W ∈ Σ | dim(W ∩ W_{0}) ≥ r + 1}

is of codimension \binom{r+2}{2} in Σ. An inequality on dimensions immediately follows: dim(\{x ∈ X | H^{0}(C, E_{x}) ≥ r + 1\}) ≥ dim(X) − \binom{r+2}{2} whenever this locus is nonempty. Thus W_{2,K}^{r}, if nonempty, has dimension at least ρ_{K}(r).

(2) The Petri map factors.

Consider the Petri map for rank two bundles

μ : H^{0}(C, E) ⊗ H^{0}(C, E^{∗} ⊗ K) → H^{0}(C, E ⊗ E^{∗} ⊗ K).

If ∧^{2}E ≅ K, then E ≅ E^{∗} ⊗ K, and the domain of the Petri map can be thought of as H^{0}(C, E)⊗H^{0}(C, E). Then μ(∧^{2}H^{0}(E)) ⊂ H^{0}(C, K), and the subspace H^{0}(C, Sym^{2}E)^{∗} ⊂ H^{0}(C, E ⊗ E)^{∗} is the Zariski tangent space to M_{C}^{s}(2, K) at E. As in the line bundle case (see [ACGH]), the Zariski tangent space to W_{2,2g−2}^{r} at E is (cokerμ)^{∗}. Thus, the Zariski tangent space to W_{2,K}^{r} at E is an intersection:

T_{W_{2,K}^{r}} (E) = (cokerμ)^{∗} ∩ H^{0}(C, Sym^{2}E)^{∗}

or equivalently, the dual of the cokernel of the factored Petri map:

μ̃ : Sym^{2}H^{0}(C, E) → H^{0}(C, Sym^{2}E)

and we obtain the same lower bound as in (1) (indeed, (2) is an infinitesimal version of (1)...see [Mu2]).

Moreover, if the factored Petri map is injective for E ∈ W_{2,K}^{r} − W_{2,K}^{r+1}, then in a neighborhood of E, W_{2,K}^{r} is smooth of the expected dimension. It is always injective if and only if (c) of the conjecture holds. Notice that as soon as \binom{r+1}{2} > g, the full Petri map has no chance of being injective at E ∈ W_{2,2g−2}^{r} since dim(∧^{2}H^{0}(C, E)) > dim(H^{0}(C, K)). We will show that this occurs already for g = 5.

(3) An open subset of W_{2,K}^{r} is the quotient of a symmetric degeneracy locus.

Let P(Ext^{1}(K, O_{C})) be the projective space parametrizing extensions of the form 0 → O_{C} → E → K → 0 modulo equivalence and scalars. If E ∈ Θ has a nowhere vanishing section, then E is in the image of the natural birational “extension map” carrying extensions to the bundle they define.

ϕ : P(Ext^{1}(K, O)) − −− > Θ

Now the locus of points in P(Ext^{1}(K, O)) such that the associated bundle E has r + 1 independent sections is easily identified as a symmetric degeneracy locus. Namely, the short exact sequence 0 → O → E → K → 0 gives rise to a long exact sequence on cohomology:

... → H^{0}(C, E) → H^{0}(C, K) →^{δ} H^{1}(C, O) → ...

By Serre duality, H^{1}(C, O_{C})^{∗} ≅ H^{0}(C, K), and the map δ is symmetric. Also by Serre duality, P(Ext^{1}(K, O)) = P(H^{0}(C, K^{2})^{∗}). As long as C is not hyperelliptic, Noether’s theorem tells us that the multiplication map Sym^{2}(H^{0}(C, K)) → H^{0}(C, K^{⊗2}) is surjective, so P(Ext^{1}(K, O)) is a subspace of P(Sym^{2}H^{0}(C, K)^{∗}).

Since dim(H^{0}(C, E)) = dim(ker(δ))+1, we see that the locus of extensions such that dim(H^{0}(C, E)) ≥ r + 1 is the restriction of the “universal” degeneracy locus D_{r} ⊂ P(Sym^{2}H^{0}(C, K)^{∗}) of symmetric maps with kernel of dimension at least r.

If E is a stable bundle with a nowhere-vanishing section, then it turns out that the pre-image of E in P(Ext^{1}(K, O)) is isomorphic to an open subset of P(H^{0}(C, E)) (see [B]). Thus the expected dimension of any component of W_{2,K}^{r} where the general bundle admits a nowhere-vanishing section is:

dim(P(Ext^{1}(K, O))) − codim(D_{r}) − r = 3g − 4 − \binom{r + 1}{2} − r = ρ_{K}(r)

3 EXISTENCE IN LOW GENERA

We will use the third approach of §2 to establish the existence of W_{2,K}^{r} when ρ_{K}(r) ≥ 0 and C is general. Namely, consider again the map

ϕ : P(Ext^{1}(K, O)) − − > Θ.

As we observed in §2, if C is not a hyperelliptic curve, then P(Ext^{1}(K, O)) ≅ P(H^{0}(C, K^{2})^{∗}) may be thought of as a projective linear subspace in the universal projective space of symmetric maps q : H^{0}(C, K) → H^{0}(C, K)^{∗}, and the intersection D_{r} ∩ P(H^{0}(C, K^{2})^{∗}) with the universal degeneracy loci give the loci of extensions 0 → O → E → K → 0 such that dim(H^{0}(C, E)) ≥ r +1. This intersection is always nonempty of dimension at least ρ_{K}(r) + r whenever ρ_{K}(r) ≥ 0.

On the other hand, let ψ : C → P(H^{0}(C, K^{2})^{∗}) be the bicanonical embedding (we assume g ≥ 3). Let Sec^{d}(C) be the secant variety defined as the closure of the union of the spans of all d-tuples of points on C. Then Sec^{g−1}(C) ⊂ P(H^{0}(C, K^{2})^{∗}) is an irreducible subvariety of dimension (g − 1) + (g − 2) = 2g − 3. This is the locus of all extensions giving rise to bundles that are not stable. Indeed, if an extension belongs to the span of a divisor D, then there is a map E → O_{C}(D). If deg(D) ≤ g −1, then by definition, E is not stable. (See [B] for more details.) Thus one way of establishing the existence of W_{2,K}^{r} is to show that D_{r} ∩P(H^{0}(C, K^{2})^{∗}) is not contained in Sec^{g−1}(C). This approach works well enough in low genera, as we will see, but seems difficult to implement in general, mainly because the dimension of Sec^{g−1}(C) grows linearly with g, while the dimensions of D_{r} ∩P(H^{0}(C, K^{2})^{∗}) for large r may remain small.

genus 3: To show: W_{2,K}^{2} ≠ ∅ (ρ_{K}(2) = 0 and ρ_{K}(3) = −4).

Here, P(H^{0}(C, K^{2})^{∗}) = P(Sym^{2}H^{0}(C, K)^{∗}) ≅ P^{5} and D_{2} is the Veronese surface in P^{5}. So we need to show that D_{2} ⊄ Sec^{2}(C). Clearly, it suffices to show that D_{2} ∩ (Sec^{2}(C) − C) = ∅. But suppose not. Then there must be a bundle E (arising from an extension in the intersection) admitting a surjective map E → O_{C}(p + q) and such that dim(H^{0}(C, E)) ≥ 3. We may then write:

0 → O(p′ + q′) → E → O(p + q) → 0

Since C is not hyperelliptic, each of O(p+q) and O(p′+q′) has only one section, and we obtain a contradiction. Note in this example that D_{2} is irreducible of dimension 2, so in fact W_{2,K}^{2} consists of a single point.

genus 4: To show: W_{2,K}^{2} ≠ ∅ (ρ_{K}(2) = 3, so dim(D_{2} ∩ P(H^{0}(C, K^{2})^{∗}) ≥ 5).

Because dim(D_{2} ∩ P(H^{0}(C, K^{2})^{∗}) ≥ dim(Sec^{3}(C)) and the latter variety is irreducible, it suffices to find a single extension in Sec^{3}(C) such that the associated bundle E has fewer than 3 independent sections. But if p+q+r is a general effective divisor of degree 3, then O(p + q + r) and K(−p − q − r) each have just one section, and any extension:

0 → K(−p − q − r) → E → O(p + q + r) → 0

gives an E with at most 2 independent sections. Such E are associated to the general extension in Sec^{3}(C), so we are done.

Note: The hyperplane P(H^{0}(C, K^{2})^{∗}) ⊂ P(Sym^{2}(H^{0}(C, K)^{∗})) is general if C is a general curve of genus 4, and it follows that W_{2,K}^{2} is irreducible, of dimension 3.

The genus 3 and 4 examples were also considered by Laszlo [L].

genus 5: To show: W_{2,K}^{3} ≠ ∅ (ρ_{K}(3) = 2, so dim(D_{3} ∩ P(H^{0}(C, K^{2})^{∗}) ≥ 5)

Here, dim(Sec^{4}(C)) = 7 but the intersection of Sec^{4}(C) − Sec^{3}(C) with the degeneracy locus D_{3} ∩ P(H^{0}(C, K^{2})^{∗}) is nonempty. We will parametrize this intersection, and show that its dimension is small enough that its closure cannot contain D_{3} ∩ P(H^{0}(C, K^{2})^{∗}). This proves existence.

Suppose E comes from an extension which lies in the intersection. Then E fits into the following exact sequence:

(∗) 0 → K(−p − q − r − s) → E → O(p + q + r + s) → 0

for some p, q, r, s ∈ C. Because h^{0}(C, E) ≥ 4, the line bundle O(p + q + r + s) must have two independent sections and the connecting homomorphism δ : H^{0}(C, O(p + q + r + s)) → H^{1}(C, K(−p − q − r − s)) must be zero.

Since C is general, G_{4}^{1} ⊂ Pic^{4}(C) is a smooth curve and for all L ∈ G_{4}^{1}, K⊗L^{−1} ≅ L. The space of extensions (∗) for a given g_{4}^{1} is parametrized by H^{1}(C, K(−2p − 2q − 2r − 2s)) ≅ H^{0}(C, O(2p + 2q + 2r + 2s))^{∗} which therefore has dimension 4. On the other hand, the map from extensions to connecting homomorphisms: H^{0}(C, O(2p + 2q + 2r + 2s))^{∗} → Sym^{2}(H^{0}(C, O(p + q + r + s))^{∗} is dual to the multiplication map, so is clearly surjective, and has a kernel of dimension one. This means that there are one non-split bundle E and one split bundle E′ fitting into each sequence (∗), such that dim(H^{0}(C, E)) = 4.

The locus of extensions 0 → O → E → K → 0 so that dim(H^{0}(C, E)) = 4 and E has a subbundle of degree four is therefore of dimension at most equal to dim(G_{4}^{1}) + dim(H^{0}(C, E)) − 1 = 4. (For the split extensions, the dimension is actually 3.) In any case, this is strictly less than the dimension of the degeneracy locus, and the proof is completed.

Note: Since a general canonical curve of genus 5 is a complete intersection of 3 general quadrics in P^{4}, the subspace P(H^{0}(C, K^{2})^{∗}) ⊂ P(Sym^{2}H^{0}(C, K)^{∗}) of codimension 3 is also general, and it follows that W_{2,K}^{3} is irreducible of dimension 2. Note that this is larger than ρ_{2g−2}(3) = 1. For higher genus, this argument for irreducibility and dimension no longer works.

genus 6: To show: W_{2,K}^{4} ≠ ∅ (ρ_{K}(4) = 0, so dim(D_{4} ∩ P (H^{0}(C, K^{2})^{∗}) ≥ 4)

(1) (D_{4} ∩ P (H^{0}(C, K^{2})^{∗}) ∩ (Sec^{5}(C) − Sec^{4}(C)) = ∅. There are no g_{5}^{2}’s on a general curve of genus 6.

(2) Suppose E is associated to an extension in (D_{4} ∩P (H^{0}(C, K^{2})^{∗})∩(Sec^{4}(C)− Sec^{3}(C)). Then as for genus 5, E must fit into an exact sequence:

(∗) 0 → K(−p − q − r − s) → E → O(p + q + r + s) → 0

where the connecting homomorphism is zero. (K(−p − q − r − s) is a g_{6}^{2}.) This time, though, the number of g_{4}^{1}’s is finite, and for each such g_{4}^{1}, the map from extensions to connecting homomorphisms is an isomorphism, so an E in (∗) with 5 independent sections must split. But such E’s have a two-dimensional family of endomorphisms, from which it follows that the dimension of the locus of extensions in the intersection is at most dim(G_{4}^{1}) + 5 − 2 = 3, and as in the genus 5 case, this implies existence.

Note: For genus six, the naive ρ_{2g−2}(4) is negative, so the nonemptiness of W_{2,K}^{4} contradicts the naive non-existence conjecture.

Note: For a general curve of genus 6 (as we have seen) and greater, it is easy to see that with the exception of genus 9 and 10, a strictly semistable bundle (i.e. a bundle with a line subbundle of degree g − 1) can never have the maximal number r + 1 of independent sections so that ρ_{K}(r) ≥ 0. This may mean that one could try to prove existence of W_{2,K}^{r} for largest r by “standard” techniques (i.e. Porteous formula or ampleness) on the compactification M_{C}(2, 2g − 2), arguing that any nonempty class so obtained must lie in M_{C}^{s}(2, 2g − 2).

The argument for genus 6 can be generalized as follows. Find the first d ≤ g − 1 so that (D_{r} ∩P(H^{0}(C, K^{2})^{∗}))∩(Sec^{d}(C)−Sec^{d−1}(C)) ≠ ∅. Then argue that for this choice of d, the dimension of the intersection is less than the expected dimension of D_{r} ∩ P(H^{0}(C, K^{2})^{∗}). Conclude that W_{2,K}^{r} ≠ ∅.

This argument works well at least up to genus 12. However, it seems not to work beyond that point, the reason being, we suspect, that the locus of extensions parametrizing unstable bundles with a large number of sections begins to have larger than expected dimension. In particular, genus 13 seems to be a very delicate case.

As additional evidence for ρ_{K}(r), we show nonexistence for small genus and ρ_{K}(r) < 0. (This is part (a) of the conjecture.)

genus 3: To show: W_{2,K}^{3} = ∅.

If a rank two bundle has 2n sections, it is immediate that it has a section vanishing at n points. Thus, if E ∈ W_{2,K}^{3}, then there is a section of E vanishing at 2 points. This contradicts stability.

genus 4: To show: W_{2,K}^{3} = ∅.

If E ∈ W_{2,K}^{3}, then E must fit into an exact sequence:

0 → O(p + q) → E → K(−p − q) → 0

But this cannot produce h^{0}(C, E) ≥ 4 since C is not hyperelliptic.

genus 5: To show: W_{2,K}^{4} = ∅.

If E ∈ W_{2,K}^{4}, then either E fits into an exact sequence:

0 → O(p + q) → E → K(−p − q) → 0

and as in genus 4, not enough sections are produced, or E fits into an exact sequence:

0 → O(p + q + r) → E → K(−p − q − r) → 0

which is even worse, since there are no g_{3}^{1}’s on a general curve of genus 5.

genus 6: To show: W_{2,K}^{5} = ∅.

Here, E ∈ W_{2,K}^{5} would have a section vanishing at 3 or 4 points. Both of these are out because there are no g_{3}^{1}’s and a g_{4}^{1} sub-line bundle wouldn’t give enough sections.

These arguments also can be extended out to at least genus 8. If we additionally use Segre’s theorem, which states that any rank two bundle E with canonical determinant must have a subbundle of degree at least [\frac{g−1}{2}] (cf. [L1]), then we can also prove nonexistence for genus 9, 11 and 13 by these arguments.

However, genus 10 and 12 are more subtle. Mukai (cf. [Mu1] or [V]) has shown that if C is a curve of genus 2s, s ≥ 5, general among the generators of Pic of a K3 surface, then there is always a stable rank 2 bundle on C with canonical determinant and s+2 independent sections. This is exactly where we need to prove non-existence for genus 10 and 12. In light of Lazarsfeld’s theorem (cf. [L2]) that such curves are Petri general for line bundles, it follows that unlike the other low genera cases, the locus of curves of genus 10 or 12 that satisfy non-existence in our conjecture will not contain the Petri general locus, if indeed it is non-empty.

4 INJECTIVITY OF THE PETRI MAP

In this section, we show that the modified Petri map is injective for small values of r. This will complete the picture for several of the low genus examples studied in §3. In particular, we prove:

Proposition 4.1: If C is general of genus g ≥ 2, E is a stable bundle with canonical determinant generically generated by its global sections and dim(H^{0}(C, E)) ≤ 5, then

μ : Sym^{2}(H^{0}(C, E)) → H^{0}(C, Sym^{2}(E))

is injective.

We note in passing that if E is not generically generated by its global sections, there is a subbundle L of E of degree less than g−1, such that H^{0}(C, L) = H^{0}(C, E). It is not hard to show that if h^{0}(C, E) ≤ 4, the map μ : Sym^{2}(H^{0}(C, L)) → H^{0}(C, L^{2}) is injective. A general curve can only admit a line bundle of degree less than g − 1 with 5 sections if g ≥ 35, so for now we can safely ignore the bundles not generated by their global sections, and conclude from the Proposition and §3 that:

The conjecture holds for general curves of genus ≤ 7

Proof of the Proposition: For h^{0}(C, E) ≤ 2, the proposition is easily verified. For h^{0}(C, E) ≥ 3, we begin by giving a geometric description of the injectivity of μ. The evaluation map:

ε : H^{0}(C, E) ⊗ O_{C} → E

induces the morphism:

Φ_{E} : C → Gr(2, H^{0}(C, E)^{∗}), p ↦ E_{p}^{∗}

Let q be a quadric hypersurface in P(H^{0}(C, E)^{∗}) and denote by Gr(1, q) the subvariety in Gr(2, H^{0}(C, E)^{∗}) of projective lines in q. Then

Claim: The map μ is injective if and only if Φ_{E}(C) is not contained in Gr(1, q) for any quadric q in P(H^{0}(C, E)^{∗}).

Proof of Claim: Let S be the universal subbundle over the Grassmannian G := Gr(2, H^{0}(C, E)^{∗}), and let i : S → H^{0}(C, E)^{∗} ⊗ O_{G} be the natural inclusion map. Consider the (injective) map:

Ψ : Sym^{2}(H^{0}(C, E)) → H^{0}(G, Sym^{2}S^{∗}), ϕ ↦ i^{∗} ∘ ϕ ∘ i.

Let q_{ϕ} be the quadric hypersurface in P(H^{0}(C, E)^{∗}) associated to the quadratic form ϕ ∈ Sym^{2}(H^{0}(C, E)). Then μ(ϕ) = 0 if and only if Φ_{E}(C) is contained in the zero locus of Ψ(ϕ), which is just Gr(1, q_{ϕ}).

If dim(H^{0}(C, E)) = 3, then Gr(1, q) consists of at most two points. By the preceeding claim, μ is injective.

If dim(H^{0}(C, E)) = 4 and q_{ϕ} is smooth, then the image of Gr(1, q_{ϕ}) under the Plücker embedding, ρ : Gr(2, H^{0}(C, E)^{∗}) → P(∧^{2}H^{0}(C, E)^{∗}), is equal to a disjoint pair of smooth plane conics (see [GH], p. 199.) It follows that if Φ_{E}(C) is contained in Gr(1, q_{ϕ}), then there exists a line bundle L in W_{g−1}^{1} such that L^{2} ⊂ (ρ ∘ Φ_{E})^{∗}O(1) ≅ K_{C} which contradicts the injectivity of the ordinary Petri map for line bundles.

If q_{ϕ} is of rank three with vertex p and Φ_{E}(C) is contained in q_{ϕ}, then Φ_{E}(C) is contained in the plane:

Γ = {Λ : ⟨p⟩ ⊂ Λ ⊂ H^{0}(C, E)^{∗}}.

But this is the zero locus of a section of the universal quotient bundle Q over G. Consequently, the pull-back to C of the kernel of the map

H^{0}(C, E)^{∗} ⊗ O_{G} → Q

has a section. That is, E^{∗} has a section, which contradicts the stability of E.

If q_{ϕ} is of rank two, then Gr(1, q_{ϕ}) is a pair of disjoint planes R_{1} and R_{2}. These planes are the zero loci of sections of S^{∗}, so Φ_{E}(C) cannot be contained in either of them. The case where q_{ϕ} is of rank one is the same.

If dim(H^{0}(C, E)) = 5 and q_{ϕ} is smooth, then the image of Gr(1, q_{ϕ}) under the Plücker embedding is given by the image of P^{3} under the Veronese embedding:

ν : P^{3} → P^{9}

Therefore, once again, if Φ_{E}(C) is contained in Gr(1, q_{ϕ}), then there exists a line bundle L in W_{d}^{1} with d ≤ g − 1 and

L^{2} ⊆ (ρ ∘ Φ_{E})^{∗}O(1) ≅ K_{C}

But no such L exists on a general curve.

The cases where q_{ϕ} is not smooth may be disposed of by arguments similar to those when dim(H^{0}(C, E)) = 4.

Acknowledgements: The authors would like to thank Peter Newstead and the VBAC for inviting us to the Brill-Noether conference in Liverpool, England in July, 1991, where the work on this paper began.

Update: Independently of the authors (and at around the same time), Shigeru Mukai formulated the same conjecture for Brill-Noether loci of rank two bundles with canonical determinant ([Mu2]). In addition, Mukai has exhibited some breathtakingly beautiful examples in small genus, where the geometry of the lowest dimensional (smooth!) loci is worked out in detail. The interested reader is urged to consult [Mu2] and [Mu3]

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