Opers of higher types, Quot-schemes and Frobenius instability loci

In this paper we continue our study of the Frobenius instability locus in the coarse moduli space of semi-stable vector bundles of rank $r$ and degree $0$ over a smooth projective curve defined over an algebraically closed field of characteristic $p>0$. In a previous paper we identified the"maximal"Frobenius instability strata with opers (more precisely as opers of type $1$ in the terminology of the present paper) and related them to certain Quot-schemes of Frobenius direct images of line bundles. The main aim of this paper is to describe for any integer $q \geq 1$ a conjectural generalization of this correspondence between opers of type $q$ (which we introduce here) and Quot-schemes of Frobenius direct images of vector bundles of rank $q$. We also give a conjectural formula for the dimension of the Frobenius instability locus.


Introduction
Let p be a prime number. Let k be an algebraically closed field of characteristic p. Let X/k be a connected, smooth projective curve over k. We will write F : X → X for the absolute Frobenius morphism of X. A foundational classical problem in the theory of vector bundles on smooth, projective curves is the following: the Quot-schemes Quot r,0 (F * (Q)) parameterizing subsheaves of given rank r and degree 0 of F * (Q) are substantially better behaved than for general rank : in fact, one of the main results of [JP15] says that if rk(Q) = 1 then the dimension of Quot r,0 (F * (Q)) is 0 if its expected dimension is 0, and its k-rational points correspond to opers (of type 1). As shown in section 6, this fails if rk(Q) > 1. The existence of these higher-dimensional components of Quot r,0 (F * (Q)) stems from certain line subbundles L → Q of sufficiently high degree, which induce natural embeddings Quot r,0 (F * (L)) → Quot r,0 (F * (Q)).
In the present paper we lay the ground work for addressing the second problem and its finer special cases. The key tool we introduce here is the notion of opers of type q and rank q (see [JP15,Definition 3.1.1], recalled in Section 4). This notion generalizes the notion of an oper (V , ∇, V • ) studied by [BD00] and consisting of a vector bundle V equipped with an integrable connection ∇ and a full flag V • satisfying some transversality conditions. An oper of type q should be thought of as the bundle analog of the parabolic induction (and its adjoint the Jacquet Functor) in the theory of automorphic forms -the parabolic in the present case is of type given by the -tuple (q, . . . , q), which corresponds to a partial flag whose associated quotients are all of rank q. Given an oper of type q and rank q we naturally obtain a rank-q bundle Q as first quotient of the oper filtration.
In the case rk(Q) > 1 we conjecture (Conjecture 7.6) a similar statement assuming that one restricts attention to "non-degenerate" subsheaves of F * (Q), i.e. excluding in particular the above-mentioned sub-Quotschemes Quot r,0 (F * (L)). More precisely, we conjecture that, if the expected dimension of Quot q,0 (F * (Q)) is 0, then dormant opers of type q = rk(Q) form a non-empty open subset of dimension 0 of the Quot-scheme Quot q,0 (F * (Q)), which has, as mentioned above, components of dimension > 0. Here dormant means that the p-curvature of the connection ∇ is zero. We check that this conjecture holds when Q is a semi-stable direct sum of q line bundles (Theorem 3.5).
Finally, somewhat independently of the previous considerations, we give a conjecture for the dimension of the Frobenius instability locus J (r) in the coarse moduli space of semi-stable rank r and degree 0 vector bundles. This conjecture says that dim J (r) ≥ (r 2 − r + 1)(g − 1) − (r − 1).
The conjecture holds for r = 2. We also conjecture that a general vector bundle in J (r) has "minimal" Harder-Narasimhan filtration (Conjecture 8.3).
Acknowledgements. We would like to thank the referee for useful comments and suggestions.

Preliminaries on Quot-schemes
In what follows, the following notations and assumptions will be in force. Let X be a smooth, projective curve of genus g ≥ 2 over an algebraically closed field k of characteristic p > 0. Let F : X → X be the absolute Frobenius morphism of X. For a vector bundle V , we shall write V * for its dual H om(V , O X ).
Suppose θ is a theta-characteristic for X. Let G 2 be the unique non-split extension of θ −1 by θ: this bundle depends on the choice of the theta-characteristic θ but our notation suppresses this dependence. For r < p let G r = Sym r−1 (G 2 ) which will be called the Gunning bundle of rank r. This is an indecomposable bundle of degree zero and trivial determinant.
We recall two formulas from [JRXY06]. Let Q be a vector bundle of rank q and slope µ(Q) on X. Then and equivalently The first is a consequence of the Riemann-Roch formula χ(V ) = deg(V ) + rk(V )(1 − g) for a vector bundle V of rank rk(V ) and degree deg(V ) on X and the fact that χ(V ) = χ(F * (V )) for the finite map F. The second is equivalent to the first.
Let Q be a stable bundle on X of rank q and slope µ(Q) = µ < 0. Let r ≥ 1 be an integer. For a coherent sheaf V and a surjection V → G to a coherent sheaf, we will say that G has codegree d if the kernel ker(V → G) has degree d. Similarly we will say that G has corank r if ker(V → G) has rank r. Let Quot r,0 (F * (Q)) be the Quot-scheme of quotients of F * (Q) of codegree 0 and corank r. If V → G is a quotient with kernel E we will habitually write [E] for the corresponding point of the relevant Quot-scheme corresponding to this quotient.
Let [E] ∈ Quot r,0 (F * (Q)) be a point of the Quot-scheme. Then we define the integer The integer e(E) is called the expected dimension of Quot r,0 (F * (Q)) at the point [E] (see [HL10, Chapter 2]).
Proposition 2.4. Let r ≥ 1 be an integer. Let Q be a vector bundle on X of rank q ≤ r − 1 and slope µ(Q) = µ.
Let [E] ∈ Quot r,0 (F * (Q)) be a point of the Quot-scheme. Then the following assertions hold. .
(2) We have e(E) = 0 if and only if (3) In particular, if r = q for some integer ≥ 2, then e(E) = 0 if and only if Proof. It is enough to prove the first assertion as the rest are immediate consequences of the first. Let [E] ∈ Quot r,0 (F * (Q)) be a point corresponding to a quotient F * (Q) → G. Then by definition the expected dimension is given by e(E) = χ(H om(E, G)). By the Riemann-Roch formula we have which gives As E has degree zero, so deg(G) = deg(F * (Q)) = qµ + q(p − 1)(g − 1). Substituting this in the above equation and simplifying the result gives the asserted formula.
which leads to a contradiction, since rk(E ) = rk(E) = r.

A finiteness theorem
We start with a lemma.
Proof. Suppose the assertion is not true. Then we have Quot r ,0 (F * (Q)) ∅. Consider a subsheaf E ∈ Quot r ,0 (F * (Q)). We will use the notation of [JP15, Lemma 3.4.2]. Let W = F * (E) and equip it with the filtration induced by the natural filtration V • on the bundle V = F * (F * (Q)). We shall denote the latter by 0 = W m+1 ⊂ W m ⊆ · · · ⊆ W 1 ⊆ W 0 = W . Let r i = rk(W i /W i+1 ). Then 1 = r 0 ≥ r 1 ≥ · · · ≥ r m ≥ 1. So r i = 1 for all i ≥ 0. Now we have the following inequalities As rk(E) = r so m = r − 1 and hence the last inequality can be written as Thus we have arrived at a contradiction.
Theorem 3.5. We consider the morphism Moreover, by [JP15, Proposition 4.5.1 and Theorem 4.1.1] the vector bundles E and E i are semi-stable and their degrees are zero. We now consider the kernel Then clearly K i ⊂ F * (L i ). Note that K i = E ∩F * (L i ) and so K i ∩K j = 0 for i j. Moreover, by our assumption on E, we have K i 0 for all i = 1, . . . , q. As the bundles E and E i are semi-stable of degree zero, the bundles K i are also semi-stable of degree zero. We now apply Lemma 3.1 to the line bundle L i of degree −( − 1)(g − 1) and use the fact that Quot rk(K i ),0 (F * (L i )) ∅ -since this Quot-scheme contains [K i ] -and we obtain that rk(K i ) ≥ . Thus we have a homomorphism which is injective and so both bundles have the same rank and degree. So this map is an isomorphism. Moreover, we observe that K i = E i for any i = 1, . . . , q. Thus we have shown that Φ induces a bijection at the level of k-rational points between q i=1 Quot ,0 (F * (L i ))(k) and Ω(Q)(k). Since by [JP15, Theorem 6.2.1] each Quot ,0 (F * (L i ))(k) is a finite set, so Ω(Q)(k) is also a finite set. Since k is algebraically closed, we deduce that dim Ω(Q) = 0.
Remark 3.6. If the line bundles L i are not distinct, one still can show finiteness of Ω(Q) defined as the residual component of the Quot-schemes Quot q,0 (F * (Q)) for any subbundleQ ⊂ Q of degree 0 and rank q − 1.

Opers of type q
We now recall the definition of our main object (see [JP15, Definition 3.1.1]).
Note that our terminology suppresses the dependence on Q, but occasionally we may need to emphasize the dependence on Q and in such contexts will refer to such a triple as an oper of type Q.
From the definition it is immediate that one has, for all 0 ≤ i ≤ − 1, isomorphisms of bundles Note that we have the following equivalence In the next proposition we will show that opers of type q exist over an algebraically closed field of any characteristic. We first recall that G denotes the Gunning bundle of rank associated to the thetacharacteristic θ. We equip G with any connection ∇ G and with its natural filtration G • . Then it is well-known that the triple (G , Proposition 4.4. Let k be any algebraically closed field. If char(k) = p > 0, we assume that p > q( − 1)(g − 1). Let S be a stable vector bundle on X with deg(S) = 0 and rk(S) = q. Then the triple is an oper of type q for any connection ∇ on G ⊗ S.
Proof. The proof is elementary. Let us note that the space of connections on G ⊗ S is non-empty, since by stability the degree-0 vector bundle S admits a connection ∇ S , so ∇ G ⊗ ∇ S is a connection on G ⊗ S. Note that, again by stability of S, the filtration V • coincides with the Harder-Narasimhan filtration of V . The assumption on p implies that no proper subbundle of the filtration G • ⊗ S admits a connection. These observations easily show that any connection ∇ on V is an oper connection. Finally we note that We say that an oper (V , ∇, V • ) of type q is nilpotent (resp. dormant) if the oper connection ∇ is nilpotent of exponent at most rk(V ) (resp. has p-curvature zero). Before proceeding further let us recall the following result of [JP15, Theorem 3.1.6] which shows that dormant opers of type q exist.
Theorem 4.5. Let Q be any vector bundle of rank q. Then the triple forms a dormant oper of type q. Here ∇ can denotes the canonical connection on the vector bundle F * (F * (Q)).
The relationship between opers of type 1 and the Quot-scheme Quot r,0 (F * (Q)) when Q is a line bundle was studied in [JP15]. The main result [JP15, Theorem 6.2.1 and Theorem 5.4.1] is the following We now give an alternative characterization of the underlying bundle V of an oper (V , ∇, V • ) of type q.
Theorem 4.7. Let X/k be a smooth, projective curve over an algebraically closed field k. Let S be a vector bundle on X. Let θ be a line bundle on X. Suppose that the following hypotheses are satisfied: Then for every integer ≥ 2 there exists a vector bundle V on X satisfying the following is the unique non-split extension of this type.
Up to an isomorphism, there is only one vector bundle V with these properties.
Remark 4.8. If rk(Q) is not divisible by p then one has an isomorphism E nd(S) = O X ⊕ E nd 0 (S). Further one then also has E nd 0 (S) * E nd 0 (S). Hence in particular the hypotheses of the theorem can easily be satisfied for any stable bundle S of degree zero and of rank coprime to the characteristic of the ground field. In particular these hypotheses are easily satisfied in characteristic zero.
We begin with the following lemma.
Hence the claim.
Proof. Clearly the claim is true for j = as V = 0. We prove the claim by descending induction on j. Suppose the lemma is true for some j 0 with i 0 + 1 < j 0 ≤ . Then we claim that the assertion is also true for j 0 − 1. As j 0 > i 0 + 1, one has the exact sequence Then applying Hom(S i 0 −1 , −) one gets By induction hypothesis the first term is zero and by Lemma 4.23 the last term is zero. Hence the middle term is zero and the claim is proved. Now we are ready to prove Theorem 4.7.
Proof of Theorem 4.7. The construction of the bundle V whose existence is asserted in the theorem is inductive. We let V = 0 and V −1 = S −1 . Now we define V −2 as the unique non-split extension which is given by the isomorphism of Lemma 4.10: By Lemma 4.18 we get Ext 1 (S −3 , S −1 ) = 0, and by Lemma 4.10 we get Ext 1 (S −3 , S −2 ) = H 1 (X, Ω 1 X ). Therefore we get Ext 1 (S −3 , V −2 ) H 1 (X, Ω 1 X ). So we define V −3 as the unique non-split extension given by this isomorphism. Then we have by construction Now we repeat this process to obtain the general construction. Suppose that for some i 0 ≥ 0, V i 0 has been constructed with the asserted properties. Then one has an exact sequence Then we claim that there is a unique non-split extension in Ext 1 (S i 0 −1 , V i 0 ) which gives V i 0 −1 . We proceed as follows: apply Hom(S i 0 −1 , −) to the above short exact sequence to get By Lemma 4.10 one as Ext 1 (S i 0 −1 , S i 0 ) H 1 (X, Ω 1 X ), and so to prove our claim it suffices to prove that Ext 1 (S i 0 −1 , V i 0 +1 ) = 0. This follows from Lemma 4.24. Repeating this process eventually one gets V 0 as the unique non-split extension This completes the proof.

Properties of the Quot-scheme Quot r,0 (F * (Q)) when q = 1
The following proposition gives some properties on the Quot-scheme Quot r,0 (F * (Q)) when Q is a line bundle. It generalizes the case r = 2, which was already shown in [ We also note that there is a gap in the proof of [JP15, Theorem 7.1.2] and therefore we shall give the proof of the following theorem with all the details.
Let Q be a line bundle of degree deg(Q) = −(r − 1)(g − 1) + d with d ≥ 0 and let C be an irreducible component of Quot r,0 (F * (Q)). We recall from Proposition 2.4(5) that the expected dimension of C is rd.
Definition 5.1. We will say that C contains a dormant oper if there exists an effective divisor D of degree d and a point [E] ∈ Quot r,0 (F * (Q(−D))) -hence [E] corresponds to a dormant oper -such that [E] ∈ C under the natural inclusion Quot r,0 (F * (Q(−D))) ⊂ Quot r,0 (F * (Q)).
Remark 5.2. We note that the above definition of C containing a dormant oper is more restrictive than the natural one: there exists a [E] ∈ C such that the triple (F * E, ∇ can , F * E • ), where the filtration F * E • is the induced filtration from F * (F * (Q)), is an oper of type 1. Note that the latter definition was used in [JP15] for r = 2. It can be checked that both definitions coincide for small d. We decompose D = x + D with x ∈ X and D effective of degree d. Let C be an irreducible component of Quot r,0 (F * (Q(−x))) ∩ C containing [E]. By induction we have dim(C ) = rd. Now we claim that codim C C ≤ r. To prove this, note that C ∅. Since C is an irreducible component of the Quot-scheme, it is equipped with a universal quotient sheaf Q over X × C .
We denote by E the kernel ker(p * X (F * (Q)) −→ Q). Since Q and p * X (F * (Q)) are C -flat, E is also C -flat and ∀c ∈ C the homomorphism E |X×{c} → F * (Q) is injective (see e.g. [HL10]). Hence, since F * (Q) is locally free, E |X×{c} is also locally free (since torsion free over a smooth curve) and by [HL10, Lemma 2.1.7] we conclude that E is locally free over X × C . Since p * X (F * (Q)) = (F × id C ) * (p * X (Q)) we obtain by adjunction a non-zero map (F × id C ) * (E) → p * X (Q), hence a non-zero global section σ of the rank-r vector bundle V := H om((F × id C ) * (E), p * X (Q)) over X ×C . It is clear that Quot r,0 (F * (Q(−x)))∩C is the zero-scheme of the section σ |{x}×C ∈ H 0 (C , V |{x}×C ) obtained after restriction. Hence codim C C ≤ r and therefore dim(C ) ≤ rd + r. On the other hand, by the dimension estimates of the Quot-schemes in Proposition 2.5(2) we have dim(C ) ≥ rd + r. Therefore dim(C ) = rd + r and we are done.
We now assume that d > 0. With the previous notation we denote by Σ the zero-scheme of the global section σ of the vector bundle V and by p X and p C projections onto X and C respectively. Thus one has a diagram We have the set-theoretical equalities Since Σ is closed and p C is a proper map, so Σ = p C (Σ) is a closed subset of the irreducible component C . So there are two possibilities: So let us suppose we are in Case (II). In this case there is at least one irreducible component of Σ, which we will again denote by Σ, which surjects onto C under p C . Hence Consider the restrictionp X : Σ → X of p X : X × C → X to Σ. First we assume thatp X is surjective. Since X is smooth and Σ is irreducible, by [Har77, Chapter III, Proposition 9.7]p X is flat. Hence by [Har77, Chapter III, Corollary 9.6] any irreducible component of the fiberp −1 X (x) has dimension equal to dim(Σ) − 1 ≥ rd − 1. By assumption C contains a dormant oper [E] and as Σ = C , there exists an x 0 ∈ X such that (x 0 , [E]) ∈ Σ. Since any irreducible component ofp −1 X (x 0 ) has dimension ≥ rd − 1, we see that the irreducible component ofp −1 X (x 0 ) containing [E] has dimension at least rd − 1 and this irreducible component is contained in Quot r,0 (F * (Q(−x 0 ))). But this contradicts part (1) with d − 1 ≥ 0.
Finally, we need to consider the case whenp X is not surjective.
Remark 5.5. The gap in the proof of [JP15, Theorem 7.1.2] is related to the fact that the intersection Quot 2,0 (F * (Q(−x))) ∩ C is not necessarily irreducible.
6. The Quot-scheme Quot r,0 (F * (Q)) is bigger than expected when q ≥ 2 We assume that q = rk(Q) ≥ 2 and that or equivalently, Note that the last inequality implies that d < (r − 1)(g − 1) − 1. With this notation we have the following result.
Proof. By [JP15, Theorem 2.3.1] there exists a line subbundle L ⊂ Q such that Note that the expression on the right-hand side is not necessarily an integer. We then introduce the quantity δ = deg L + (r − 1)(g − 1). Then clearly Quot r,0 (F * (L)) ⊂ Quot r,0 (F * (Q)) and by Proposition 2.5 (2) dim Quot r,0 (F * (L)) ≥ rδ. Therefore in order to show the proposition it will be enough to show the inequality δ > d. We have δ = deg L + (r − 1)(g − 1), (6.5) Now we observe that the inequality which holds by assumption on d.
Remark 6.8. In particular if e(E) = 0 the Quot-scheme Quot r,0 (F * (Q)) has some positive-dimensional components, as already observed in the previous section, see also Theorem 3.5.
Since we have an injective map W i /W i+1 → V i /V i+1 and both the bundles are of the same rank, we obtain This gives The last sum evaluates to Thus one sees that deg(W i /W i+1 ) = deg(V i /V i+1 ). Therefore the above injective maps are isomorphisms: W i /W i+1 V i /V i+1 for i = 0, . . . , − 1. This proves the assertion.
One important consequence of this lemma is the following fundamental result.
Theorem 7.2. Let Q be a semi-stable vector bundle with deg(Q) = −q( − 1)(g − 1) and rk(Q) = q for some integer ≥ 2. We put r = q and use the notation of Lemma 7.1.
On the other hand we have assumed that p > r(r − 1)(g − 1). Therefore we have arrived at a contradiction. Thus the vector bundle W is semi-stable. Now if W does not map injectively into F * (Q), then the image is a subsheaf of some degree d ≥ 1 and rank ≤ r − 1 and hence it has slope ≥ 1 r−1 . Again by [JP15, Proposition 4.2.1] one sees that F * (Q) does not have any subsheaves of suitably positive degree and of rank ≤ r − 1. So W → F * (Q). Proposition 7.3. Using the above notation, the set of points [W ] ∈ Quot r,0 (F * (Q)) satisfying W = {0} is an open subset.
Proof. The Quot-scheme Quot r,0 (F * (Q)) comes equipped with a universal quotient G defined over the product X × Quot r,0 (F * (Q)) which is flat over Quot r,0 (F * (Q)): is locally free, it is flat over Quot r,0 (F * (Q)) and therefore for any point y ∈ Quot r,0 (F * (Q)) the homomorphism W |X × {y} → F * (Q) is injective. Thus W |X × {y} is locally free for any y, which in turn implies that W is locally free. Hence, with the notation of Lemma 7.1 we obtain a homomorphism between locally free sheaves over X × Quot r,0 (F * (Q)) Φ : F * (W ) → p * X (V /V ). By Lemma 7.1 the condition W = {0}, with W = F * (W |X×{y}), is equivalent to Φ X×{y} being an isomorphism. But the set of points y ∈ Quot r,0 (F * (Q)) where Φ X×{y} is not an isomorphism is clearly a closed subset.
We recall that by Theorem 7.2 the open subscheme Ω(Q) parameterizes dormant opers (W , ∇, W • ) of type q with W 0 /W 1 Q.
We now list some evidence for this conjecture: (1) If q = 1, the conjecture is true. This is shown in [JP15] -see Theorem 4.6. Note that in this case we have equality Ω(Q) = Quot r,0 (F * (Q)).
(2) The conjecture is true for a decomposable bundle of the form Q = ⊕ q i=1 L i , where the L i are q distinct line bundles of degree deg(L i ) = −( − 1)(g − 1). This is shown in Theorem 3.5. We note that in this case points of Ω(Q) correspond to direct sums of q opers of type 1.
Let M(r) be the coarse moduli space of semi-stable vector bundles of rank r and degree 0 over X. Let J (r) ⊂ M(r) be the closed subscheme of M(r) parameterizing semi-stable vector bundles E such that F * (E) is not semi-stable. The closed subscheme J (r) will be referred to as the Frobenius instability locus. The purpose of this section is to make a conjecture on the dimension of J (r). Before stating the conjecture we recall some facts on the structure of J (r).
Let M(q, −1) be the moduli space of semi-stable bundles of rank q and degree −1 over X. Note that as we are in the coprime case any semi-stable bundle of rank q and degree −1 is also stable. Let U be the universal bundle over X × M(q, −1). Consider the relative Quot-scheme π : Quot r,0 ((F × id M(q,−1) ) * U ) −→ M(q, −1) with fiber π −1 (Q) = Quot r,0 (F * (Q)) over a stable bundle Q ∈ M(q, −1).
For q = 1, . . . , r − 1 we denote by J q ⊂ J (r) the closure of the forgetful map where J s (r) denotes the subset of stable vector bundles E such that F * (E) is not semi-stable.
In order to compute the dimension of J q we need to know the following dimensions (1) dim Quot r,0 (F * (Q)) for general Q ∈ M(q, −1).
We therefore focus on the case q = 1. Let Q be a line bundle of degree −1 and let C be an irreducible component of Quot r,0 (F * (Q)) containing a dormant oper. Then by Theorem 5.3 dim C = r((r −1)(g −1)−1) and for a general vector bundle [E] ∈ C the map f E : F * (E) → Q obtained by adjunction is surjective. Let us denote the kernel of f E by S = ker f E . Note that deg(S) = 1 for general [E].
With this notation we make the following Conjecture 8.1. For a general vector bundle E ∈ C the bundle S is semi-stable.
As a consequence of this conjecture we obtain that for a general E ∈ C the filtration 0 ⊂ S ⊂ F * (E) is the Harder-Narasimhan filtration of F * (E). Therefore the quotient F * (E) Q = F * (E)/S is uniquely determined and dim Hom(E, F * (Q)) = 1. This means that the forgetful map α 1 is generically injective on components C containing dormant opers.
Remark 8.2. Conjecture 8.1 holds trivially for r = 2 and the lower bound of the dimension of J (2) was already worked out in [JP15, Theorem 7.1.2].