The normal bundle of a general canonical curve of genus at least 7 is semistable

Let $C$ be a general canonical curve of genus $g$ defined over an algebraically closed field of arbitrary characteristic. We prove that if $g \notin \{4,6\}$, then the normal bundle of $C$ is semistable. In particular, if $g \equiv 1$ or $3$ mod $6$, then the normal bundle is stable.


Introduction
Let k be an algebraically closed field of arbitrary characteristic. Let C be a nonsingular, irreducible, non-hyperelliptic curve of genus g ≥ 3 defined over k. Then the canonical linear system K C embeds C in P g−1 . The image is called a canonical curve of genus g. Canonical curves of genus g lie in an irreducible component of the Hilbert scheme of curves of genus g in P g−1 . Studying the properties of canonical curves is an essential tool in curve theory.
Given a vector bundle V on C of rank r and degree d, recall that the slope of V is defined by µ(V ) ∶= d r . The bundle V is called semistable if, for every proper subbundle W , we have µ(W ) ≤ µ(V ). The bundle is called stable if the inequality is always strict.
Since stable bundles are the atomic building blocks of all vector bundles on a curve, it is important to ask if naturally-defined vector bundles on canonical curves, such as the restricted tangent bundle T P g−1 C or the normal bundle N C , are stable. The first of the these is straightforward: the restricted tangent bundle of a general canonical curve of genus g ≥ 3 is always stable. In fact, the restricted tangent bundle of a general Brill-Noether curve of any degree d and genus g ≥ 2 in P r is stable unless (d, g) = (2r, 2) [FL22]. On the other hand, the normal bundle can fail to be stable in low genus (cf. Remark 1).
Aprodu, Farkas and Ortega [AFO16] conjectured that once the genus is sufficiently large, the normal bundle of a general canonical curve is stable. Previously, this was only known for g = 7 [AFO16] and for g = 8 [B17]. The proofs of these two results use explicit models of low genus canonical curves due to Mukai, and thus do not generalize to large genus. In this paper, we prove: Theorem 1.1. Let C be a general canonical curve of genus g ∈ {4, 6} defined over an algebraically closed field of arbitrary characteristic. Then the normal bundle of C is semistable.
The rank of N C is g − 2 and the degree of N C is 2(g 2 − 1). Hence, In particular, if g − 2 and 6 are relatively prime, the semistability of N C implies the stability of N C . We thus obtain the following corollary.
Corollary 1.2. If g ≡ 1 or 3 (mod 6), then the normal bundle of the general canonical curve of genus g is stable.
Remark 1. When g = 3, the canonical curve is a plane quartic curve. Hence, N C ≅ O C (4) and is stable. When g = 5, the general canonical curve is a complete intersection of three quadrics. Hence, N C ≅ O C (2) ⊕3 . In particular, N C is semistable but not stable. When g = 4 or 6, N C is unstable, as we now explain. When g = 4, the canonical curve is a complete intersection of type (2, 3). The normal bundle of C in the quadric is a destabilizing line subbundle of N C of degree 18. When g = 6, the general canonical curve is a quadric section of a quintic del Pezzo surface. The normal bundle of C in this del Pezzo surface gives a degree 20 destabilizing line subbundle of N C .
We will prove Theorem 1.1 by specializing a canonical curve to the union of an elliptic normal curve E of degree g and a g-secant rational curve R of degree g − 2 meeting E quasi-transversely in g points. In §3, we describe this degeneration and the Harder-Narasimhan (HN) filtration of N E∪R R . In §4, we will prove that N E∪R E is semistable. This suffices to prove Theorem 1.1 when g is odd by [CLV22,Lemma 4.1], because N E∪R R is balanced in this case. When g is even, N E∪R R is not balanced. However, we have an explicit geometric understanding of the HN-filtration. In this case, we give two proofs of Theorem 1.1, one using the strong Franchetta Conjecture (see §4), and an elementary proof using the explicit HN-filtration and induction on g (in §5 and §6).
where codim F (F p 1 ∩ F p 2 ) refers to the codimension of the intersection in either F p 1 or F p 2 (which are equal since dim F p 1 = dim F p 2 ). Note that if F is pulled back from C, then µ adj C (F ) = µ(F ). We say that V is (semi)stable if for all subbundles F ⊂ ν * V , The advantage of this definition is that it specializes well.
Proposition 2.2. [CLV22, Proposition 2.3] Let C → ∆ be a family of connected nodal curves over the spectrum of a discrete valuation ring, and V be a vector bundle on C . If the special fiber V 0 = V 0 is (semi)stable, then the general fiber V * = V ∆ * is also (semi)stable.
Lemma 2.3. [CLV22, Lemma 4.1] Suppose that C = X ∪ Y is a reducible curve and V is a vector bundle on C such that V X and V Y are semistable. Then V is semistable. Furthermore, if one of V X or V Y is stable, then V is stable. Given a vector bundle V on a scheme X, an effective Cartier divisor D ⊂ X, and a subbundle → F ] is naturally isomorphic to V on the complement of the divisor D. In this way we can easily define multiple modifications V [D 1 when the supports of D 1 and D 2 are disjoint.
When the supports of the D i meet, subbundles of V D i are insufficient to define multiple modifications. In this context, we always assume that F i extends to a subbundle of V in an open neighborhood U i of D i . If F 2 U 2 ∖D 1 extends to a subbundle over all of U 2 then it does so uniquely and . The general situation of multiple modification is studied in [ALY19,§2]. In this paper when we need multiple modifications, the extension will be clear, and so we won't need this general framework.

A simplifying special case of elementary modifications is when
we still have an explicit description along D: More generally, if V sits in an exact sequence then we obtain an induced exact sequence with the modification V [D + → F ] that captures how the subbundle F sits with respect to the sequence (2). We will only make use of the following two special cases of this. First suppose that F ∩ S is flat over the base X. In this case (2) induces the exact sequence Second, suppose that X = C is a smooth curve and F ⊂ V is a line subbundle. By combining modifications with disjoint supports, it suffices to consider the case that D = np for a point p ∈ C. Let k ′ be the order to which the fiber of F is contained in the fiber of S in a neighborhood of p. If F is a subbundle of S, then k ′ = ∞. Let k = min(k ′ , n). In this case (2) induces the exact sequence where F is the saturation of the image of F in Q. In the special cases of k ′ = 0 or ∞ the two sequences (3) and (4) agree.
We will primarily work with elementary modifications of the normal bundle of a curve C ⊂ P r towards pointing bundles, whose definition we now recall. Given any linear space Λ ⊂ P r , the projection π from Λ, when restricted to C, is unramified on an open U Λ ⊂ C. If U Λ is dense in C and contains C sing , then the relative tangent sheaf of the map π uniquely extends to a rank (dim Λ + 1) subbundle of N C , which we denote by N C→Λ and call the pointing bundle towards Λ.
The pointing bundle exact sequence is When Λ ⊂ Ψ are nested subspaces, we have an analogous pointing bundle exact sequence where Ψ is the projection of Ψ from Λ. We abbreviate and write N C [p + → Λ] ∶= N C [p + → N C→Λ ] for modifications towards pointing bundles.
Suppose that C is a curve on a smooth variety X, and M is any smooth subvariety meeting C quasi-transversely at a point p. Then we write where T p M maps to N C p via the quotient map T p X → N C p . Observe that when M is itself a linear space through p, If M ∩ C = {p 1 , p 2 , . . . , p n }, with all points of intersection quasi-transverse, then we write Our interest in modifications towards pointing bundles is rooted in the following result of Hartshorne-Hirschowitz, describing the normal bundle of a nodal curve.
Lemma 2.4 ([HH83, Corollary 3.2]). Let X ∪ Y be a connected nodal curve in P r . Then Finally, we recall that the normal bundle of a curve can be related to the normal bundle of its proper transform in a blowup via modifications. The simplest case is that of a smooth curve lying on a smooth variety C ⊂ X, and a blowup β∶ Bl Y X → X along a smooth subvariety Y ⊂ X meeting C quasi-transversely at a single point p. Then the normal bundles of C in X, and of its proper transform in the blowup, are related as follows: Via the rules for combining modifications, these formulas immediately imply several generalizations. We will need the following case: Suppose that Y ′ ⊂ Y is a smooth subvariety, also passing through p. Write t for the natural rational map from the exceptional divisor of Bl Y ′ X to the exceptional divisor of Bl Y X. Then, for any smooth subvariety M of the image of t meeting C at p: 2.3. The Farey sequence. Recall that the N -Farey sequence is the sequence of fractions whose denominators are bounded by N in lowest terms. We refer the reader to [HW79] for the properties of the Farey sequence.
Lemma 2.5. Let V be a vector bundle of slope µ(V ) = p q in lowest terms and suppose that 0 → S → V → Q → 0 is an exact sequence of vector bundles such that either µ(S) is an adjacent q-Farey fraction to µ(V ) with gcd(deg S, rk S) = 1, or similarly for Q. If both S and Q are stable, then any destabilizing subsheaf of V is isomorphic to either S or Q.
Proof. Suppose that V has degree ep and rank eq for some e ≥ 1. Then the slope of the other bundle (µ(Q) or µ(S), respectively) is an adjacent eq-Farey fraction; this can be seen using the following two standard properties of adjacent Farey fractions: • Two rational numbers in lowest terms, p 1 q 1 and p 2 q 2 , are adjacent in the max(q 1 , q 2 )-Farey sequence if and only if det p 1 p 2 q 1 q 2 = ±1.
• In this case, they are adjacent in the q-Farey sequence for any max(q 1 , q 2 ) ≤ q < q 1 + q 2 , and the next fraction appearing between them is p 1 + p 2 q 1 + q 2 .
There are four cases to consider: µ(S) or µ(Q) is the next or previous eq-Farey fraction. Up to replacing the sequence with its dual, it suffices to consider the two cases that µ(S) or µ(Q) is the next Farey fraction. Let F be any subsheaf of V . Then F has a filtration If µ(S) is the next Farey fraction: Since F ∩ S is a subsheaf of S, we have µ(F ∩ S) ≤ µ(S) with equality only if F contains S. Since µ(V ) is the previous eq-Farey fraction to µ(S), if equality does not hold, then Lemma 2.6. Suppose that V is a family of vector bundles on a positive-genus curve C parameterized by a rational base B. Suppose that, Since B is rational, this map is constant.
On the other hand, we may specialize to the fiber over b i . As we approach along any arc, F b limits to one of S i or Q i (based on which one has slope greater than µ(V )) by Lemma 2.5. Therefore, c 1 (F ) extends to a regular map in a neighborhood of b i . Our assumption that c 1 (S 1 ) ≠ c 1 (S 2 ) (and so also c 1 (Q 1 ) ≠ c 1 (Q 2 )) then gives a contradiction.
2.4. Natural bundles on a genus 1 curve. Let E be a genus 1 curve. We say that a map f ∶ Pic a E → Pic b E is natural if for any automorphism θ∶ E → E, the following diagram commutes: Proof. Translation by a point of order a is the identity on Pic a E, and so must also be on Pic b E.

Our degeneration
Let E ⊂ P g−1 be an elliptic normal curve. Let H ≃ P g−2 be a general hyperplane and let Γ ∶= E ∩H be the hyperplane section of E. Let R be a general rational curve of degree g − 2 in H, meeting E quasi-transversely at the points of Γ. Then by [LV22,Lemma 5.7], the curve E∪R is a Brill-Noether curve of degree 2g − 2 and genus g; i.e., it is a degeneration of a canonical curve.
Lemma 3.1 ([LV22, Lemma 5.8 and Proposition 13.7]). We have By [CLV22, Lemma 4.1], when g is odd, it suffices to show that N E∪R E is semistable to conclude that the normal bundle of a general canonical curve is semistable. This is addressed in Section 4. When g is even, we will need to know that N E∪R E is semistable, and also that certain modifications of N E∪R E , related to the Harder-Narasimhan (HN) filtration of N E∪R R , are semistable. We conclude this section with a brief geometric description of the HN-filtration, expanding on [LV22, Section 13].
3.1. The HN-filtration when g is even. In this section, we suppose that g = 2n + 2 is even. We first recall without proof some results we will need from [LV22, Section 13]. Suppose that E ⊂ P 2n+1 is an elliptic normal curve. Let p 1 + ⋯ + p 2n+2 be a general section of O E (1). Let q 1 , . . . , q 2n+2 be general points on P 1 . By [LV22, Lemma 13.1], there are exactly two degree n + 1 maps sending p j to q j for all 1 ≤ j ≤ 2n + 2 (see also [CPS21] and [FLi22] for more general results of the type). Together, these define a map which is birational onto an (n 2 − 1)-nodal curve of bidegree (n + 1, n + 1) [LV22, Lemma 13.2], none of whose nodes lie on the diagonal.
Let S denote the blowup of P 1 × P 1 at the n 2 − 1 nodes of f (E), with total exceptional divisor F , and write f ∶ E ↪ S for the resulting embedding. By [LV22,Lemma 13.3], the line bundle Write π i ∶ S → P 1 for the two projections onto each factor of P 1 × P 1 . As computed in [LV22], The map S → PH 0 (L) ≃ PH 0 (L E ) ≃ P 2n+1 given by L thus factors through the balanced scrolls embedded by the relative O(1), and is hence an embedding. Let R denote the diagonal of P 1 × P 1 , viewed as a divisor on S. By construction, R meets E at p 1 , . . . , p 2n+2 . Along R, the bundle L R has degree 2n, and hence maps R into a hyperplane in P 2n+1 . The reducible curve E ∪ R is a degeneration of a canonical curve. Finally, we recall a construction of Zamora [Z99, Lemma 1.1] of a rank 4 quadric in P 2n+1 containing E, and show that it also contains the scrolls Σ 1 and Σ 2 . Let s 1 , s 2 be a basis for the linear system giving rise to the first map f 1 ∶ E → P 1 and let t 1 , t 2 be a basis for the linear system giving rise to the second map f 2 ∶ E → P 1 . Then the s i ⊗ t j are sections of O(1, 1) E = L E , and we may therefore view them as linear functions on the P 2n+1 . Furthermore, as a section of L ⊗2 E , det This determinant defines a rank 4 quadric Q ⊂ P 2n+1 containing E. Changing the bases s 1 , s 2 or t 1 , t 2 corresponds to a row/column operation, so this quadric is independent of the choice of basis. To see that the quadric contains Σ 1 , we will show that it contains every fiber P n = Span(f −1 1 (x)) for x ∈ P 1 . Choose a basis so that the first element s 1 vanishes on f −1 1 (x). Thus the linear functions corresponding to s 1 ⊗ t 1 and s 1 ⊗ t 2 vanish along Span(f −1 1 (x)) in P 2n+1 , and hence the quadric Q contains this plane. Varying x, we see that Q contains Σ 1 . Similarly Q contains Σ 2 . Putting all of this together, we can summarize this situation with the following setup: Setup 3.2. Given an elliptic curve E ⊂ P 2n+1 and two maps f i ∶ E → P 1 , we obtain the following inclusions: Moreover, projection from Q sing induces maps Σ i → P 1 ×P 1 ⊂ P 3 , whose composition with projection onto the ith P 1 factor is the structure map for the projective bundle. In particular, the composition of the inclusion S ↪ Q with projection from Q sing is identified with the blowup map S → P 1 × P 1 .
The maps in Setup 3.2 give rise to a filtration of N E∪R , Proposition 3.3 ([LV22, Proposition 13.7]). The restriction of (7) to R is the HN-filtration of N E∪R R .
Remark 2. In [LV22, Proposition 13.7], the middle piece of the filtration does not have a geometric description. Instead, it is described as "N E∪R Σ 1 + N E∪R Σ 2 " -which is equal to N E∪R Q since it is contained in it, and has the same rank and degree. Proof. The data in (3.2) is determined by the following choices: (1) A basis (up to common scaling) for H 0 (O E (1)), which determines the embedding E ⊂ P 2n+2 . The choice of a basis of a vector space depends on a rational base.
(2) An unordered pair of line bundles f * i O P 1 (1) that sum to O E (1). The space of line bundles of a fixed degree on E can be identified with E. This choice corresponds to the fiber of the map a∶ (E × E) S 2 → E given by addition over O E (1). The surface (E × E) S 2 is a ruled surface over E, so this choice is rational.
(3) Two sections (up to common scaling) of each of these line bundles (defining f i ∶ E → P 1 ). As in (1), this choice depends on a rational base.
We conclude that the set of possible S, R, Σ 1 , Σ 2 , Q in Setup 3.2 varies in a rational base.

Semistability of the restriction to E
In this section, we show that the restricted normal bundle N E∪R E , where E ∪ R ⊂ P g−1 is the degenerate canonical curve introduced in Section 3, is semistable.
We will first show that N E∪R E is "close-enough-to-semistable" that no naturally defined destabilizing subbundles could exist. We have that The fractional part of the slope depends on g modulo 6. Write Lemma 4.2. The bundle N E∪R E has no subbundles of slope greater than g + 3 + 1 k and no quotient bundles of slope less than g + 3 + 1 k+1 . We will deduce this from taking m = 0 in the following more general statement.
has no subbundles of slope greater than g+3+ 1 k , and no quotient bundles of slope less than g+3+ 1 k+1 . In the course of proving Lemma 4.3, we will need the following result.
Lemma 4.4. Let n ≥ 2 be an integer and suppose that Λ ⊂ P g−1 is a quasi-transverse n-secant (n − 1)-plane to E. Let q ∈ Λ be a general point. Suppose that R ⊂ Λ is a general rational curve of degree n − 1 through E ∩ Λ and q. Let y be a general point on E. Then the modified pointing bundle is stable of slope g + 3 + 1 n . Proof. We will prove this by induction on n. Specialize q to one of the points p where R meets E. If n > 2, then the pointing bundle exact sequence (5) towards p induces the sequence as in (3) and (4). The subbundle N E→p (y) is isomorphic to O E (1)(2p + y), which is stable of slope g + 3. The quotient is a twist of another instance of our problem in P g−2 . We may therefore assume by induction that it is stable of slope g + 3 + 1 n−1 . Since c 1 (O E (1)(2p + y)) depends on the choice of the point p, we conclude by Lemma 2.6 that the general fiber is semistable (and hence stable) as desired.
It suffices, therefore, to treat the base case of n = 2. In this case, R = Λ is a 2-secant line pp ′ , and after specializing as above, the pointing bundle exact sequence towards p is In this case the subbundle and quotient bundles are stable line bundles of slopes g + 4 and g + 3 respectively. Again, applying Lemma 2.6, we conclude that the general fiber is semistable (and hence stable) as desired.
Proof of Lemma 4.3. Our argument will be by backwards induction on m. The base case of m = 5 is Lemma 4.5 below, so we suppose m ≤ 4. We first prove the upper bound on the slope of a subbundle by exhibiting a degeneration that lies in an exact sequence with a subbundle that is stable of slope exactly g + 3 + 1 k and quotient which satisfies our inductive hypothesis. Let Λ 1 ≃ P k−1 ⊂ P g−1 be the span of the first k points p 1 , . . . , p k of E ∩ R. Let Λ 2 ≃ P g−k−2 be the span of the last g − 1 − k points p k+2 , . . . , p g . Since the remaining point p k+1 is constrained to lie in the hyperplane spanned by the other points, there is a unique line L through p k+1 that meets both Λ 1 and Λ 2 . Let x 1 and x 2 denote the points where L meets Λ 1 and Λ 2 , respectively. Let R 1 be a general rational curve in Λ 1 of degree k − 1 through p 1 , . . . , p k , x 1 , and let R 2 be a general rational curve in Λ 2 of degree g − k − 2 through p k+2 , . . . , p g , x 2 . Then is a degeneration of R. It suffices to prove that N E [p 1 + ⋯ + p g−m + ↝ R ○ ] has no subbundles of slope greater than g + 3+ 1 k to prove the lemma. Consider the pointing bundle exact sequence for pointing towards the subspace Λ 1 : In order to use Lemma 4.4 to show that x 1 ] is stable of slope g + 3 + 1 k , we need that, as the points p k+1 , . . . , p g vary, the point x 1 is general in Λ 1 . That is, there are no obstructions to lifting a deformation of the point x 1 to a deformation of the plane Λ ′ 2 ∶= Λ 2 , x 1 (maintaining the necessary incidences with E). These obstructions live in H 1 (Λ ′ 2 , N ), where the bundle N is the kernel of the map The key numerical input is 2k ≤ g, which follows from m ≤ 4. Since Λ ′ 2 is the complete intersection of the k hyperplanes spanned by Λ ′ 2 and all but one of the tangent lines where P is a punctual sheaf (and hence h 1 (P ) = 0). Moreover, the evaluation map ev in (1) x 1 ∪p k+1 ∪⋅⋅⋅∪p i ∪⋅⋅⋅∪pg → 0 is surjective on global sections, since the points x 1 ∪ p k+1 ∪ ⋅ ⋅ ⋅ ∪p i ∪ ⋅ ⋅ ⋅ ∪ p g form a basis for the plane Λ ′ 2 , and h 1 (O Λ ′ 2 (1)) = 0. Therefore is another case of our inductive hypothesis with one fewer modification occurring at the points of incidence of R 2 with E (with a larger value of k if ǫ = 5 − m). The result now follows from our inductive hypothesis. Now we turn to the lower bound on the slope of any quotient. We will exhibit a specialization that lies in an exact sequence with a subbundle that is stable of slope exactly g + 3 + 1 k+1 and a quotient bundle which satisfies our inductive hypothesis. We will modify the same argument by letting Λ 1 be the k-dimensional span of p 1 , . . . , p k+1 and letting Λ 2 be a the (g − k − 3)-dimensional span of p k+3 , . . . , p g . As above, there is a unique line L through the remaining point p k+2 that meets both Λ 1 (at a point x 1 ) and Λ 2 (at a point x 2 ). We define R 1 and R 2 analogously to above. In the pointing bundle exact sequence towards Λ 1 : the subbundle is stable of slope g + 3 + 1 k+1 by Lemma 4.4 (using the same argument to ensure generality of x 1 ), and the quotient is a twist of another case of our inductive hypothesis in P g−k−2 (with a smaller value of k if ǫ = 0), with one fewer modification occurring along R 2 . This completes the inductive step. All that remains is therefore to verify the base case, which is Lemma 4.5 below.
Lemma 4.5. Suppose that E ⊂ P g−1 is an elliptic normal curve, and R is a degree g − 2 rational curve meeting E at p 1 , . . . , p g quasi-transversely. Then is stable of slope g + 3 + 1 g−2 . Proof. We will prove this by induction on g. The base case is g = 5, in which case N ′ E = N E is stable by [EiL92]. Otherwise, when g ≥ 6, the bundle N ′ E is modified at p 1 . Let Λ ≃ P g−3 be the span of p 2 , . . . , p g−1 . Let L be the line through p 1 and p g that meets Λ at a point x. Let R ′ be a rational curve of degree g − 3 through p 2 , . . . , p g−1 , x. Then R ○ = R ′ ∪ L is a degeneration of R. Consider the specialization of N ′ E . Consider the pointing bundle exact sequence for pointing towards p g : The subbundle has slope g + 3 exactly. Since R ′ is a rational curve of degree g − 3 meeting E at p 2 , . . . , p g−1 , x, the quotient bundle is a twist of an instance the same problem in P g−2 . By induction it is stable. Moreover, c 1 (N E→pg (p 1 )) depends on the ordering of p 1 , p 2 , . . . , p g . Hence by Lemma 2.6, the general fiber N ′ E is semistable (thus stable) as desired. We complete the proof by appealing to the naturality of the maximal destabilizing subbundle, and using the following purely combinatorial lemma. Then there are no integers r, d satisfying 1 ≤ r < 6k + ǫ, and (8) Proof. Suppose such integers d and r exist. Clearing denominators, (9) and (10) yield: (6k + ǫ)d − (6k + ǫ + 2)(6k + ǫ + 3)r > 0 −kd + (6k 2 + kǫ + 5k + 1)r ≥ 0 −(k + 1)d + (6k 2 + kǫ + 11k + ǫ + 6)r ≥ ǫ − 6. Adding 6 − ǫ times the second of these inequalities to ǫ times the third yields (6k + ǫ)d − (6k + ǫ + 2)(6k + ǫ + 3)r ≤ ǫ(6 − ǫ).
This completes the proof.
Proof of Theorem 4.1. Let (d, r) be the degree and rank of the maximal destabilizing subbundle of N E∪R E . Since this naturally-defined bundle depends only on the choice of O E (1) plus choices varying in a rational base, its determinant gives a natural map Pic g E → Pic d E. By Lemma 2.7, the degree d is divisible by g. By Lemma 4.2, the slope d r is at most g + 3 + 1 k , with quotient bundle having slope at least g + 3 + 1 k+1 . By Lemma 4.6, no such integers d and r exist, and hence no destabilizing bundles exist, when g ∈ {4, 6}.
Proof of Theorem 1.1 in odd genus. Let C be a general canonical curve of odd genus g ≥ 3. By [CLV22, Lemma 4.1], semistability of N C follows from the semistability of N E∪R E and N E∪R R . The first of these is Theorem 4.1; the second is Lemma 3.1.
Proof of Theorem 1.1 in even genus using the Strong Franchetta Conjecture. The proof of Theorem 1.1 in even genus is considerably harder. Here we will give an argument using the Strong Franchetta Conjecture proved by Harer [H83] and Arbarello and Cornalba [AC87,AC98] in characteristic 0 and Schröer [S03] in characteristic p. In next two sections, we will give an elementary proof.
Suppose that the normal bundle of the general canonical curve is unstable. Specialize to E ∪ R as in Section 3. If g ≥ 8, then N E∪R E is semistable by Theorem 4.1, and any destabilizing subbundle of N E∪R R of rank r has slope at most µ(N E∪R R ) + 1 r by Lemma 3.1. Consequently, if g ≥ 8, then the maximal destabilizing subbundle F of N C would satisfy On the other hand, by the Strong Franchetta Conjecture, det F is a multiple of the canonical bundle. We conclude that the degree of F is s(2g − 2) for some integer s. Since the slope of the normal bundle of a canonical curve is (g + 1)(2g − 2) (g − 2), we obtain the inequality or upon rearrangement, Since (s − r)(g − 2) − 3r is an integer, this is a contradiction. Hence, N C is semistable for the general canonical curve.

Degeneration so that Q sing meets E
In order to give an elementary proof of Theorem 1.1 in the even genus case using the explicit description of the HN-filtration given in Section 3.1, we will show in Section 6 that it suffices to bound the slopes of subbundles of N E Q [2Γ + → N E S ]. To achieve such a bound, we will utilize a further degeneration in which E meets the singular locus Q sing of the rank 4 quadric Q described in Section 3.1 in two points {x 1 , x 2 }. The basic inductive strategy will be to degenerate in this way, and then examine the sequence obtained by projection from the line x 1 x 2 . If we do this carefully, the quotient will be another instance of our Setup 3.2 in P 2n−1 . In this section, we construct this degeneration and prove that the projection exact sequence behaves as desired. In the next section, we will use this to complete our inductive proof of Theorem 1.1 in the even genus case.
We will construct this degeneration from an instance (E, R, S, Q) of Setup 3.2 in P 2n−1 . The basic strategy will be to construct a degenerate instance of Setup 3.2 by specializing the smooth elliptic curve of type (n + 1, n + 1) on (the blowup of) P 1 × P 1 to the union of a smooth elliptic curve of type (n, n) union a (1, 0) curve and a (0, 1) curve. Write Γ = E ∩ R. Recall that via the given maps f 1 and f 2 , E maps to P 1 × P 1 ; R corresponds to the diagonal in P 1 × P 1 . We illustrate this below.
x 1 We take x 1 , x 2 ∈ E so that f 1 (x 1 ) = f 2 (x 2 ), and write p = (f 1 (x 1 ), f 2 (x 2 )). Let L 1 = f 1 (x 1 ) × P 1 and L 2 = P 1 × f 2 (x 2 ) denote the corresponding lines of the ruling (which meet at p). Let ∆ denote the remaining set (not including {x 1 , x 2 }) of points where one of the L i meets E, together with p and the nodes of E. Construct the blowup S ○ of P 1 × P 1 at ∆, and write The pair (S, E) consisting of a surface S and divisor E as in Setup 3.2 admits a degeneration to (S ○ , E ○ ∪ L 1 ∪ L 2 ) as an abstract pair of a surface with a divisor. Under the complete linear series O S ○ (n, n)(− ∑ q∈∆ F q ) , the lines L i get contracted to the points x i ; thus, in P 2n+1 , the curve E limits to E ○ embedded in P 2n+1 as an elliptic normal curve. In the limit, the linear series corresponding to the maps f i acquire basepoints at x i on E ○ . Blowing up at x 1 and x 2 to extend the maps across the central fiber, the limiting maps have degree n on E ○ and 1 on the corresponding exceptional lines L i . Equivalently, they are induced by projection of E ○ ∪ L 1 ∪ L 2 onto the two P 1 factors.
We now show that O S ○ (n, n)(− ∑ q∈∆ F q ) is basepoint free. Let ∆ denote the nodes of E. By the discussion in §3.1, the linear series O S (n − 1, n − 1)(− ∑ q∈∆ F q ) is basepoint free. Pulling back to S ○ , we conclude that O S ○ (n − 1, n − 1)(− ∑ q∈∆ F q ) is basepoint free. Multiplying by the equations of the lines L 1 and L 2 , we see that any basepoints of O S ○ (n, n)(− ∑ q∈∆ F q ) must lie on the lines L i . Since O S ○ (n, n)(− ∑ q∈∆ F q ) L i has degree zero, if there is a base point on L i , then the linear series must identically vanish on L i . An easy dimension count rules this possibility out.
Under this complete linear series O S ○ (n, n)(− ∑ q∈∆ F q ) , the image of R ○ is of degree 2n − 1, and F p is mapped to the line which meets E ○ at x 1 and x 2 (and which also meets the other component R ○ ). The images of E ○ , R ○ ∪ F p , and S ○ , along with the cone Q ○ over Q with vertex x 1 x 2 , give a degeneration of (E, R, S, Q) in our Setup 3.2 as subschemes of P 2n+1 .
Consider (E, R, S, Q) limiting to (E ○ , R ○ ∪ F p , S ○ , Q ○ ). The above description shows that x 1 and x 2 are limits of pointsx 1 ,x 2 ∈ Γ ∶= E ∩ R. Write Γ − = Γ ∖ {x 1 ,x 2 }. The limit of Γ − is identified with Γ. Our next task is to determine the flat limit of the bundles This is subtle precisely because E ○ passes through Q ○ sing (in particular the flat limit is not just To do this, define Explicitly, B is the graph of the rational map Q ⇢ P 1 × P 1 given by projection from Q sing , and similarly for B ○ . As in Setup 3.2, the composition of the map S → Q with this projection is the blowup map S → P 1 × P 1 , and similarly for S ○ . The exceptional divisor of B is isomorphic to [Q sing ≃ P 2n−3 ] × P 1 × P 1 , and similarly for B ○ . The line x 1 x 2 naturally embeds in Q ○ sing (coinciding with the image of F p ) and the lines L i naturally embed in P 1 × P 1 . The flat limit of the (proper transform of) E in B specializes to the curve E ○ ∪ L 1 ∪ L 2 in B ○ . In this limit, the pointsx 1 andx 2 limit to p 1 ∈ L 1 and p 2 ∈ L 2 , where, as above, p i = L i ∩ F p . This setup is illustrated in the following picture. The points in the limit of Γ (namely, Γ ∪ {p 1 , p 2 }) are circled.
Lemma 5.1. Let (E, R, S, Q) be a general instance of Setup 3.2 in P 2n+1 . Then admits a specialization to Proof. Since E does not meet Q sing , we have This bundle fits into a flat family N whose central fiber is The L i are lines in the exceptional divisor of types (0, 1, 0) and (0, 0, 1), respectively. In particular, their normal bundles in the exceptional divisor are trivial, and so their normal bundles in is obtained by making two positive modifications. Since the restriction of the projection from Q ○ sing to E ○ has degree 2n (i.e., equal to the degree of E) by construction, E ○ must meet Q sing at x 1 and x 2 , both with multiplicity 1. In the blowup, E ○ is therefore transverse to the exceptional divisor at the x i , so the positive modification at To identify the positive subbundle, note that there is a unique subbundle of N L i that is isomorphic to O L i (1), and that one such subbundle is Consider the modification Away from the central fiber, we have N ′ ≃ N . The central fiber N ′ 0 therefore gives another flat limit of the bundle . But by construction, N ′ 0 has trivial restriction to L 1 and L 2 by (1). Blowing down L 1 and L 2 , we conclude that a flat limit of the bundles Our final goal is to relate this to projection from the line x 1 x 2 . By construction, this projection map sends (E ○ , R ○ , S ○ , Q ○ ) in P 2n+1 to (E, R, S, Q) in P 2n−1 . We accomplish this by rewriting N ○ in terms of the normal bundle of the proper transform of E ○ in sing , there is a natural map from the exceptional divisor of B ○ − to the exceptional divisor of B ○ . Write M i for the preimage of x 1 x 2 × L i in the exceptional divisor of B ○ − . Then by (6) in Section 2.2, we have Explicitly, the exceptional divisor of B ○ − is isomorphic to where the M i are the (2n − 3)-planes of the rulings of Q corresponding to L i .
Note that x 1 x 2 × p is contained in M 1 and M 2 , and is contracted to the point p ∈ Q under projection. Moreover, M i is transverse (not just quasi-transverse!) to E at x i . Projection from x 1 x 2 therefore induces the exact sequence

Completing the proof in even genus
Let g = 2n + 2 be even. We consider the degenerate canonical curve E ∪ R ⊂ P 2n+1 introduced in Section 3. In this section, we leverage the geometric description of the HN-filtration of N E∪R R given in Section 3.1 and the semistability of N E∪R E proved in Section 4 to prove that the normal bundle of a general canonical curve of even genus is semistable.
Let S, Σ 1 , Σ 2 , and Q be as in Setup 3.2. We first reduce to proving a bound on the slopes of certain subbundles of N E∪R Q E .
Proposition 6.4. Suppose that Condition 6.1 is satisfied. Then N E∪R is semistable.
Proof. Let ν∶ E ⊔ R → E ∪ R denote the normalization, and G ⊆ ν * N E∪R be any subbundle. By Lemma 3.1 and Theorem 4.1, we have Combining these, we have with the stronger bound unless G is actually a subbundle of N E∪R and N E∪R S R ⊂ G R ⊂ N E∪R Q R . In other words, we are immediately done unless G is a subbundle of N E∪R and G R contains the positive factor O P 1 (g + 2) and is contained in next piece of the HN-filtration O P 1 (g + 2) ⊕ O P 1 (g + 1) ⊕(g−4) . We therefore assume that these hold. The restriction G E is thus a subbundle of N E∪R E with N E∪R S Γ ⊂ G Γ . Write G ′ for the kernel of the map from G E to N Q E : , which is stable of slope 2n + 2. On the other hand, the kernel G ′ ⊆ N E∪R Q E has slope by Condition 6.1. Thus G E also has slope bounded by 2n + 5 + 3 n − 1 rk G . Hence and N E∪R is semistable.
Our goal is therefore to prove that Condition 6.1 holds for all n ≥ 3. In fact, we will prove that Condition 6.2 holds for all n ≥ 3, since this implies that Condition 6.1 holds. While Condition 6.2 is stated for all subbundles of N E Q [2Γ + → N E S ], it suffices to check the slope bound for the finitely many Harder-Narasimhan pieces.
Lemma 6.5. Let N be a vector bundle on an irreducible curve C with HN-filtration Let B(r, d) be any (affine) linear function whose coefficient of d is nonnegative. If B(0, 0) ≤ 0 and form the vertices of a convex polygon in the (r, d) plane. For any subbundle F ⊆ N , the pair (rk F, deg F ) is in this polygon. The assumption that B(r, d) ≤ 0 for all vertices implies that it is also true for any point of the convex polygon.
Corollary 6.6. Suppose that for each HN-piece V of Then Condition 6.2 holds.
Proof. Apply Lemma 6.5 with The final input is the specialization of (E, R, S, Q) constructed in Section 5, giving rise to the exact sequence (12). Using this, we will prove the following numerical proposition, which is the heart of our inductive proof.
Proof. We will use the notation and results of Section 5. In particular, let (E, R, S, Q) be a general instance of Setup 3.2 in P 2n−1 . Let x 1 , x 2 be points on E such that f 1 (x 1 ) = f 2 (x 2 ). Then in Section 5 we constructed a specialization (E ○ , R ○ ∪ F p , S ○ , Q ○ ) of a general instance (E, R, S, Q) of Setup 3.2 in P 2n+1 , such that E ○ meets Q ○ sing in the points x 1 , x 2 . We writex 1 ,x 2 for points on E limiting to x 1 , x 2 .
Applying Corollary 6.6, it suffices to check that Condition 6.2 holds for each piece of the HN- Since the HN-pieces are natural, their degrees are multiples of 2n+2 by Lemma 2.7. Let F 0 ⊂ N E Q [2Γ + → N E S ] be any such subbundle of rank r and degree a multiple of 2n+2.
We now utilize the specialization constructed in Section 5. Write N ○ for the bundle appearing in Lemma 5.1, which is a flat limit of the bundle N E Q [2Γ −x 1 −x 2 + → N E S ]. This bundle sits in the exact sequence Let F ○ ⊆ N ○ be the saturation of the flat limit of F . Then rk F ○ = rk F = r and deg F ○ ≥ deg F .
Substituting into (13), we get which proves the proposition when F intersects the kernel of φ nontrivially.
Case 2: F ○ is isomorphic to its image under φ. Identifying F ○ with its image under φ, we have F ○ ⊂ N E Q [2Γ + → N E S ](x 1 + x 2 ) and r ≤ 2n − 3.
To do this, we will use the fact that naturality of the HN-pieces implies that 2n + 2 divides deg(F 0 ). If there are no integers k satisfying the inequality (15) 3r n < k ≤ 3r n − 1 , then (14) holds, as deg(F 0 ) is an integer. Hence, we assume that there is an integer k satisfying (15). First, suppose n ≥ 6. We claim that the width of the interval (15) is strictly less than 1. Indeed, since r ≤ 2n − 3 and n ≥ 6, 3r n − 1 − 3r n = 3r n 2 − n ≤ 6n − 9 n 2 − n < 1.
If k − 2ℓ ≤ 0, the left inequality is violated. If k − 2ℓ ≥ 1, then the left fraction is nonpositive, which contradicts our observation that the width of this interval is strictly less than 1. For 4 ≤ n ≤ 5, we complete the proof by checking directly that there are no integers k satisfying the conditions (16) 3r n < k ≤ 3r n − 1 , 0 < r ≤ 2n and 3r + k − 1 ≡ 0 (mod 2n + 2).
To finish, it suffices to deal with the base case: Proposition 6.8. Condition 6.2 holds in P 7 .
Proof. In this case n = 3 and we want that every subbundle F of N E Q [2Γ + → N E S ] has slope at most 12 + 7 rk F . To prove this, we use the normal bundle exact sequence for E ⊂ S ⊂ Q. Since S is the complete intersection of Σ 1 and Σ 2 in Q, we have that Since 4 is not divisible by 8, by Lemma 2.7, the degree 4 maps giving rise to the scrolls Σ 1 and Σ 2 are exchanged by monodromy. Hence the two scrolls are exchanged by monodromy, and therefore the two bundles N S Σ i E have degree 24 and the same profile of Jordan-Hölder factors. We will first show that N S Σ i E is semistable of slope 12.
The bundle N E [Γ + → N E S ] = N E [ + ↝ R] has slope 12 and satisfies interpolation by [LV22] (in the language of that paper, this is the inductive hypothesis I(8, 1, 7, 0, 1) and the tuple (8, 1, 7, 0, 1) is good). Because any bundle with integral slope that satisfies interpolation is semistable, this bundle is semistable (see, for example, [V18, Remark 1.6].) Consider the normal bundle exact sequence The line subbundle N E S (Γ) has degree 8. First suppose that one (and hence both) of N S Σ i E had a line subbundle of slope at least 15. Then the full preimage in N E Q [Γ + → N E S ] would be a bundle of slope at least 38 3 = 12 + 2 3. Since this is also a subbundle of N E [Γ + → N E S ], it contradicts the semistability of N E [Γ + → N E S ]. Hence every line subbundle of N S Σ i E is of degree at most 14. It suffices, therefore, to rule out the possibility that N S Σ i E is a direct sum of line bundles of degrees 14, 10 or 13, 11. In either of these cases, the sum of the two positive subbundles would be of degree 28 or 26. Since 8 does not divide 28 or 26, this is impossible by Lemma 2.7. Hence N S Σ i E is semistable.
We now turn to the normal bundle exact sequence involving the double modification and consider how F sits with respect to this sequence. If F does not contain N E S (2Γ), then µ(F ) ≤ 12 ≤ 12 + 7 rk F . If F contains N E S (2Γ), then µ(F ) ≤ 16 1 rk F + 12 rk F − 1 rk F ≤ 12 + 4 rk F ≤ 12 + 7 rk F .