2.1 Counterexamples in all characteristics

We start with the five counterexamples to Theorem 1.4 that occur in all characteristics:

$$\begin{align*}(d, g, r) \in \{(5,2,3), (6,4,3), (6,2,4), (7,2,5),(10,6,5)\}.\end{align*}$$

In each of these cases, we will construct a certain surface S containing C, and see that S prevents Theorem 1.2 (or the generalization (1.3) thereof) from holding. Indeed, if S cannot be made to pass through the requisite number of points (or be made incident to the requisite linear spaces), then C cannot either. Since Theorem 1.4 implies Theorem 1.2 (and the generalization (1.3)), this implies that these five cases must also be counterexamples to Theorem 1.4.

2.1.1 The family $(d,g,r)=(r+2, 2, r)$

Let C be a curve of genus $2$ and L be a line bundle of degree $r+2$ on C. Write $f \colon C \to \mathbb {P}^1$ for the hyperelliptic map. Then is a vector bundle of rank $2$ on $\mathbb {P}^1$ with

$$\begin{align*}\chi(E) = \chi(L) = r+1.\end{align*}$$

By Riemann–Roch, $c_1(E) = r-1$ . The inclusion $L \to f^*f_* L$ embeds C in the projective bundle $\mathbb {P} E^\vee $ so that $O_{\mathbb {P} E^\vee }(1)|_C \simeq L$ . Therefore, the image of C in $\mathbb {P}^r$ under the complete linear series for L lies on the image S of $\mathbb {P} E^\vee $ under the complete linear series for $\mathcal {O}_{\mathbb {P} E^\vee }(1)$ . The surface S is a scroll of degree equal to

$$\begin{align*}[\mathcal{O}_{\mathbb{P} E^\vee}(1)]^2 = - c_1(E^\vee) \cdot \mathcal{O}_{\mathbb{P} E^\vee}(1) = r-1.\end{align*}$$

By [Reference Coskun8, Lemma 2.6], the dimension of the space of such scrolls is $r^2 + 2r - 6$ .

• If $\boldsymbol{(d, g, r) = (5, 2, 3)}$ Then $r^2 + 2r - 6 = 9$ . Since it is $1$ condition for a surface in $\mathbb {P}^3$ to pass through a point, S cannot pass through more than nine general points. This contradicts (1.1), which predicts that C should be able to pass through $10$ general points.

• If $ \boldsymbol{(d, g, r) = (6, 2, 4)}$ Then $r^2 + 2r - 6 = 18$ . Since it is two conditions for a surface in $\mathbb {P}^4$ to pass through a point and one condition to meet a line, S cannot pass through nine general points while meeting a general line. This contradicts equation (1.3), which predicts that C should be able to pass through nine general points points while meeting a general line.

• If $ \boldsymbol{(d, g, r) = (7, 2, 5)} $ Then $r^2 + 2r - 6 = 29$ . Since it is three conditions for a surface in $\mathbb {P}^5$ to pass through a point, S cannot pass through more than nine general points. This contradicts equation (1.1), which predicts that C should be able to pass through $10$ general points.

2.2 Rational curves in characteristic $2$

In this section, we explain the final infinite family of counterexamples to Theorem 1.4 that occurs only in characteristic $2$ . This phenomenon was already observed for space curves in [Reference Coskun, Larson and Vogt9] in relation to semistability. We begin by describing which vector bundles on a rational curve satisfy interpolation in terms of the Birkhoff–Grothendieck classification.

Lemma 2.2. The bundle $E = \bigoplus _i \mathcal {O}_{\mathbb {P}^1}(e_i)$ satisfies interpolation if and only if for all $i, j$ ,

$$\begin{align*}|e_i - e_j| \leq 1\quad \text{and} \quad e_i \geq -1.\end{align*}$$

Proof. For any $n \geq 0$ ,

$$ \begin{align*} h^0(E(-n)) = 0 \quad &\Leftrightarrow\quad n \geq 1 + \max(e_i) \\ h^1(E(-n)) = 0 \quad &\Leftrightarrow \quad n \leq 1 + \min(e_i). \end{align*} $$

One of these two conditions holds for all $n \geq 0$ if and only if $|e_i - e_j| \leq 1$ for all i and j, and the second of these holds for $n=0$ if and only if $e_i \geq -1$ for all i.

As a consequence of the Euler sequence, the conormal bundle of C sits in the exact sequence

$$\begin{align*}0 \to N_C^\vee(1) \to \mathcal{O}_C^{\oplus r+1} \to \mathscr{P}^1(\mathcal{O}_C(1)) \to 0, \end{align*}$$

where $\mathscr {P}^1(\mathcal {O}_C(1))$ is the bundle of first principle parts of $\mathcal {O}_C(1)$ . If the characteristic is $2$ , and we write $\pi \colon C \to C^{(2)}$ for the (relative) Frobenius morphism, then we have

$$\begin{align*}\mathscr{P}^1(\mathcal{O}_C(1)) \simeq \pi^*\pi_*\mathcal{O}_C(1).\end{align*}$$

Therefore, $ N_C^\vee (1)$ is isomorphic to the pullback of a bundle under the Frobenius morphism, so every entry of its splitting type is even.

Lemma 2.3. Assume that the characteristic of the ground field is $2$ . Let $C \subset \mathbb {P}^r$ be a rational curve of degree d. Then $N_C$ satisfies interpolation only if

$$\begin{align*}d \equiv 1 \pmod{r-1}.\end{align*}$$

Proof. Suppose that $N_C \simeq \bigoplus _{i = 1}^{r - 1} \mathcal {O}_{\mathbb {P}^1}(e_i)$ . Since $N_C^\vee (1) \simeq \bigoplus _{i = 1}^{r - 1} \mathcal {O}_{\mathbb {P}^1}(d - e_i)$ , and every entry of its splitting type is even, every $e_i$ satisfies $e_i \equiv d$ mod $2$ . Applying Lemma 2.2, we conclude that $N_C$ can only satisfy interpolation if all $e_i$ are equal. This implies

$$\begin{align*}(r + 1)d - 2 = c_1(N_C) \equiv (r - 1)d \pmod{2(r - 1)},\end{align*}$$

and therefore $d \equiv 1$ mod $r - 1$ as desired.

3.1 Elementary modifications of vector bundles

In this section, we give a brief overview of the key properties of elementary modifications of vector bundles. Our presentation will roughly follow the more detailed exposition given in Sections 2–4 of [Reference Atanasov, Larson and Yang2].

Definition 3.1. Let E be a vector bundle on a scheme X, and $D \subset X$ be a Cartier divisor and $F \subset E|_D$ be a subbundle of the restriction of E to D. We define the negative elementary modification of E along D towards F by

We then define the (positive) elementary modification of E along D towards F as

By construction, a modification of E along D is naturally isomorphic to E when restricted to the complement of D. If $D_1$ and $D_2$ are disjoint, then we may easily make sense of multiple modifications such as by working locally. However, if $D_1$ and $D_2$ meet, then we do not have enough data to even define multiple modifications: For example, if $D_1 = D_2 = D$ and $F_1 = F_2 = F$ , then we should have , so we must know how F extends over $2D$ . To sidestep these issues, we suppose when defining multiple modifications – at least along divisors that meet – that we are given not just a subbundle $F_i$ of $E|_{D_i}$ , but a subbundle $F_i$ of $E|_{U_i}$ where $U_i \subset X$ is an open neighborhood of $D_i$ .

We first construct the modification , which is naturally isomorphic to E on $X {\smallsetminus} D_1$ , and so in particular on $U_2 {\smallsetminus} D_1$ . If the data of a subbundle $F_2 \subset E|_{U_2 {\smallsetminus} D_1}$ extend to $U_2$ , then it does so uniquely, and we may modify along $D_2$ towards this extension. However, this subbundle may not extend to $U_2$ . The following situation where it does will include all situations we shall need in this paper.

By [Reference Atanasov, Larson and Yang2, Proposition 2.17], we can transfer modification data as above when it is treelike. That is, given modification data M for E such that $\{(D, U, F)\} \cup M$ is treelike, we obtain modification data $M'$ for . In this way, we inductively define the multiple modification $E[M]$ for treelike modification data M. This is independent of the order in which the modifications from M are applied [Reference Atanasov, Larson and Yang2, Proposition 2.20].

Example 3.4. A simplifying special case is when F is a direct summand of E. Writing $E \simeq F \oplus E'$ ,

In nice cases, a short exact sequence of vector bundles induces a short exact sequence of modifications. To make this more precise, consider a short exact sequence

(3.1) $$ \begin{align} 0 \to S \to E \to Q \to 0. \end{align} $$

For example, first suppose that $F \cap S$ is flat over the base X. Then equation (3.1) induces the short exact sequence

(3.2)

A more interesting example is when the base is a curve $X = C$ , and $F \subset E$ is a line subbundle, and $D = np$ , where $p \in C$ is a smooth point. Then we obtain an induced sequence for the modification , as follows. Define $k'$ to be the order to which F is contained in S in a neighborhood of p. In other words, if F is not contained in S, this is the length of the subscheme $\mathbb {P} S \cap \mathbb {P} F$ in $\mathbb {P} E$ ; if F is contained in S, this is $\infty $ . Let $k = \max (k', n)$ . Then equation (3.1) induces the short exact sequence

(3.3)

where $\overline {F}$ is the saturation of the image of F in Q. When $k' = \infty $ or $k' = 0$ , this agrees with equation (3.2).

3.2 Pointing bundles

Given an unramified map $f \colon C \to \mathbb {P}^r$ , the sheaf $N_f = \ker (f^*\Omega _{\mathbb {P}^r} \to \Omega _X)^\vee $ is a vector bundle, which we refer to as the normal bundle of the map f . In almost all cases that we shall consider, f will be an embedding, in which case $N_f = N_C$ coincides with the normal bundle of the image.

We will primarily deal with modifications of $N_f$ towards pointing subbundles $N_{f \to \Lambda }$ , whose definition we now recall. Let $\Lambda \subset \mathbb {P}^r$ be a linear space of dimension $\lambda $ . Let $\pi _{\Lambda } \circ f$ denote the composition of f with the projection map

$$\begin{align*}\pi_{\Lambda} \colon \mathbb{P}^r \dashrightarrow \mathbb{P}^{r-\lambda -1}. \end{align*}$$

Let $U =U_{\Lambda }$ denote the open locus of $C {\smallsetminus} (\Lambda \cap C)$ where $\pi _{\Lambda } \circ f$ is unramified; explicitly this is the locus of points of C whose tangent space does not meet $\Lambda $ . Assuming that U is dense and contains the singular locus of C, we may define $N_{f \to \Lambda }$ as the unique subbundle of $N_f$ whose restriction to U is

$$\begin{align*}\ker(N_f|_U \to N_{\pi_{\Lambda} \circ f}|_U).\end{align*}$$

The notation $N_{f \to \Lambda }$ is evocative of the geometry of sections of $N_{f \to \Lambda }$ : Informally speaking, they ‘point towards’ the subspace $\Lambda \subset \mathbb {P}^r$ . When f is an embedding, we write $N_{C \to \Lambda } = N_{f \to \Lambda }$ . If the projection $(\pi _\Lambda \circ f) \colon C \to \mathbb {P}^{r-\lambda -1}$ is unramified, then $N_{f \to \Lambda }$ sits in the pointing bundle exact sequence

(3.4) $$ \begin{align} 0 \to N_{f \to \Lambda} \to N_f \to N_{\pi_\Lambda \circ f} (\Lambda \cap C) \to 0. \end{align} $$

The same definitions work for families of curves in a projective bundle. For a treatment in this more general setting, see [Reference Atanasov, Larson and Yang2, Section 5].

The simplest case, and our primary interest, is when $\Lambda = p$ is a point in $\mathbb {P}^r$ . In this case, by [Reference Atanasov, Larson and Yang2, Propositions 6.2 and 6.3], we have the following explicit descriptions:

• • If $p \in \mathbb {P}^r$ is a general point (in which case $U_p = C$ ), then $N_{f \to p} \simeq f^*\mathcal {O}_{\mathbb {P}^r}(1)$ .

• • If $p \in C$ is a general point (in which case $U_p = C {\smallsetminus} p$ ), then $N_{f \to p} \simeq f^*\mathcal {O}_{\mathbb {P}^r}(1)(2p)$ .

When modifying towards a pointing bundle, we use the simpler notation

For two points $p, q \in C$ , we also define the more compact notation

We now restate a foundational result of Hartshorne–Hirschowitz, which describes the normal bundle of a nodal curve in projective space, in this language of pointing bundles. Let $X \cup _{\Gamma } Y$ be a reducible nodal curve. For each point $p_i \in \Gamma $ , let $q_i$ denote any point on $T_{p_i}Y {\smallsetminus} p_i$ . For simplicity, we introduce the following notation. For any subset $\Gamma ' = \{p_1, \dots p_n\} \subseteq \Gamma $ , we write

When $\Gamma ' = \Gamma $ is the full set of points where $X $ and Y meet, we simplify further and write

(We analogously define

and

.)

Proposition 3.5 [Reference Hartshorne and Hirschowitz18, Corollary 3.2]

As above, let $X \cup Y \subseteq \mathbb {P}^r$ be a reducible nodal curve. Then

3.3 Interpolation for vector bundles

3.3.1 Interpolation for bundles on nodal curves

Let C be a nodal curve, and let E be a nonspecial vector bundle on C. Twisting down by a point can only decrease $h^0$ and can only increase $h^1$ . Therefore, E satisfies interpolation provided that there exist two divisors $D_+$ and $D_-$ for which

(3.5) $$ \begin{align} h^0(E(-D_+)) = 0, \quad h^1(E( - D_-)) = 0, \quad \text{and} \quad \deg D_+ - \deg D_- \leq 1. \end{align} $$

Alternatively, if E has rank r, then twisting down by a point either decreases $h^0$ by r or increases $h^1$ . Thus, E satisfies interpolation if and only if, for every $n> 0$ , some collection of n points impose the expected number of conditions:

$$\begin{align*}h^0(E(-p_1 - \cdots - p_n)) = \max(0, h^0(E) - rn) \qquad \text{for some}\ p_1, \dots, p_n.\end{align*}$$

More generally, we can use this idea to define interpolation for a space of sections of a vector bundle. Given $V \subseteq H^0(E)$ , write

We say that $V \subseteq H^0(E)$ satisfies interpolation if $H^1(E) = 0$ and, for every $n> 0$ , there are n points $p_1, \dots , p_n$ such that

$$\begin{align*}\dim V(-p_1 - \cdots - p_n) = \max(0,\dim V - rn).\end{align*}$$

The following basic result allows us to reduce interpolation for a vector bundle E on a reducible curve $X \cup _{\Gamma } Y$ to interpolation for a space of sections on one component.

Lemma 3.6 [Reference Larson and Vogt23, Lemma 2.10]

Let E be a vector bundle on $X \cup _{\Gamma } Y$ . If the restriction map on Y

$$\begin{align*}\operatorname{res}_{Y,\Gamma} \colon H^0(Y, E|_Y) \to E|_\Gamma\end{align*}$$

is injective and the space of sections $\{\sigma \in H^0(X, E|_X) : \sigma |_\Gamma \in \operatorname {Im}(\operatorname {res}_{Y, \Gamma })\}$ has dimension $\chi (E)$ and satisfies interpolation, then E satisfies interpolation.

The main case of interest in this paper is when $Y = R$ is a rational curve and $E|_R$ is perfectly balanced, that is, $E|_R \simeq \mathcal {O}_{\mathbb {P}^1}(a)^{\oplus r}$ for some $a \in \mathbb {Z}$ .

Lemma 3.7. Let $C \cup _{\Gamma } R$ be a nodal curve with R rational, and let E be a vector bundle on $C \cup R$ with $E|_R$ perfectly balanced of slope at least $\#\Gamma - 1$ . If $E|_C$ satisfies interpolation, then E satisfies interpolation.

Proof. Let D be an effective divisor of degree $\mu (E|_R) - \#\Gamma + 1$ supported on $R {\smallsetminus} \Gamma $ . It suffices to prove that $E(-D)$ satisfies interpolation, as we now explain. Let $D_+$ and $D_-$ be the two divisors satisfying equation (3.5) for the vector bundle $E(-D)$ . Then the divisors $D_+ + D$ and $D_-+D$ satisfy equation (3.5) for the vector bundle E. The bundle $E(-D)|_R$ is perfectly balanced of slope $\#\Gamma - 1$ , and so $\operatorname {res}_{R, \Gamma }$ is an isomorphism. The result now follows from Lemma 3.6.

3.3.2 Interpolation and twists

If E satisfies interpolation, then as in the proof of Lemma 3.7, the twist $E(D)$ by any effective divisor D also satisfies interpolation. Conversely, we have the following.

Lemma 3.8 [Reference Atanasov, Larson and Yang2, Proposition 4.12]

Suppose that E is a vector bundle on a genus g curve such that

$$\begin{align*}\chi(E) \geq \operatorname{rk}(E) \cdot g.\end{align*}$$

If there exists an effective divisor D for which $E(D)$ satisfies interpolation, then E also satisfies interpolation.

3.3.3 Interpolation and modifications

Consider a vector bundle E and its modification . Given sufficient generality of either p or F, or if the slope $\mu (E) \in \mathbb {Z}$ , it is sometimes possible to deduce that satisfies interpolation from the assumption that E satisfies interpolation.

Lemma 3.9. Let E be a vector bundle on C, let $p \in C$ be a smooth point and let $F \subseteq E|_p$ . If E satisfies interpolation and $\mu (E) \in \mathbb {Z}$ , then satisfies interpolation.

Proof. Since $\mu (E) \in \mathbb {Z}$ , there is an effective divisor D with

$$\begin{align*}H^0(E(-D)) = 0 \quad \text{and} \quad H^1(E(-D)) = 0.\end{align*}$$

Then

Definition 3.10. We say that a collection of subspaces $\{W_b\}_{b \in B}$ of a vector space V are linearly general if, for any subspace $U \subset V$ , there is some $b \in B$ so that $W_b$ is transverse to U.

Lemma 3.11 [Reference Atanasov, Larson and Yang2, Proposition 4.10]

Let E be a vector bundle on C, and let $p \in C$ be a smooth point. Let $\{F_b\}_{b \in B}$ be a collection of subspaces of $E|_p$ that all contain a fixed subspace $F_0$ . Suppose that both E and $E[p \to F_0]$ satisfy interpolation. If the collection $\{F_b/F_0\}_{b \in B}$ is linearly general in $E|_p/F_0$ , then for some $b\in B$ , the positive modification satisfies interpolation.

Lemma 3.12 [Reference Atanasov, Larson and Yang2, Proposition 4.21 for $n=1$ ]

Suppose E satisfies interpolation, $L \subset E$ is a nonspecial line subbundle, and the quotient $Q = E/L$ also satisfies interpolation. If $\mu (L) \leq \mu (H)$ , then satisfies interpolation.

Remark 3.13. While [Reference Atanasov, Larson and Yang2] assumes characteristic zero, none of the specific results we quote from [Reference Atanasov, Larson and Yang2] use this assumption – except for Proposition 4.21. This proposition states that if $\mu (L) \leq \mu (H) + n - 1$ , then satisfies interpolation. The proof uses that vanishing at $np$ imposes n conditions on sections of any linear series. This is true in any characteristic when $n=1$ , but fails in positive characteristic for $n> 1$ . Since we will use Proposition 4.21 of [Reference Atanasov, Larson and Yang2] only when $n = 1$ , we do not need a restriction on the characteristic.

Lemma 3.14. Let E be a vector bundle on an irreducible curve C. Let $p_1, \ldots , p_n \in C$ be points, and let $L_i \subseteq E|_{p_i}$ be one-dimensional subspaces. Suppose that both E and

satisfy interpolation. Then for any $0 <m <n$ , there is a collection of distinct indices $i_1, \dots , i_m$ such that

satisfies interpolation.

Proof. By induction on n we reduce to the case $m=n-1$ . Write and $N = \lceil \chi (E') / \operatorname {rk} E'\rceil $ . Let $D_N$ and $D_{N-1}$ be general divisors of degrees N and $N - 1$ , respectively. Since E and $E'$ both satisfy interpolation and $\chi (E) < \chi (E')$ , we have

$$\begin{align*}h^0(E'(-D_N)) = 0, \quad h^0(E'(-D_{N-1})) \neq 0, \quad \text{and} \quad h^0(E(-D_{N-1})) < h^0(E'(-D_{N-1})).\end{align*}$$

Let . Since $\chi (E_i) = \chi (E') - 1$ , it suffices to show $h^0(E_i(-D_{N-1})) < h^0(E'(-D_{N-1}))$ for some i. This follows from the fact that

$$\begin{align*}\bigcap_i H^0(E_i(-D_{N-1})) = H^0(E(-D_{N-1})) \subsetneq H^0(E'(-D_{N-1})).\\[-49pt] \end{align*}$$

3.3.4 Interpolation and short exact sequences

Lemma 3.15. Consider an exact sequence

$$\begin{align*}0 \to S \to E \to Q \to 0\end{align*}$$

of vector bundles on an irreducible curve C. Suppose that S and Q satisfy interpolation and

(3.6) $$ \begin{align}\mu(S) \leq \lfloor \mu(Q) \rfloor + 1 \quad \text{and} \quad \mu(Q) \leq \lfloor \mu(S) \rfloor +1.\end{align} $$

Then E also satisfies interpolation.

Proof. Since S and Q are nonspecial, E is nonspecial. By (3.6), there exists an integer $n \in \mathbb {Z}$ such that $\mu (S)$ and $\mu (Q)$ are contained in the closed interval $[n, n+1]$ . Since (3.5) is satisfied for $D_+$ a general divisor of degree $n + 1$ , and $D_-$ a general divisor of degree n, we conclude that E satisfies interpolation as desired.

We will most often use this result in the special case in which S is a line subbundle of E.

Corollary 3.16. Suppose that $S \subset E$ is a nonspecial line subbundle and $|\mu (S) - \mu (E)| < 1$ . If the quotient $Q = E/S$ satisfies interpolation, then E does as well.

Proof. By assumption $-1 < \frac {\deg (Q) - \operatorname {rk}(Q)\deg (S)}{\operatorname {rk}(Q)+1} < 1.$ Hence, we have strict inequalities

$$\begin{align*}\deg(S) < \mu(Q) + \frac{\operatorname{rk}(Q) + 1}{\operatorname{rk}(Q)} \quad \text{and} \quad \mu(Q) < \deg(S) + \frac{\operatorname{rk}(Q) + 1}{\operatorname{rk}(Q)},\end{align*}$$

which imply the required inequalities in Lemma 3.15, since $\mu (Q)$ is a $[1/\operatorname {rk}(Q)]$ -integer.

4.1 Base cases

We can reduce the number of base cases by extending Theorem 1.4 to $r=1$ and $r=2$ . For $r=2$ , we replace $N_C$ with the normal sheaf $N_f$ , where $f \colon C \to \mathbb {P}^2$ is a general BN-curve. In this case, adjunction implies that $N_f = K_C \otimes f^*\mathcal {O}_{\mathbb {P}^2}(3)$ is a nonspecial line bundle, and therefore satisfies interpolation. For $r=1$ , we only consider the case where $f \colon C \to \mathbb {P}^1$ is an isomorphism, so $N_f = 0 $ satisfies interpolation.

4.2 first strategy: degeneration of C

The first inductive strategy we will use is degeneration of C to reducible curves $X \cup Y$ . In Section 5 we will study certain such degenerations, for which Y has a prescribed form, and we can thus relate interpolation for $N_C$ to interpolation for certain modifications of $N_X$ .

Since the sectional monodromy group of a general BN-curve always contains the alternating group [Reference Kadets20], and in particular is $(r+1)$ -transitive, it makes sense to talk of a general $(r+1)$ -secant rational curve of degree $r-1$ in a hyperplane. While the following hypothesis does not encompass all modifications that might appear using this method, it includes those modifications that will play the most central role in our inductive argument:

Hypothesis 4.1 ( $I(d, g, r, \ell , m)$ )

Let $C \subset \mathbb {P}^r$ be a general BN-curve of degree d and genus g. Let $u_1, v_1, \dots , u_\ell , v_\ell $ be $\ell $ pairs of general points on C. Let $R_1, \dots , R_m$ be m general $(r+1)$ -secant rational curves of degree $r - 1$ (contained in hyperplanes transverse to C). Then the modification

of the normal bundle of C satisfies interpolation.

A central complicating factor is that the inductive hypothesis $I(d, g, r, \ell , m)$ is not always true. The following definition describes a set of tuples $(d, g, r, \ell , m)$ for which we will prove that it holds.

We conclude Section 5 by using this first strategy to show that if $I(d, g, r, 0, m)$ holds for every good tuple $(d, g, r, 0, m)$ , then Theorem 1.4 holds except possibly for rational curves or canonical curves of even genus.

4.3 Second Strategy: Limits of Modifications and Projection

The basic issue with the first strategy described above is that every time we apply it, we get more modifications. In order to make an inductive argument work, we need a second inductive strategy that decreases the number of modifications.

Hypothesis $I(d,g,r,\ell , m)$ asserts that

satisfies interpolation. Let p be a general point on C. The pointing bundle exact sequence induces the exact sequence

(4.1)

where $\overline {C}$ and $\overline {R}_j$ denote the images of C and $R_j$ , respectively under projection from p. In order to apply Corollary 3.16 to relate interpolation for the original bundle $N_C'$ to interpolation for the quotient bundle, the sequence must be close to balanced. The failure of the sequence to be balanced is related to the quantity

We first apply these ideas in Section 6 to treat the family of good tuples $(d, g,r, 0, 0)$ with ${\delta (d, g, r, 0,0)=1}$ , which are difficult from the perspective of our more uniform inductive arguments.

More generally, in order to make this sequence sufficiently close to balanced, we will appropriately specialize the points on C at which the modifications occur. To illustrate this idea in the simplest possible case, assume here that $\ell \geq \lfloor \delta \rfloor $ . Since the points $v_1, \dots , v_{\lfloor \delta \rfloor }$ are general on C, we may specialize them all to the point p. This induces the specialization of $N_C'$ to

Using equation (3.1), the pointing bundle exact sequence becomes

By our auspicious choice to specialize exactly ${\lfloor \delta \rfloor }$ points to p, this sequence is now close enough to balanced to reduce to proving interpolation for the quotient bundle. Furthermore, since $u_1, \dots , u_{\lfloor \delta \rfloor }$ are general, the modification at p in the quotient is linearly general and we can erase it by Lemma 3.12. It therefore suffices to prove interpolation for

which evidently has fewer modifications. However, there are two basic issues with this argument:

1. 1. In general, we might not have $\ell \geq \lfloor \delta \rfloor $ .

2. 2. Since $\overline {R}_i$ is still an $(r+1)$ -secant curve of degree $r-1$ , the argument does not reduce to another case of our inductive hypothesis.

To surmount both of these two difficulties, we will need to specialize the $R_i$ as well.

This second strategy will be fleshed out in Sections 7 and 8: In Section 7, we will study how to specialize the $R_i$ so that they can also contribute modifications to $N_{C \to p}$ . Then in Section 8, we will refine the basic argument outlined above to use these degenerations of the $R_i$ as well, and also explain further degenerations that will be necessary to reduce to another case of our inductive hypothesis.

4.4 Outline of the remainder of paper

Section 9 is a brief interlude in which we use the inductive arguments of Section 8 to treat the case of rational curves not implied by $I(d,g,r,\ell ,m)$ for good tuples. We also explain the counterexamples to Theorem 1.4 that are not counterexamples to Theorem 1.2. At this point, we will have reduced both Theorem 1.4 and Theorem 1.2 to $I(d,g,r,\ell , m)$ for good tuples, as well as Theorem 1.4 for canonical curves of even genus, which we treat at the end of the paper in Section 13. The intervening sections 10–12 inductively prove $I(d,g,r,\ell ,m)$ for good tuples.

In Section 10, we complete a purely combinatorial analysis, in which we show that the inductive arguments of Section 8 can be applied to reduce $I(d, g, r, \ell , m)$ for all good tuples to

• • $I(d, g, r, \ell , m)$ for a certain large but finite list of sporadic cases $(d, g, r, \ell , m)$ with $r \leq 13$ .

• • The infinite family of tuples $(d, g, r, 0, 0)$ with $\delta = 1$ , which was already treated in Section 6.

In Section 11, we give a more complicated, yet more flexible, inductive argument in the style of those in Section 8 and verify by exhaustive computer search that it reduces the finitely many sporadic cases identified above to a managable list of $30$ base cases. These base cases are treated by ad-hoc techniques in Section 12.

5.1 Peeling off one-secant lines

Our most basic degeneration will be when $D = L$ is a quasitransverse one-secant line. (Recall that two subschemes X and Y of a scheme Z are transverse (respectively, quasitransverse) at a point $p \in X \cap Y$ if the natural map of tangent spaces $T_pX \oplus T_pY \to T_pZ$ is surjective (respectively, either injective or surjective).) If C has degree d and genus g, then $C \cup L$ has degree $d+1$ and genus g. Write $v \in L {\smallsetminus} \{u\}$ for any other point on the line L.

Proof. Generalizing C, we may suppose C is a general BN-curve. We will show $H^1(T_{\mathbb {P}^r}|_{C \cup L}) =0$ , which implies that the map $C \cup L \to \mathbb {P}^r$ may be lifted as $C \cup L$ is deformed to a general curve.

Since C is a general BN-curve, $H^1(T_{\mathbb {P}^r}|_C)=0$ by the Gieseker–Petri theorem. Furthermore, we have $H^1(T_{\mathbb {P}^r}|_L(-u)) =0$ because $T_{\mathbb {P}^r}|_L \simeq \mathcal {O}_{\mathbb {P}^1}(2) \oplus \mathcal {O}_{\mathbb {P}^1}(1)^{\oplus (r-1)}$ . This implies $H^1(T_{\mathbb {P}^r}|_{C \cup L}) = 0$ as desired, using

$$\begin{align*}0 \to T_{\mathbb{P}^r}|_L(-u) \to T_{\mathbb{P}^r}|_{C \cup L} \to T_{\mathbb{P}^r}|_C \to 0.\\[-38pt] \end{align*}$$

When C is nonspecial with genus small relative to r, we can combine Lemma 5.2 with Lemma 3.8 to reduce to a positive modification of $N_C'$ .

Proof. Since

, it suffices by Lemma 3.8 to show that

(5.1)

We have

If $g \leq r + 6$ , then

$$\begin{align*}(r-1)g - 4g = (r-5)g \leq (r-5)(r+6) \leq r(r+1),\end{align*}$$

and the desired inequality (5.1) holds.

5.2 Peeling off one-secant lines

Our next basic degeneration will be to the union of a curve C and a quasitransverse one-secant line L, meeting C at points u and v. If C has degree d and genus g, then $C \cup L$ has degree $d+1$ and genus $g+1$ .

We generalize Lemma 5.2 to one-secant lines. In slightly greater generality, let $p \in L {\smallsetminus} \{u, v\}$ be a point, and $\Lambda $ be a linear space disjoint from the span of the tangent lines to C at u and v.

As in Corollary 5.3, we can reduce to a positive modification of $N_C'$ .

Proof. We have

As in the proof of Corollary 5.3, if $g \leq r+6$ , then $4g + r(r+1)+2 \geq (r-1)g$ , and so

(5.2)

Hence, by Lemma 3.8,

satisfies interpolation.

5.3 Peeling off rational normal curves in hyperplanes

The final of our basic degenerations is to the union of a BN-curve C of degree at least $r+1$ in $\mathbb {P}^r$ , and an $(r+1)$ -secant rational curve R of degree $r-1$ contained in a hyperplane H. If C has degree d and genus g, then $C \cup R$ has degree $d + r-1$ and genus $g + r$ . Observe that

$$\begin{align*}\rho(d,g,r) = \rho(d + r -1, g+r, r) + 1.\end{align*}$$

Proof. Generalizing C, we may suppose C is a general BN-curve.

If $g> 0$ , then we can specialize C to the union $C' \cup L$ , where $C'$ is a general BN-curve of degree $d-1$ and genus $g-1$ , and L is a general one-secant line. Otherwise, if $g = 0$ , then we can specialize C to the union $C' \cup L$ , where $C'$ is a general BN-curve of degree $d-1$ and genus g, and L is a general one-secant line. Either way, we can arrange for one of the points p of $C \cap R$ to specialize onto L, and the rest to specialize onto $C'$ .

Write . Note that this is a set of $r + 1$ or $r+2$ points on $C'$ . We will show the following:

1. (a) The curve $C' \cup L \cup R$ is a smooth point of the Hilbert scheme.

2. (b) The curve $L \cup R$ can be smoothed to a rational normal curve M while preserving the points of incidence with $C'$ .

3. (c) The curve $C' \cup M$ is in the Brill–Noether component.

By part (a), the curves $C \cup R$ and $C' \cup M$ are both generalizations of a smooth point of the Hilbert scheme. Hence, they are in the same component, which must be the Brill–Noether component by part (c).

Beginning with part (b), write N for the subsheaf of $N_{L \cup R}$ whose sections fail to smooth the node p. It suffices to show that $H^1(N(-\Gamma ))=0$ . This follows from the exact sequence

$$\begin{align*}0 \to N_{R \cup L}|_R(-p - \Gamma) \to N(-\Gamma) \to N|_L(-\Gamma) \to 0,\end{align*}$$

together with the isomorphisms

$$ \begin{align*} N_{R \cup L}|_R &\simeq N_{R/H} \oplus \mathcal{O}_R(1)(p) \simeq \mathcal{O}_{\mathbb{P}^1}(r+2)^{\oplus r-2} \oplus \mathcal{O}_{\mathbb{P}^1}(r), \\ N|_L &\simeq N_L \simeq \mathcal{O}_{\mathbb{P}^1}(1)^{\oplus r-1}. \end{align*} $$

Since $N_{L \cup R}(-\Gamma )$ is a positive modification of $N(-\Gamma )$ , we also have $H^1(N_{L \cup R}(-\Gamma )) = 0$ . Similarly, $N_{C' \cup L \cup R}|_{C'}$ is a positive modification of $N_{C'}$ . As $C'$ is a general BN-curve, the Gieseker–Petri theorem implies that $H^1(N_{C' \cup L \cup R}|_{C'})=0$ . Hence, using the exact sequence

$$\begin{align*}0 \to N_{L \cup R}(-\Gamma) \to N_{C' \cup L \cup R} \to N_{C' \cup L \cup R}|_{C'} \to 0, \end{align*}$$

we see that $H^1(N_{C' \cup L \cup R}) = 0$ , and part (a) follows.

Finally, for part (c), we will show that $H^1(T_{\mathbb {P}^r}|_{C' \cup M})= 0$ , and hence the map from $C' \cup M$ to $\mathbb {P}^r$ can be lifted as $C' \cup M$ is smoothed to a general curve. This vanishing follows from the exact sequence

$$\begin{align*}0 \to T_{\mathbb{P}^r}|_M(-\Gamma) \to T_{\mathbb{P}^r}|_{C' \cup M} \to T_{\mathbb{P}^r}|_{C'} \to 0, \end{align*}$$

the isomorphism $T_{\mathbb {P}^r}|_M \simeq \mathcal {O}_{\mathbb {P}^r}(r+1)^{\oplus r} $ , and the Gieseker–Petri theorem ( $H^1(T_{\mathbb {P}^r}|_{C'}) = 0$ ).

Our next goal is to study the restricted normal bundle , which is of slope $r + 2$ . In most cases, this bundle is perfectly balanced (equivalently semistable):

Lemma 5.8. Unless r is odd and C is an elliptic normal curve, is perfectly balanced, that is

(5.3)

If r is odd and C is an elliptic normal curve, then is ‘almost balanced’, that is, is isomorphic to one of the two bundles:

$$\begin{align*}\mathcal{O}_{\mathbb{P}^1}(r + 2)^{\oplus (r - 1)} \quad \text{or} \quad \mathcal{O}_{\mathbb{P}^1}(r + 3) \oplus \mathcal{O}_{\mathbb{P}^1}(r + 2)^{\oplus (r - 3)} \oplus \mathcal{O}_{\mathbb{P}^1}(r + 1).\end{align*}$$

Proof. Write d and g for the degree and genus of C. First, we reduce to the cases where C is nonspecial, that is, where $d \geq g + r$ . To do this, when $d < g + r$ , we inductively specialize C to a union $C' \cup D$ , where $C'$ is a general BN-curve of degree $d' \geq r + 2$ . Since in particular $d' \geq r + 1$ , we may specialize the points where R meets C onto $C'$ , thereby replacing C by $C'$ , which is not an elliptic normal curve since $d' \geq r + 2$ . To find such a specialization, we break into cases as follows:

1. 1. If $d < g + r$ and $\rho (d, g, r)> 0$ (which forces $d \geq 2r + 1 \geq r + 3$ ), we apply Lemma 5.4 to degenerate C to the union of a general BN-curve $C'$ of degree $d - 1$ and genus $g - 1$ , with a one-secant line D.

2. 2. If $d < g + r$ and $\rho (d, g, r) = 0$ , but C is not a canonical curve (which forces $d \geq 3r \geq 2r + 2$ ), we claim that we may specialize C to the union of a general BN-curve $C'$ of degree $d - r$ and genus $g - r - 1$ , with an $(r + 2)$ -secant rational normal curve D. Indeed, as in the proof of Lemma 5.1, we have that $H^1(T_{\mathbb {P}^r}|_{C' \cup D})=0$ by combining $H^1(T_{\mathbb {P}^r}|_{C'})=0$ (from the Gieseker–Petri theorem) and $H^1(T_{\mathbb {P}^r}|_D(-D \cap C')) = 0$ (from $T_{\mathbb {P}^r}|_D \simeq \mathcal {O}_{\mathbb {P}^1}(r + 1)^{\oplus r}$ ).

3. 3. If C is a canonical curve, we claim that we may specialize C to the union of a general BN-curve $C'$ of degree $r + 2$ and genus $2$ , with a r-secant rational curve of degree $r - 2$ . Indeed, we glue an abstract curve of genus $2$ to $\mathbb {P}^1$ at r general points, and map it to projective space via the complete linear series for the dualizing sheaf.

For C nonspecial, we will prove the lemma by induction on r. Let $\Gamma $ be a collection of $r + 1$ points where C meets R; for $r \geq 3$ , this is exactly the intersection $C \cap R$ , and so . However, we can extend the lemma to cover the case $r = 2$ as well, by replacing equation (5.3) with the assertion that . With this formulation, the base case of $r=2$ is clear since is a line bundle of degree $4$ . The base case $r=3$ is [Reference Coskun, Larson and Vogt9, Lemma 4.2].

For the inductive step, we suppose $r \geq 4$ . Let $H'$ be a general hyperplane transverse to H. We will degenerate C to a union $C' \cup L$ , where $C' \subset H'$ is a general BN-curve of degree $d-1$ and genus g. By hypothesis, $d\geq r+1$ , and so $C'$ meets H in at least r points, which are in linear general position in $H \cap H' \simeq \mathbb {P}^{r - 2}$ since the sectional monodromy group of a general curve always contains the alternating group [Reference Kadets20]. Since $\operatorname {Aut} \mathbb {P}^{r-2}$ acts transitively on r-tuples of points in linear general position, we can apply an automorphism so that $r-1$ of these points are on R and the final point p lies on a one-secant line L to R.

We claim that $C' \cup L$ is a BN-curve of degree d and genus g, as $H^1(T_{\mathbb {P}^r}|_{C' \cup L})=0$ . Indeed, $H^1(T_{\mathbb {P}^r}|_{C'}) = 0$ because $C'$ is nonspecial (and $T_{\mathbb {P}^r}$ is a quotient of $\mathcal {O}_{\mathbb {P}^r}(1)^{\oplus r+1}$ ). Furthermore, since $T_{\mathbb {P}^r}|_L \simeq \mathcal {O}_{\mathbb {P}^1}(2) \oplus \mathcal {O}_{\mathbb {P}^r}(1)^{\oplus r}$ , we have $H^1(T_{\mathbb {P}^r}|_L(-p)) = 0$ . The result now follows by considering

$$\begin{align*}0 \to T_{\mathbb{P}^r}|_L(-p) \to T_{\mathbb{P}^r}|_{C' \cup L} \to T_{\mathbb{P}^r}|_{C'} \to 0.\end{align*}$$

Call u and v the points on R where L is one-secant. Projecting from L induces an exact sequence

The curve $\overline {R}$ is again a rational curve of degree $r-3$ in a hyperplane that is incident to $\overline {C'}$ at $r-1$ points. Furthermore, if C is not an elliptic normal curve, then neither is $\overline {C'}$ . Applying our inductive hypothesis (for $\mathbb {P}^{r - 2}$ ), in combination with the above exact sequence, completes the proof.

It is natural to ask what happens when C is an elliptic normal curve and r is odd. This case is only necessary to treat the special family of canonical curves of even genus. When we treat that case in Section 13, we will show that is not perfectly balanced in this case. Moreover, we will give a geometric construction of its Harder–Narasimhan filtration.

5.4 Reduction to good tuples

In this section, we show that, apart from rational curves and canonical curves of even genus, all other cases of Theorem 1.4 follow from $I(d,g,r, 0, m)$ for good tuples with $\ell = 0$ .

Proof. Let C be a general BN-curve of degree d and genus g in $\mathbb {P}^r$ with $g \geq r$ . Let $u_1, v_1, \ldots , u_\ell , v_\ell $ be general points on C. Let $R_1, \dots , R_m$ be general $(r+1)$ -secant rational curves of degree $r-1$ . The statement $I(d, g, r, \ell , m)$ asserts that

satisfies interpolation. Combining the assumptions that $\rho (d,g,r)\geq 0$ and $g \geq r$ , we see that

$$\begin{align*}d \geq r + \frac{rg}{r+1} \geq r + \frac{r^2}{r + 1}> 2r - 1.\end{align*}$$

We may therefore prove $I(d, g, r, \ell , m)$ by peeling off an additional $(r+1)$ -secant rational curve $R_{m+1}$ of degree $r-1$ . That is, we specialize the curve C as in Lemma 5.7 to the union of a general BN-curve $C'$ of degree $d-r+1$ and genus $g-r$ and a rational curve $R_{m+1}$ , in such a way that the points of $C \cap (R_1 \cup \cdots \cup R_m)$ and the $u_i$ and $v_i$ specialize onto $C'$ . By virtue of specializing the auxilliary points onto $C'$ , we have

Since we assume that $(d,g,r)\neq (2r, r+1, r)$ if r is odd, Lemma 5.8 implies that $N_{C' \cup R_{m+1}}|_{R_{m+1}}$ is perfectly balanced. By Lemma 3.7, it suffices to prove that $N_{C' \cup R_{m+1}}|_{C'}$ satisfies interpolation. This restriction is

which satisfies interpolation by our assumption that $I(d-r+1, g-r, r,\ell ,m+1)$ holds.

7.1 Setup

Let n be an integer satisfying $0 \leq n \leq r - 1$ and $n \equiv r - 1$ mod $2$ . Let $p, q_1, q_2, \ldots , q_{r - 1} \in C$ be points such that $2p + q_1 + \cdots + q_{r - 1}$ lies in a hyperplane H. Assume that $2p + q_1 + \cdots + q_{r - 1}$ is otherwise in linear general position, that is, $p + q_1 + \cdots + q_{r - 1}$ and each $2p + q_1 + \cdots + q_{i - 1} + q_{i + 1} + \cdots + q_{r - 1}$ spans H. When C is a general nonspecial BN-curve, we claim this assumption is satisfied if p is general and H is a general hyperplane containing $2p$ . Indeed, in this case, the projection $\overline {C}$ from p is a general BN-curve, and p remains a general point on $\overline {C}$ . Since the sectional monodromy group of a general curve always contains the alternating group [Reference Kadets20], the corresponding points of $\overline {C}$ are in linear general position in the projection of H.

For i between $1$ and n, write $L_i$ for the line joining p and $q_i$ . For j from $1$ e to , let $Q_j$ be a plane conic passing through p, $q_{n + 2j - 1}$ and $q_{n + 2j}$ . The following diagram illustrates the $L_i$ and $Q_j$ :

Define

We will study when $R^\circ $ is a limit of $(r + 1)$ -secant rational normal curves $R^t$ in hyperplanes that are $(r + 1)$ -secant to C, in such a way that exactly two points of secancy limit together to p while the remaining points of secancy limit to $q_1, q_2, \ldots , q_{r - 1}$ . For this, it is evidently necessary to have a containment of Zariski tangent spaces $T_p C \subset T_p R^\circ $ . In what follows, we will show that this condition is sufficient.

Suppose that $T_p C \subset T_p R^\circ $ and

. Then the tangent line to $L_i$ (respectively to $Q_j$ ) at p gives a distinguished point $a_i$ (respectively $b_j$ ) in

Write $\Gamma = \{a_1, \ldots , a_n, b_1, \ldots , b_{n'}\}$ , which is a collection of m points in $\Lambda $ . Our linear generality assumption on $2p + q_1 + \cdots + q_{r - 1}$ implies that $\Gamma $ is linearly general in $\Lambda $ . Indeed, a linear dependence between $a_{i_1}, \ldots , a_{i_\alpha }$ , $b_{j_1}, \ldots , b_{j_\beta }$ in $\Lambda $ implies a linear dependence between $2p, q_{i_1}, \ldots , q_{i_\alpha }, q_{n + 2j_1 - 1}, q_{n + 2j_1}, \ldots , q_{n + 2j_\beta - 1}, q_{n + 2j_\beta }$ in H. Let $T \subset \Lambda $ be a general rational normal curve in $\Lambda $ passing through $\Gamma $ , and let M be a general one-secant line to T. Our argument will furthermore show that we can choose the family $R^t$ so that the modifications along $R^t$ at the points approaching p limit to M, that is, so that

fits into a flat family whose central fiber is

.

7.2 The construction

To construct the desired family $R^t$ , let B be the spectrum of a DVR, with special point $t = 0$ . Consider the blowup of $\mathbb {P}^r \times B$ along $C \times 0$ . The special fiber X over $0$ has two components: The first is isomorphic to the blowup $\operatorname {Bl}_C \mathbb {P}^r$ , and contains the proper transform $\hat {R}^\circ = \hat {L}_1 \cup \cdots \cup \hat {L}_n \cup \hat {Q}_1 \cup \cdots \cup \hat {Q}_{n'}$ of $R^\circ $ . The second is isomorphic to the normal cone $\mathbb {P}(N_C \oplus \mathcal {O}_C)$ , and contains the special fiber of the proper transform of $C \times B$ , which coincides with $\mathbb {P} \mathcal {O}_C \subset \mathbb {P}(N_C \oplus \mathcal {O}_C)$ and is isomorphic to C. The two components meet along $\mathbb {P} N_C$ , and the intersection $\hat {R}^\circ \cap \mathbb {P} N_C$ is the finite set of points $\Gamma \cup \Gamma '$ , where $\Gamma = \{a_1, \ldots , a_n, b_1, \ldots , b_{n'}\}$ lies in the fiber over p, and $\Gamma ' = \{q_1', q_2', \ldots , q_{r - 1}'\}$ contains one point $q_i'$ in the fiber over each $q_i$ . The following diagram illustrates the central fiber X:

Let $\ell _k \subset \mathbb {P}(N_C \oplus \mathcal {O}_C)|_{q_k}$ denote the line joining $q_k$ to $q_k'$ . These lines are pictured as dotted vertical lines in the above diagram.

Write $\Delta = M \cap T$ . The linear series defines a map $\mathbb {P}^1 \simeq T \to \mathbb {P} V \simeq \mathbb {P}^{m-1}$ of degree m, which identifies the two points of $\Delta $ to a common point. Fix an embedding $\mathbb {P} V \hookrightarrow \mathbb {P}(N_C \oplus \mathcal {O}_C)$ , which agrees with the identification $\mathbb {P} H^0(\mathcal {O}_T(1)) \simeq \Lambda $ , and sends this common point to . Composing these maps, we obtain a map $f \colon \mathbb {P}^1 \simeq T \to \mathbb {P}(N_C \oplus \mathcal {O}_C)$ , which is pictured as the dotted curve in the above diagram. By construction, f passes through $\Gamma $ and is nodal at p. Moreover, composing projection from p with f is the identity on T, and projection from p sends the Zariski tangent space of the image of f at p to M.

We now glue $\hat {R}^\circ $ to f and the $\ell _i$ , that is, we consider the map $F \colon \hat {R}^\circ \cup T \cup \ell _1 \cup \cdots \cup \ell _{r - 1} \to X$ defined by the natural inclusions on $\hat {R}^\circ $ and the $\ell _i$ , and by f on T. To complete the argument, it suffices to deform F to the general fiber in a way that preserves its incidence to C. To check that this is possible, we just need to check that the corresponding obstruction space vanishes, that is, that .

7.3 the normal space $\mathbb {P} N_C|_p$

One tool that we will use – both to show in the next section that , and in the following section to analyze the transformation – is the natural identification of $\mathbb {P} N_C|_p$ with the projection of $\mathbb {P}^r$ from $T_p C$ . Under this projection, $\overline {q}_1, \overline {q}_2, \ldots , \overline {q}_{r - 1}$ are a collection of $r - 1$ points, which are general subject to the condition of being contained in a hyperplane H. The conics $Q_j$ project to the lines through $\overline {q}_{n + 2j - 1}$ and $\overline {q}_{n + 2j}$ . The image $\overline {p}$ of p is identified with the osculating $2$ -plane, which coincides with $\mathbb {P} N_{C\to p}|_p \in \mathbb {P} N_C|_p$ . Under this identification, the points $a_i$ are identified with $\overline {q}_i$ , and the points $b_j$ lie on $\overline {Q}_j$ . The following diagram illustrates this setup:

7.4 Vanishing of

Given a vector bundle E on a reducible curve $X \cup _{\Gamma } Y$ , recall that the Mayer–Vietoris sequence is

$$\begin{align*}0 \to E \to E|_X \oplus E|_Y \to E|_{\Gamma} \to 0.\end{align*}$$

Applying this to , we obtain

(7.1)

For each of the direct summands in the middle term, we both show that $H^1$ vanishes, and extract information about the image of its global sections in $T_{\mathbb {P} N_C}|_{\Gamma \cup \Gamma '}$ . We then combine this information to show that the rightmost map is surjective on global sections.

Proof. Both statements follow from

, which can in turn be deduced from

using the isomorphisms

$$\begin{align*}N_{\ell_k / \mathbb{P}(N_C \oplus \mathcal{O}_C)|_{q_k}} \simeq \mathcal{O}_{\ell_k}(1)^{r-1} \quad \text{and} \quad N_{\mathbb{P}(N_C \oplus \mathcal{O}_C)|_{q_k} / \mathbb{P}(N_C \oplus \mathcal{O}_C)} \simeq \mathcal{O}_{\mathbb{P}(N_C \oplus \mathcal{O}_C)|_{q_k}}.\\[-41pt] \end{align*}$$

Proof. The map $\pi _p \circ f \colon T \to T$ is the identity map, so the pointing bundle exact sequence yields a surjection

which we may further compose with the surjection $N_T \to N_{\Lambda / \mathbb {P} N_C}|_T$ coming from the inclusion $T \subset \Lambda $ . Define K via the exact sequence

By considering the diagram

and noting that $H^0(N_\Lambda ) \simeq H^0(N_\Lambda |_T)$ , it suffices to show that

$$\begin{align*}H^1(K(-\Gamma)) = H^1(N_{\Lambda / \mathbb{P} N_C}|_T) = 0.\end{align*}$$

The first vanishing statement follows from the pointing bundle sequence

$$\begin{align*}0 \to N_{f \to C}(-\Delta) \simeq \mathcal{O}_T(\Delta + \Gamma) \to K \to N_{T/\Lambda} \simeq \mathcal{O}_T(\Gamma)^{\oplus (m - 3)} \to 0.\end{align*}$$

The second vanishing statement follows from the sequence:

$$\begin{align*}0 \to N_{\Lambda / \mathbb{P} N_C|_p}|_T \simeq \mathcal{O}_T(1)^{r - m} \to N_{\Lambda / \mathbb{P} N_C}|_T \to N_{\mathbb{P} N_C|_p / \mathbb{P} N_C}|_T \simeq \mathcal{O}_T \to 0.\\[-42pt] \end{align*}$$

We finally consider the bundle

$$\begin{align*}N_{\hat{R}^\circ / \operatorname{Bl}_C \mathbb{P}^r} \simeq \bigoplus_{i = 1}^n N_{\hat{L}_i / \operatorname{Bl}_C \mathbb{P}^r} \oplus \bigoplus_{j = 1}^{n'} N_{\hat{Q}_j / \operatorname{Bl}_C \mathbb{P}^r}.\end{align*}$$

To describe the images of

$$\begin{align*}H^0(N_{\hat{L}_i / \operatorname{Bl}_C \mathbb{P}^r}) \to T_{\mathbb{P} N_C}|_{a_i} \quad \text{and} \quad H^0(N_{\hat{Q}_j / \operatorname{Bl}_C \mathbb{P}^r}) \to T_{\mathbb{P} N_C}|_{b_j},\end{align*}$$

we define

$$\begin{align*}A_i = T_{q_i} C \quad \text{and} \quad B_j = \langle T_{q_{n+2j - 1}} C, T_{q_{n+2j}} C\rangle,\end{align*}$$

for the tangent line or span of tangent lines, and $\overline {A}_i$ and $\overline {B}_j$ for their projections from $T_p C$ .

Proof. We have an exact sequence

Using this, the kernel in equation (2) is the image of

in $T_{\mathbb {P} N_C}|_{a_i}$ . Moreover, the normal bundle exact sequence for $L_i$ in the span $\overline {L_i A_i}$ gives

and $N_{L_i / \overline {L_i A_i}}(-p)|_p$ is identified with $T_{a_i}\overline {A}_i$ under projection from $T_pC$ .

The same diagram chase as in Lemma 7.2 implies that it suffices to show:

$$\begin{align*}H^1(N_{L_i / \overline{L_i A_i}}(-2p)) = H^0(N_{\overline{L_i A_i} / \mathbb{P}^r}|_{L_i} (- q_i-p)) = H^1(N_{\overline{L_i A_i} / \mathbb{P}^r}|_{L_i} (- q_i-p)) = 0.\end{align*}$$

These statements follow from the isomorphisms

$$\begin{align*}N_{L_i / \overline{L_i A_i}}(-2p) \simeq \mathcal{O}_{L_i}(-1) \quad \text{and} \quad N_{\overline{L_i A_i} / \mathbb{P}^r}|_{L_i} (-p - q_i) \simeq \mathcal{O}_{L_i}(-1)^{r - 2}. \end{align*}$$

Proof. We will imitate the proof of Lemma 7.3. We have an exact sequence

Moreover, the normal bundle exact sequence for $Q_j$ in the span $\overline {Q_j B_j}$ gives

and

is identified with $T_{b_j}\overline {B}_j$ under projection from $T_pC$ . Our goal is therefore to show both

and

$$\begin{align*}H^0(N_{\overline{Q_j B_j} / \mathbb{P}^r}|_{Q_j} (- q_{n+2j-1} - q_{n+2j}-p)) = H^1(N_{\overline{Q_j B_j} / \mathbb{P}^r}|_{Q_j} (- q_{n+2j-1} - q_{n+2j}-p)) = 0.\end{align*}$$

The first vanishing statement follows from the exact sequence:

The second vanishing statement follows from the isomorphism

$$\begin{align*}N_{\overline{Q_j B_j} / \mathbb{P}^r}|_{Q_j} (- q_{n+2j-1} - q_{n+2j}-p) \simeq O_{\mathbb{P}^1}(-1)^{\oplus (r - 4)}. \end{align*}$$

Combining these lemmas, we immediately see that $H^1$ of the middle terms in the sequence (7.1) vanish. We now see that the rightmost map of equation (7.1) is surjective on global sections, as follows. First, we apply Lemma 7.1 to handle the points of $\Gamma '$ ; this reduces our problem to showing the surjectivity of

Applying Lemmas 7.3 and 7.4, we see that the composition to $(T_C|_p)^m$ is surjective. It thus suffices to show that the image contains the kernel of $T_{\mathbb {P} N_C}|_\Gamma \to (T_C|_p)^m$ , that is, $T_{\mathbb {P} N_C|_p}|_\Gamma $ .

Since removing any point from $\Gamma $ yields a linearly independent collection of points, any deformation in $\mathbb {P} N_C|_p$ of all but one point of $\Gamma $ lifts to a deformation of $\Lambda $ . Combining Lemma 7.2 with Lemma 7.3, we therefore conclude that the image contains each $T_{\langle \Lambda , \overline {A}_i\rangle }|_\Gamma $ . Similarly, combining Lemma 7.2 with Lemma 7.4, we conclude that the image contains each $T_{\langle \Lambda , \overline {B}_j\rangle }|_\Gamma $ . Since the $T_{\langle \Lambda , \overline {A}_i\rangle }|_\Gamma $ and $T_{\langle \Lambda , \overline {B}_j\rangle }|_\Gamma $ span $T_{\mathbb {P} N_C|_p}|_\Gamma $ , the desired result follows.

7.5 The transformation

We next show that $ M$ is ‘suitably generic’ in $\mathbb {P} N_C|_p$ .

Lemma 7.5(2) is sharp, in the sense that the conclusion is always false if $n = 2$ (the subspace M is never transverse to $\Lambda = \overline {q_3 q_4 \cdots q_{r - 1}}$ ). Nevertheless, there is a variant that does hold for $n = 2$ . By part (1), the general such M is disjoint from $\overline {p} = \mathbb {P} N_{C \to p}|_p \in \mathbb {P} N_C|_p$ . We may therefore ask for the weaker conclusion that the image of M is linearly general in the quotient $\mathbb {P} (N_C / N_{C \to p})|_p$ , that is, that M is transverse to any $\Lambda $ containing $\overline {p}$ . In this case, the analog of Lemma 7.5(2) holds apart from a single counterexample.

Proof. By assumption, $r-1\equiv n=2$ mod $2$ ; hence, r is odd. If C is not an elliptic curve, then since $d \geq r + 1$ , either $d \geq r + 2$ or $(d, g) = (r + 1, 0)$ . We consider these two cases separately.

Case 1: ${\boldsymbol{d}} \geq {\boldsymbol{r}} + \mathbf {2}$ . Let $\Lambda \subset \mathbb {P} N_C|_p$ be any codimension $2$ plane containing $\overline {p}$ . We will show that M can be chosen disjoint from $\Lambda $ . Since any $(r - 1 - n) + 1 = r - n=r-2$ points lie in a hyperplane, $\overline {q}_1$ is a general point on $\overline {C}$ and is therefore not contained in $\Lambda $ . Let $H \simeq \mathbb {P}^{r-3}$ be a general hyperplane containing $\overline {q}_3, \overline {q}_4, \ldots , \overline {q}_{r - 1}$ . Since $\overline {p} \notin H$ and $\overline {p} \in \Lambda $ , it follows that $\Lambda $ is transverse to H. As $d \geq r + 2$ , the hyperplane section $H \cap \overline {C}$ contains two points $\{x, y\}$ distinct from $\overline {q}_1, \overline {q}_3, \overline {q}_4, \ldots , \overline {q}_{r - 1}$ . Since the sectional monodromy group of a general curve always contains the alternating group [Reference Kadets20], the points $\{x, y, \overline {q}_1, \overline {q}_3, \overline {q}_4, \ldots , \overline {q}_{r - 1}\}$ are in linear general position. For any $k \geq 0$ , there is a unique k-plane in $\mathbb {P}^{2k+2}$ meeting each of $k+2$ lines in linear general position. Applying this with $k = (r-7)/2$ , we see that there is a unique $[(r - 3)/2]$ -plane $\Lambda _x\subset H$ containing $\overline {q}_1$ and x, and meeting each of the lines $\overline {Q}_j$ . If $\overline {q}_2 = x$ , then by this uniqueness, $\Lambda _x$ coincides with the projection of $T_p R^\circ $ . Similarly define $\Lambda _y$ . Because M can be linearly general in either $\Lambda _x$ or $\Lambda _y$ , it suffices to show that one of $\Lambda _x$ or $\Lambda _y$ contains a line disjoint from $\Lambda $ .

Note that $\Lambda _x \cap \overline {Q}_1$ is the projection of x from $\langle \overline {q}_1, \overline {Q}_2, \ldots , \overline {Q}_{n'}\rangle $ onto $\overline {Q}_1$ , and similarly for $\Lambda _y \cap \overline {Q}_1$ . It follows that $\Lambda _x \cap \overline {Q}_1 \neq \Lambda _y \cap \overline {Q}_1$ , and thus that $\langle \Lambda _x, \Lambda _y \rangle $ contains $\overline {Q}_1$ . Similarly, $\langle \Lambda _x, \Lambda _y \rangle $ contains all other $\overline {Q}_i$ . By inspection, $\langle \Lambda _x, \Lambda _y \rangle $ contains $\overline {q}_1$ , x, and y. Therefore, $\langle \Lambda _x, \Lambda _y \rangle = H$ . In particular, $\langle \Lambda _x, \Lambda _y \rangle $ is a distinct hyperplane from $\langle \Lambda , \overline {q}_1\rangle $ . Without loss of generality, $\Lambda _x$ contains a point $z \notin \langle \Lambda , \overline {q}_1\rangle $ . Then $\langle z, \overline {q}_1 \rangle $ gives the desired line contained in $\Lambda _x$ and disjoint from $\Lambda $ .

Case 2: $({\boldsymbol{d}}, {\boldsymbol{g}}) = ({\boldsymbol{r}} + \mathbf {1}, \mathbf {0})$ . Since $\overline {f} \colon \mathbb {P}^1 \simeq C \to \mathbb {P}^{r - 2}$ is a general rational curve of degree $r - 1$ , it suffices to verify that M is linearly general for a particular choice of $\overline {f}$ . We may therefore take

$$\begin{align*}\overline{f}(t) = \left[t^2 + 1 : t : \frac{t - p_3}{t - q_3} : \frac{t - p_4}{t - q_4} : \cdots : \frac{t - p_{r - 1}}{t - q_{r - 1}}\right],\end{align*}$$

where $p_i \in \mathbb {P}^1$ are general. For $3 \leq i \leq r-1$ , we have $\overline {f}(q_i) = [0: \cdots : 0 : 1 : 0 : \cdots : 0]$ , where the $1$ occurs in the ith position. (The interested reader may verify that this is not actually a specialization, that is, the general rational curve of degree $r - 1$ in $\mathbb {P}^{r - 2}$ is of this form after applying automorphisms of the source and target.)

Let $H = H_s$ be a generic hyperplane passing through $\overline {q}_3, \overline {q}_4, \ldots , \overline {q}_{r - 1}$ , defined by the ratio of the first two coordinates being equal to s. Note that H meets $\overline {f}(\mathbb {P}^1)$ at two other points $\overline {q}_1 = \overline {f}(q_1)$ and $\overline {q}_2 = \overline {f}(q_2)$ . The parameters $q_1$ and $q_2$ are the solutions of the equation $t + t^{-1} = (t^2 + 1)/t = s$ . The projection $\Lambda _s$ of $T_p R^\circ $ is the unique $[(r - 3)/2]$ -plane $\Lambda _s$ containing $\overline {q}_1$ and $\overline {q}_2$ and meeting each of the lines $\overline {Q}_i$ . We will show that, for $s \in \mathbb {P}^1$ generic, $\Lambda _s$ is transverse to any fixed subspace $\Lambda $ of codimension $2$ containing $\overline {p}$ . Hence, a general line $M \subseteq \Lambda _s$ is disjoint from $\Lambda $ .

To show this, we calculate $\Lambda _s$ explicitly. Since $\Lambda _s$ is unique, it suffices to exhibit a particular $[(r - 3)/2]$ -plane containing $\overline {q}_1$ and $\overline {q}_2$ and meeting each of the lines $\overline {Q}_i$ . We claim that we may take:

$$\begin{align*}\Lambda_s = \langle \alpha(s), \beta_1(s), \beta_2(s), \ldots, \beta_{(r - 3)/2}(s)\rangle,\end{align*}$$

where

$$ \begin{align*} \alpha(s) &= \bigg[s : 1 : \frac{p_3 q_4 s - p_3 - q_4}{q_3 q_4 - 1} : \frac{p_4 q_3 s - p_4 - q_3}{q_3 q_4 - 1} : \\ &\qquad\qquad\qquad\qquad \cdots : \frac{p_{r-2} q_{r-1} s - p_{r-2} - q_{r-1}}{q_{r-2} q_{r-1} - 1} : \frac{p_{r-1} q_{r-2} s - p_{r-1} - q_{r-2}}{q_{r-2} q_{r-1} - 1} \bigg] \\ \beta_i(s) &= \left[0 : 0 : \cdots : 0 : 0 : \frac{p_{2i+1}}{q_{2i + 1}} \cdot \frac{s - q_{2i+1} - p_{2i + 1}^{-1}}{s - q_{2i+1} - q_{2i+1}^{-1}} : \frac{p_{2i+2}}{q_{2i + 2}} \cdot \frac{s - q_{2i+2} - p_{2i + 2}^{-1}}{s - q_{2i+2} - q_{2i+2}^{-1}} : 0 : 0 : \cdots : 0 : 0\right]. \end{align*} $$

Here, the nonzero entries of $\beta _i(s)$ occur in the $(2i + 1)$ st and $(2i + 2)$ nd coordinates. Indeed, $\Lambda _s$ meets $\overline {Q}_i$ at $\beta _i(s)$ , so it suffices to check that $\Lambda _s$ contains $\overline {f}(t)$ when $s = t + t^{-1}$ . This follows from the following identity, which may be verified by separately considering the first coordinate, the second coordinate, the $(2i + 1)$ st coordinate, and the $(2i + 2)$ nd coordinate:

$$\begin{align*}\overline{f}(t) = t \cdot \alpha(t + t^{-1}) - \sum_i \frac{(q_{2i + 1} t - 1)(q_{2i + 2} t - 1)}{q_{2i + 1} q_{2i + 2} - 1} \cdot \beta_i(t + t^{-1}).\end{align*}$$

This establishes that $\Lambda _s$ is given by the above explicit formula, as claimed.

From the above explicit formulas for $\alpha $ and the $\beta _i$ , it is evident that $\alpha $ is an isomorphism from $\mathbb {P}^1$ onto a line L, and the $\beta _i$ are quadratic maps from $\mathbb {P}^1$ onto lines $M_i$ such that $L, M_1, M_2, \ldots , M_{(r - 3)/2}$ are linearly independent and span $\mathbb {P}^{r - 2}$ . In fact, the above formulas for the $\beta _i$ imply that, up to changing coordinates on the $M_i$ , the $\beta _i$ are independently general quadratic maps – so in particular distinct (from themselves and from $\alpha $ ). Since the image of $\overline {f}$ does not lie in any union of proper linear subspaces, and $\Lambda $ must meet $\overline {p}$ (which is a general point on the image of $\overline {f}$ ), all that remains is to prove Lemma 7.7 below.

8.1 Outline of inductive arguments

In order to indicate the specializations and projections of the original BN-curve C, we introduce the following notation. Write $C(0,0;0) = C$ for our original general BN-curve of degree d and genus g in $\mathbb {P}^r$ . More generally, the notation $C(a, b; c)$ will denote a curve obtained from $C(0,0;0)$ by peeling off a one-secant lines (as described in equation (1) below), peeling off b two-secant lines (as described in (2) below), and projecting from c general points on the curve (as described in equation (4) below). In particular, $C(a,b;c)$ is a BN-curve of degree $d-a-b-c$ and genus $g-b$ in $\mathbb {P}^{r-c}$ . The inductive arguments we will give will make use of the following six key ingredients:

1. 1. (cf. Section 5.1) We peel off a one-secant line, that is, we degenerate $C(a, b; c)$ to $C(a + 1, b; c) \cup L$ , where L is a one-secant line to $C(a + 1, b; c)$ , meeting $C(a + 1, b; c)$ at a point we will call x. In this case, we write y for some point in $L {\smallsetminus} \{x\}$ . We always do this specialization so that all marked points determining the modification data specialize onto $C(a + 1, b; c) {\smallsetminus} \{x\}$ .

2. 2. (cf. Section 5.2) We peel off a one-secant line, that is, we degenerate $C(a, b; c)$ to $C(a, b + 1; c) \cup L$ , where L is one-secant to $C(a, b + 1; c)$ , meeting $C(a, b + 1; c)$ at points we will denote $\{z, w\}$ . We always do this specialization so that all marked points determining the modification data specialize onto $C(a, b + 1; c) {\smallsetminus} \{z, w\}$ .

3. 3. We specialize the modification data. For the modifications , we use the technology developed in Section 7. For the remaining modifications, we specialize the marked points determining the modification data (which start out general).

4. 4. We project from a point $p \in C(a, b; c)$ . Namely, if we write $C(a, b; c + 1)$ for the projection of $C(a, b; c)$ from p, then the pointing bundle exact sequence induces (cf. equation (3.3)) an exact sequence

   $$\begin{align*}\qquad 0 \to N_{C(a, b; c) \to p}(\text{mods to}\ p) \to N_{C(a, b; c)}[\text{mods}] \to N_{C(a, b; c + 1)}[\text{residual mods}](p) \to 0.\end{align*}$$
   If the number n of modifications towards p satisfies $|n - \delta | < 1$ , then by Corollary 3.16 interpolation for $N_{C(a, b; c)}[\text {mods}]$ follows from interpolation for $N_{C(a, b; c + 1)}[\text {residual mods}]$ . More generally, if n satisfies $|n - \delta | \leq 1 - \frac {\epsilon }{r - 1}$ , then we may iterate this construction (i.e., first specialize as desired and then project) a total of $\epsilon $ times.

5. 5. We erase modifications that are linearly general. Namely, suppose that one of our modifications is linearly general. Then interpolation for follows from interpolation for N by Lemma 3.11. More generally, if M is not linearly general, but contains some subspace $M_0$ and is linearly general in the quotient $N|_p / M_0$ , then interpolation for follows from interpolation for N and by Lemma 3.11.

6. 6. We specialize any remaining $R_i$ to pass through the center of projection. In more detail, suppose that we projected from a point p, and that prior to this step, $R_i$ remains general; write $\overline {R}_i$ for the projection of $R_i$ from p. Specializing $R_i$ to pass through p then induces the specialization of $\overline {R}_i$ to a union $R^{\prime }_i \cup L$ , where $R^{\prime }_i$ is an r-secant rational normal curve in a hyperplane (the projection from p of the hyperplane containing $R_i$ ), and L is a line passing through p and a point $t \in R^{\prime }_i$ . This has the effect of replacing the modification with the modifications . By Lemma 8.1 below, $R^{\prime }_i$ is a general r-secant rational normal curve in a hyperplane, and $t \in R_i'$ is a general point. The modification is therefore in a linearly general direction, and can be erased as above. In other words, at least when no other modifications are made at p, the combined effect of these steps is to replace with (which fits well with our inductive hypothesis).

8.2 Main inductive arguments

We begin with the following proposition, which applies this method without utilizing specialization (2) (peeling off a one-secant line) and which specializes the $R_i$ as in Section 7. Since this is the first application of the method described above, we include some additional explanations which serve to clarify this method, and will be omitted in subsequent applications.