Proof. Choose a sufficiently ample line bundle L on C. For $q_1\neq q_2\in \nu ^{-1}(p)$ , consider the subspace

$$\begin{align*}V = \nu^* \mathrm{H}^0(L) + \mathrm{H}^0(\nu^* L \otimes {\mathcal O}_{C^\nu}(-q_1-q_2)) \subset \mathrm{H}^0(\nu^* L). \end{align*}$$

Then $s_1(q_1) = s_2(q_2)$ for all $s_1,s_2\in V$ . Let $f: C^\nu \to G \subset {\mathbb {P}} V^*$ be the morphism given by the linear series V. Clearly, $\nu $ factors through f. For L sufficiently ample, G has a node $q = f(q_1) = f(q_2)$ over p as the only singularity.

Proof. The first claim is an application of the usual Arbarello–Cornalba lemma in the case of K3 surfaces (see, e.g., [Reference Dedieu and SernesiDS17]), whereas the second and third follow essentially from the first (see [Reference Chen, Gounelas and LiedtkeCGL19, §2] and the proof of [Reference Chen, Gounelas and LiedtkeCGL19, Lemma 6.3]).

Proof. As the singularity at p is locally reducible, from Lemma 2.2 we may take $f:\widetilde {C}\to X$ to be a partial normalisation of C which has one node over the point p and is smooth otherwise. In particular, $[f]\in \overline {\mathscr {M}}_{g+1}(X, {\mathcal O}(C))$ . Let M be an irreducible component of $\overline {\mathscr {M}}_{g+1}(X,{\mathcal O}(C))$ containing $[f]$ . From [Reference Chen, Gounelas and LiedtkeCGL19, Theorem 2.11], $\dim M\geq g+1$ . Consider now the moduli map

Let $D_M$ be an irreducible component of $M\cap \phi ^{-1}(\partial \overline {\mathscr {M}}_{g+1})$ containing $[f]$ , where $\partial \overline {\mathscr {M}}_{g+1} = \overline {\mathscr {M}}_{g+1} - \mathscr {M}_{g+1}$ is the boundary divisor of $\overline {\mathscr {M}}_{g+1}$ .

For a general point, $[h]\in D_M$ , $h: \Gamma \to X$ is a stable map such that $\Gamma $ is an integral curve of geometric genus g with a node and $h(\Gamma )$ and C lie on the same component of $V_{C,g}$ . Since C deforms in the expected dimension, $\dim D_M \le g$ , and hence, $D_M \subsetneq M$ . On the other hand, since $\partial \overline {\mathscr {M}}_{g+1}$ is a ${\mathbb {Q}}$ -Cartier divisor, $D_M$ has codimension one in M. We must have

$$\begin{align*}g + 1\le \dim M = \dim D_M + 1 \le g + 1, \end{align*}$$

and hence, $\dim M = g+1$ . This proves that for a general point $h:\Gamma \to X$ of M, $D = h(\Gamma )$ is an integral curve of geometric genus $g+1$ that deforms in the expected dimension.

Since C deforms with maximal moduli, there exists an irreducible component D of $ \phi ^{-1}(\partial \overline {\mathscr {M}}_{g+1})$ containing $[f]$ such that $\dim \phi (D) = g$ . Let $M'$ be an irreducible component of $\overline {\mathscr {M}}_{g+1}(X,{\mathcal O}(C))$ containing D. Since $\phi (M')$ is not contained in $\partial \overline {\mathscr {M}}_{g+1}$ , we conclude

$$\begin{align*}g + 1 = \dim M' \ge \dim \phi(M') \ge \dim \phi(D) +1 = g+1, \end{align*}$$

and hence, $\dim \phi (M') = g+1$ . Therefore, for a general point $h:\Gamma \to X$ of $M'$ , $D = h(\Gamma )$ is an integral curve of geometric genus $g+1$ that deforms with maximal moduli.

Proof. The result follows by induction, Proposition 2.7 and the fact that a general deformation of a nodal curve will be nodal and as such has unramified normalisation morphism, hence deforms in the expected dimension from [Reference Chen, Gounelas and LiedtkeCGL19, Proposition 2.9].

Proof. Let $\widehat {D} = \sum _{i=1}^n \Gamma _i$ , where $\Gamma _i$ are the irreducible components of $\widehat {D}$ . From the exact sequences

for $m=1,\ldots ,n$ , we obtain

$$\begin{align*}\begin{aligned} \mathop{\mathrm{ch}}\nolimits(\Omega^1_{\widehat{X}}(\log \widehat{D})) &= \mathop{\mathrm{ch}}\nolimits(\Omega^1_{\widehat{X}}) + \sum_{m=1}^n \mathop{\mathrm{ch}}\nolimits({\mathcal O}_{\Gamma_m})\\ &= \mathop{\mathrm{ch}}\nolimits(\Omega^1_{\widehat{X}}) + \sum_{m=1}^n (\mathop{\mathrm{ch}}\nolimits({\mathcal O}_{\widehat{X}})-\mathop{\mathrm{ch}}\nolimits({\mathcal O}_{\widehat{X}}(-\Gamma_m)))\\ &= K_{\widehat{X}} + \widehat{D} + \frac{1}2(K_{\widehat{X}}^2 - 2c_2(\widehat{X}) - \sum_{m=1}^n \Gamma_m^2), \end{aligned} \end{align*}$$

where $\mathop {\mathrm {ch}}\nolimits (\bullet )$ is the Chern character. It follows that

$$\begin{align*}\begin{aligned} c_2(\Omega^1_{\widehat{X}}(\log \widehat{D})) &= c_2(\widehat{X}) + \frac{1}2(K_{\widehat{X}} + \widehat{D})^2 - \frac{1}2 K_{\widehat{X}}^2 + \frac{1}2\sum_{m=1}^n \Gamma_m^2\\ &= c_2(\widehat{X}) + (K_{\widehat{X}} + \widehat{D}) \widehat{D} - \sum_{1\le i<j\le n} \Gamma_i \Gamma_j. \end{aligned} \end{align*}$$

Note that further blowing up $\widehat {X}$ at a singularity of $\widehat {D}$ does not change $c_2(\Omega ^1_{\widehat {X}}(\log \widehat {D}))$ . The minimal log resolution of $(X,D)$ does not blow up all singularities of D in case that D is reducible: If D has an ordinary double point at p where two components of D meet transversely, we do not need to blow up X at p. On the other hand, we can choose to blow up X at such p since it does not change $c_2(\Omega ^1_{\widehat {X}}(\log \widehat {D}))$ . This has the advantage of streamlining our argument. Hence, we choose a log resolution $(\widehat {X}, \widehat {D})$ of $(X,D)$ which is minimal with the properties that $\widehat {D}$ has simple normal crossings and the proper transforms of the components of D are disjoint from each other.

Let us write

$$\begin{align*}\widehat{D} = \sum_{i=1}^n \Gamma_i = \Delta + \sum_{p\in D_s} E_p, \end{align*}$$

where $\Delta $ is the proper transform of D under $\pi : \widehat {X}\to X$ and $E_p = \pi ^{-1}(p)$ for $p\in D_s$ , where $D_s$ is the set of singularities of D. Clearly, $E_p$ is a tree of smooth rational curves for all $p\in D_s$ . Then the above equality takes the form

$$\begin{align*}\begin{aligned} c_2(\Omega^1_{\widehat{X}}(\log \widehat{D})) &= c_2(\widehat{X}) + (K_{\widehat{X}} + \Delta) \Delta + \sum_{p\in D_s} (K_{\widehat{X}} + E_p) E_p + \sum_{p\in D_s} \Delta E_p\\ &\quad - \sum_{p\in D_s} \sum_{\substack{1\le i<j\le n\\ \Gamma_i\cup \Gamma_j\subset E_p}} \Gamma_i \Gamma_j. \end{aligned} \end{align*}$$

Since $\Delta $ is the normalisation of D,

$$\begin{align*}\begin{aligned} (K_{\widehat{X}} + \Delta) \Delta &= 2p_a(\Delta) - 2 = 2p_a(D) - 2 - 2 \sum_{p\in D} \delta_p\\ &= (K_X + D) D - 2 \sum_{p\in D} \delta_p. \end{aligned} \end{align*}$$

For every $p\in D_s$ , $p_a(E_p) = 0$ , and hence,

$$\begin{align*}\sum_{p\in D} (K_{\widehat{X}} + E_p) E_p = -2 \sum_{p\in D_s} 1. \end{align*}$$

It is also clear that $\Delta E_p$ equals the number of local branches of D at $p\in D_s$ . Therefore,

$$\begin{align*}\sum_{p\in D_s} \Delta E_p = \sum_{p\in D_s} \gamma_p. \end{align*}$$

Since $E_p$ is a tree of smooth rational curves,

$$\begin{align*}\sum_{\substack{1\le i<j\le n\\ \Gamma_i\cup \Gamma_j\subset E_p}} \Gamma_i \Gamma_j = |E_p| - 1 \end{align*}$$

for $p\in D_s$ , where $|E_p|$ is the number of irreducible components of $E_p$ . Finally,

$$\begin{align*}c_2(\widehat{X}) = c_2(X) + \sum_{p\in D_s} |E_p|. \end{align*}$$

Combining all the above, we obtain equation (3.3).

Proof. As in the proof of Lemma 3.2, further blowing up $\widehat {X}$ at a singularity of $\widehat {D}$ does not change $(K_{\widehat {X}} + \widehat {D})^2$ . So we choose a log resolution $(\widehat {X}, \widehat {D})$ of $(X,D)$ which is minimal with the properties that $\widehat {D}$ has simple normal crossings and the proper transforms of the components of D are disjoint from each other.

The proof of Lemma 3.2 already gives

(3.5) $$ \begin{align} (K_{\widehat{X}} + \widehat{D}) \widehat{D} = (K_X + D) D - \sum_{p\in D_s} \mu_p + \sum_{p\in D_s} (\gamma_p - 1). \end{align} $$

From now on, we denote $K_{\widehat {X}/X}=K_{\widehat {X}}-\pi ^*K_X$ . Then, equation (3.5) and the fact that $K_{\widehat {X}/X}.\pi ^*F=0$ for any divisor F on X yield

$$\begin{align*}\begin{aligned} (K_{\widehat{X}} + \widehat{D})^2 - (K_X + D)^2 &= - \sum_{p\in D_s} \mu_p + \sum_{p\in D_s} (\gamma_p - 1) + (K_{\widehat{X}}^2 - K_X^2)\\ &\quad + \sum_{p\in D_s} K_{\widehat{X}} E_p + (K_{\widehat{X}} \Delta - K_X D)\\ &= - \sum_{p\in D_s} \mu_p + \sum_{p\in D_s} (\gamma_p - 1) + K_{\widehat{X}/X}^2\\ &\quad + \sum_{p\in D_s} K_{\widehat{X}} E_p + K_{\widehat{X}} (\Delta - \pi^* D). \end{aligned} \end{align*}$$

Thus, equation (3.4) holds as long as we can prove

(3.6) $$ \begin{align} (\gamma_p - 1) + K_{\widehat{X}} E_p + (K_{\widehat{X}/X}^2)_p + (K_{\widehat{X}} (\Delta - \pi^* D))_p \ge \frac{\mu_p}{m_p} \end{align} $$

for all $p\in D_s$ . The problem is local, so we work in a formal neighbourhood of a point $p\in D_s$ in X. For simplicity, we drop the subscript p in all notation so that $m = m_p$ , $\mu = \mu _p$ , $\gamma = \gamma _p$ and $E = E_p$ .

We can factor $\pi : \widehat {X}\to X$ into a sequence of blowups:

where each $\pi _{i,i-1}: X_i\to X_{i-1}$ is the blowup of $X_{i-1}$ at one point for $i=1,2,\ldots ,a$ . Let $\pi _{i,j} = \pi _{j+1,j} \circ \pi _{j+2,j+1} \circ \ldots \circ \pi _{i,i-1}$ be the birational map $X_i\to X_j$ for $0\le j < i\le a$ , and let $F_i$ be the exceptional divisor of $\pi _{i,i-1}: X_i\to X_{i-1}$ for $i=1,2,\ldots ,a$ . Then

$$\begin{align*}\begin{aligned} K_{\widehat{X}/X} &= \pi_{a,1}^* F_1 + \pi_{a,2}^* F_2 + \ldots + \pi_{a,a-1}^* F_{a-1} + F_a\\ \Delta &= \pi^* D - m_1 \pi_{a,1}^* F_1 - m_2 \pi_{a,2}^* F_2 - \ldots - m_{a-1}\pi_{a,a-1}^* F_{a-1} - m_a F_a \end{aligned} \end{align*}$$

for some $m_i\in {\mathbb {Z}}_+$ satisfying that

$$\begin{align*}m = m_1 = \max_{1\le i \le a} m_i. \end{align*}$$

It follows (see, e.g., [Reference de Jong and PfisterdJP00, Theorem 5.4.13]) that

$$\begin{align*}\begin{aligned} \mu + \gamma - 1 = 2\delta &= \sum_{i=1}^a m_i(m_i - 1)\\ K_{\widehat{X}/X}^2 + K_{\widehat{X}} (\Delta - \pi^* D) &= \sum_{i=1}^a (m_i - 1). \end{aligned} \end{align*}$$

Therefore, equation (3.6) holds provided that we can prove

(3.7) $$ \begin{align} (\gamma - 1) + K_{\widehat{X}} E\ge 0. \end{align} $$

We recall that $E = \pi ^{-1}(p)$ is a tree of smooth rational curves. Thus, from the adjunction formula,

$$\begin{align*}K_{\widehat{X}} E = (K_{\widehat{X}} + E) E - E^2 = -2 - E^2 \ge -1, \end{align*}$$

where $E^2\le -1$ because the components of E have negative definite intersection matrix. So we have equation (3.7) if $\gamma \ge 2$ . Otherwise, $\gamma = 1$ , i.e., D has a locally irreducible or unibranch singularity at p. We claim that $K_{\widehat {X}} E \ge 0$ in this case.

Let $E_i = \pi _{i,0}^{-1}(p)$ for $i=1,2,\ldots ,a$ . Then $E_1 = F_1$ and $K_{X_1} E_1 = -1$ . If $\pi _{i,i-1}: X_i \to X_{i-1}$ is the blowup of $X_{i-1}$ at a smooth point of $E_{i-1}$ , then

$$\begin{align*}E_i = \pi_{i,i-1}^* E_{i-1} \text{ and } K_{X_i} E_i = K_{X_{i-1}} E_{i-1}. \end{align*}$$

Otherwise, if $\pi _{i,i-1}: X_i \to X_{i-1}$ is the blowup of $X_{i-1}$ at a singular point of $E_{i-1}$ , then

$$\begin{align*}E_i = \pi_{i,i-1}^* E_{i-1} - F_i \text{ and } K_{X_i} E_i = K_{X_{i-1}} E_{i-1} + 1. \end{align*}$$

In conclusion, we have

$$\begin{align*}K_{X_1} E_1 = -1 \text{ and } K_{X_i} E_i = \begin{cases} K_{X_{i-1}} E_{i-1} & \text{if } \pi_{i,i-1}(F_i) \not\in (E_{i-1})_{\text{sing}}\\ K_{X_{i-1}} E_{i-1} + 1 & \text{if } \pi_{i,i-1}(F_i) \in (E_{i-1})_{\text{sing}} \end{cases} \end{align*}$$

for $2\le i\le a$ . Therefore, $K_{\widehat {X}} E = K_{X_a} E_a \ge 0$ as long as one of $\pi _{i,i-1}$ is the blowup of $X_{i-1}$ at a singular point of $E_{i-1}$ . For a locally irreducible singularity $p\in D_s$ , it is easy to see that $\pi _{a,a-1}: X_a \to X_{a-1}$ blows up $X_{a-1}$ at a singular point of $E_{a-1}$ . Consequently, $K_{\widehat {X}} E \ge 0$ when $\gamma = 1$ . This proves equation (3.7) and hence equation (3.6), giving equation (3.4).

Proof. Suppose that D only has locally irreducible singularities. Then

(3.9) $$ \begin{align} (K_X + D) D - \sum_{p\in D} \mu_p = (K_X + D) D - 2\sum_{p\in D} \delta_p = 2g - 2. \end{align} $$

By equation (3.8) and $\mathrm {c}_2(X)=24$ , we have

(3.10) $$ \begin{align} D^2 - \sum_{p\in D} \left(1 - \frac{1}{m_p}\right) \mu_p \le 66 + 6g. \end{align} $$

Combining equations (3.9) and (3.10), we have

(3.11) $$ \begin{align} \sum_{p\in D} \frac{\mu_p}{m_p}\le 68 + 4g. \end{align} $$

On the other hand,

(3.12) $$ \begin{align} \mu_p \ge m_p(m_p - 1) \end{align} $$

for all $p\in D$ . Putting equations 3.9–3.12 together gives

$$\begin{align*}\begin{aligned} 68 + 4g\ge \sum_{p\in D} \frac{\mu_p}{m_p} \ge \sum_{p\in D} \left( \sqrt{\mu_p + \frac{1}4} - \frac{1}2 \right) \ge \sqrt{D^2 + \frac{9}4 - 2g} - \frac{1}2, \end{aligned} \end{align*}$$

where we note that we have used that the function $f(x)=\sqrt {x+\frac 14}-\frac 12$ vanishes at 0 and has everywhere negative second derivative, hence is concave and $\sum f(x_i) \geq f(\sum x_i)$ for positive real $x_i$ . It follows that $D^2 \le 4690 + 550g + 16g^2$ . Therefore, D has at least one locally reducible singularity if $D^2> 4690 + 550g + 16g^2$ .

Proof. Suppose that $D_1$ and $D_2$ meet at a unique point q. Applying equation (3.8) to $(X, D = D_1 + D_2)$ , we have

(3.13) $$ \begin{align} D^2 - \sum_{p\in D} \left(1 - \frac{1}{m_{D,p}}\right)\mu_{D,p}\le 72 + 3(D^2 - \sum_{p\in D} \mu_{D,p}), \end{align} $$

where we use $\mu _{C,p}$ and $m_{C,p}$ to denote the pseudo-Milnor number and multiplicity of a reduced curve C at p, respectively.

Note the following simple facts for $i=1,2$ , $p\in D$ and $D_1\cap D_2=\{q\}$ as above

(3.14) $$ \begin{align} \begin{aligned} \mu_{D,p} &= \mu_{D_1,p} + \mu_{D_2,p} + 2 (D_1.D_2)_p - 1 \\ &= \mu_{D_1,p} + \mu_{D_2,p} -1 \text{ if }p\neq q\\ \mu_{D,q} &= \mu_{D_1,q} + \mu_{D_2,q} + 2D_1D_2 - 1\\ m_{D,p} &= m_{D_1,p} + m_{D_2,p} \le M:= \sqrt{D_1^2 + \frac{9}4} + \sqrt{D_2^2 + \frac94} + 1. \end{aligned} \end{align} $$

Combining equations (3.13) and (3.14), we obtain

$$\begin{align*}\begin{aligned} 75 - 3\sum_{i=1}^2 \sum_{p\in D_i} \mu_{D_i,p} =&\ 72 + 3 (D^2 - \sum_{p\in D} \mu_p) - 3(D_1^2 + D_2^2)\\ \ge&\ D^2 - \sum_{p\in D} \left(1 - \frac{1}{m_{D,p}}\right)\mu_{D,p} - 3(D_1^2 + D_2^2)\\ \ge&\ 2(D_1D_2 - D_1^2 - D_2^2) - \sum_{p\in D} \left(1 - \frac{1}M\right)\mu_{D,p}\\ =&\ \frac{2}M D_1D_2 - 2(D_1^2 + D_2^2) + \frac{M-1}{M}\left(1-\sum_{i=1}^2 \sum_{p\in D_i} \mu_{D_i,p}\right). \end{aligned} \end{align*}$$

Hence,

$$\begin{align*}75 \ge \frac{2}M D_1D_2 + \frac{M-1}{M} - 2(D_1^2 + D_2^2), \end{align*}$$

and the proposition follows.

Proof of Theorem A

Let us first prove it for $g = 1$ .

By [Reference Chen, Gounelas and LiedtkeCGL19, Theorem A], there are infinitely many integral rational curves $C_n$ on X. Suppose that $C_n^2$ is unbounded. Then $C_n$ has a locally reducible singularity by Proposition 3.4 for $C_n^2$ sufficiently large. Such $C_n$ can be deformed to a nonisotrivial family of curves of geometric genus 1 by Proposition 2.7.

Suppose that $C_n^2\le c$ for all n. We claim that

(3.15) $$ \begin{align} \varlimsup_{\min(m,n)\to \infty} C_m C_n = \infty. \end{align} $$

Fixing $N\in {\mathbb {Z}}_+$ , since $\mathop {\mathrm {rank}}\nolimits _{\mathbb {Z}}\mathrm {Pic\ }(X)\le 20$ , $C_N,C_{N+1},\ldots ,C_{N+20}$ are linearly dependent in $\mathrm {Pic\ }(X)_{\mathbb {Q}}$ . Suppose that

(3.16) $$ \begin{align} a_0 C_N + a_1 C_{N+1} + \ldots + a_{20} C_{N+20} = 0 \end{align} $$

in $\mathrm {Pic\ }(X)$ for some integers $a_i$ , not all zero. Since $C_i$ are effective, $a_i$ cannot be all positive or negative. Let us rewrite equation (3.16) as

$$\begin{align*}F = \sum_{a_i>0} a_i C_{N+i} = - \sum_{a_j < 0} a_j C_{N+j}. \end{align*}$$

Since $C_N, C_{N+1}, \ldots , C_{N+20}$ are distinct integral curves, it is easy to see that F is nef. This implies that there are only finitely many integral rational curves R such that $FR = 0$ , since if $F^2=0$ , then F can only be zero on the (up to 24) singular fibres of the elliptic fibration induced by F, and if $F^2>0$ , then from the Hodge index theorem the orthogonal space $F^\perp $ in the effective cone is negative definite and spanned by finitely many $-2$ -curves. Hence, there exists $m\ge N$ such that $FC_m \ge 1$ . Then $C_m + 2F$ is nef and big, and hence,

$$\begin{align*}\lim_{n\to\infty} (C_m+2F) C_n = \infty. \end{align*}$$

Thus, there exists $C\in \{C_N,C_{N+1},\ldots ,C_{N+20}, C_m\}$ such that $C C_n$ is unbounded. This proves equation (3.15).

By Proposition 3.5, $C_m$ and $C_n$ meet at (at least) two distinct points for $C_m C_n$ sufficiently large since $C_m^2\le c$ and $C_n^2 \le c$ . There are infinitely many such pairs $C_m$ and $C_n$ by equation (3.15), and

$$\begin{align*}\varlimsup_{\min(m,n)\to \infty} (C_m + C_n)^2 = \infty. \end{align*}$$

Such $C_m\cup C_n$ can be deformed to a nonisotrivial family of curves of geometric genus 1 by Proposition 2.9, which as pointed out above will have unbounded self-intersection. This proves the theorem for $g=1$ . The remaining cases follow from Propositions 2.7 and 3.4 by induction.

Hypothesis 4.2. There exists an unramified morphism $f:E\to X$ from a smooth genus 1 curve which deforms in the expected dimension and with maximal moduli.

Proof. We maintain the notation for $Y,L$ from the beginning of this section. Suppose for a contradiction that L is ${\mathbb {Q}}$ -effective. Let m be the smallest positive integer such that $mL$ is effective, and let $G\in |mL|$ . We write

$$\begin{align*}G = \sum b_i D_i, \end{align*}$$

where $D_i \in |a_i L + \pi ^* F_i|$ are the irreducible components of G for some $a_i \in {\mathbb {N}}$ and some divisors $F_i\in \mathrm {Pic\ }(X)$ , and $b_i\in {\mathbb {Z}}_+$ is the multiplicity of $D_i$ in G. Since $mL=\sum a_i b_i L + \sum b_i \pi ^*F_i$ , we obtain that

(4.1) $$ \begin{align} \sum b_i F_i=0 \text{ in } \mathrm{Pic\ }(X). \end{align} $$

Let $C\subset X$ be an integral curve of geometric genus 1 as given by Hypothesis 4.2. From the assumption, there exists an irreducible curve $B\subset |C|$ with C as member and such that every curve $\Gamma \in B$ is of geometric genus 1.

When $a_i = 0$ , $F_i$ is necessarily effective and $CF_i \ge 0$ . Note also that there exists at least one i such that $CF_i\le 0$ and $a_i> 0$ since otherwise, $CF_i> 0$ for all $a_i> 0$ and so $\sum CF_i> 0$ , contradicting equation (4.1).

From now on, we denote by $a=a_i$ , $D = D_i$ and $F = F_i$ so that $a_i> 0$ and $CF_i \le 0$ .

From the assumption, the general deformation of the normalisation of C is an immersion. We henceforth replace C by a general member of B and let $\nu : E = C^\nu \to X$ be its normalisation, i.e., we have that $\nu ^* \Omega ^1_X \to \Omega ^1_E$ is a surjection. As the kernel is torsion-free on a smooth curve, it is a line bundle, and by taking determinants we see that it must be isomorphic to $(\Omega ^1_E)^\vee \cong {\mathcal O}_E$ . This leads to the exact sequence

(4.2)

where ${\mathscr N}_\nu $ is the normal bundle of $\nu $ . From our assumption and the following lemma, the above sequence does not split.

Proof. If $f:\mathcal {C}\to B$ the family with B a smooth projective curve and E the generic fibre of f, then a section $\Omega ^1_E\to \nu ^*\Omega ^1_X$ also induces a splitting of

on some open subset $U\subset B$ . Dualising this sequence and pushing forward to U, we get a split sequence whose first coboundary map in cohomology is the Kodaira–Spencer map. Hence, this map is necessarily zero so the family over U is isotrivial.

Since $aL+F$ is effective, $\mathrm {H}^0(S^a \Omega ^1_X \otimes {\mathcal O}_X(F)) \ne 0$ and as C is a general member of a covering family of curves on X, we see that

$$\begin{align*}\mathrm{H}^0(E, S^a \nu^* \Omega^1_X \otimes {\mathcal O}_E(\nu^* F)) \ne 0 \end{align*}$$

as otherwise a global section of $S^a \Omega ^1_X \otimes {\mathcal O}_X(F)$ would vanish everywhere. By equation (4.2), $S^a \nu ^* \Omega ^1_X \otimes {\mathcal O}_E(\nu ^* F)$ has a filtration

$$\begin{align*}0\subsetneq E_1\subsetneq E_2\subsetneq \cdots \subsetneq E_{a+1}:=S^a \nu^* \Omega^1_X \otimes {\mathcal O}_E(\nu^* F) \end{align*}$$

with graded pieces all isomorphic to ${\mathcal O}_E(\nu ^* F)$ . If the global section ${\mathcal O}_E\to E_{a+1}$ from above vanishes when mapped to $E_{a+1}/E_a={\mathcal O}_E(\nu ^* F)$ , then it must induce a nonzero global section of $E_a$ . By induction, one of the quotients $E_i/E_{i-1}$ must have a nonzero global section, and hence, $\mathrm {H}^0({\mathcal O}_E(\nu ^* F)) \ne 0$ . On the other hand, $CF \le 0$ and $\deg \nu ^* F \le 0$ . So we necessarily have ${\mathcal O}_E(\nu ^* F) = {\mathcal O}_E$ .

This proves that for all i satisfying $a_i> 0$ and $C F_i \le 0$ we have ${\mathcal O}_E (\nu ^* F_i) = {\mathcal O}_E$ , and hence, $CF_i = 0$ . For the remaining i, we clearly have $CF_i \ge 0$ . Therefore, we conclude that $C F_i = 0$ for all i from equation (4.1). In summary, we have

• ○ if $a_i> 0$ , ${\mathcal O}_E (\nu ^* F_i) = {\mathcal O}_E$ ,

• ○ if $a_i = 0$ , $F_i$ is effective and $CF_i = 0$ .

As exact sequence (4.2) does not split,

(4.3) $$ \begin{align} h^0(E, S^n \nu^* \Omega^1_X) = 1 \end{align} $$

for all $n\in {\mathbb {Z}}_+$ .

Work now again with a fixed i so that $a_i>0$ as above, keeping the notation $D,F,a$ . Since D is reduced, $Y_p = \pi ^{-1}(p)$ meets D transversely for $p\in X$ general, and as C is a general member of a covering family of curves on X, also $Y_p = \pi ^{-1}(p)$ meets D transversely for $p\in C$ general. Let now $R = E\times _X Y \cong \mathrm {Proj}\:(S^\bullet (\nu ^* \Omega ^1_X))$ with diagram

Since $Y_p$ and D meet transversely for $p\in C$ general, $R_q$ and $\rho ^* D$ meet transversely for $q\in E$ general, where $R_q$ is the fibre of R over q.

Note that $\rho ^* D$ is a section of $a \rho ^* L$ . From equation (4.3), $h^0(R, n \rho ^* L) = 1$ for all $n\geq 0$ , and so we must have $\rho ^* D = a \Gamma $ , where $\Gamma $ is the unique section of $\rho ^* L$ . Then we must have $a = 1$ because $R_q$ and $\rho ^* D$ meet transversely for $q\in E$ general.

Hence, we have concluded that $a_i = 0$ or $1$ for all i. If there are two distinct components $D_i$ and $D_j$ of G such that $a_i = a_j = 1$ , then $\rho ^* D_i = \rho ^* D_j = \Gamma $ . Therefore,

$$\begin{align*}D_i \cap \pi^{-1}(C) = D_j\cap \pi^{-1}(C)\end{align*}$$

for $C\in B$ general, and hence, $D_i = D_j$ . Consequently, G has only one component $D_i$ with $a_i = 1$ , and so we have $\mathrm {H}^0(\Omega ^1_X \otimes {\mathcal O}_X(F)) \ne 0$ for some $F\in \mathrm {Pic\ }(X)$ such that $-F=\sum F_i$ is effective. As $\mathrm {H}^0(\Omega ^1_X \otimes {\mathcal O}_X(F))\subset \mathrm {H}^0(\Omega ^1_X)$ , we obtain a contradiction from the case $m=1$ , namely Theorem 4.1.

Question 4.6. Let $f:C\to X$ be a morphism from a smooth projective curve of genus $g\geq 1$ to a K3 surface over an algebraically closed field. If f deforms in the expected dimension, is a general deformation of f unramified?

Proof. Taking cohomology of the sequence

the Kodaira–Spencer map $\mathrm {H}^0(C,N_f)\to \mathrm {H}^1(C, T_C)$ must be injective, as it is the induced differential to the moduli map and C deforms with maximal moduli. This implies that $\mathrm {H}^0(C, f^*T_X)=0$ , but as C deforms to cover X, we obtain the result.

Question 5.2. Let X be a K3 surface over an algebraically closed field. Is it true that for any ample divisor $D\in \mathrm {Pic\ }(X)_{\mathbb {Q}}$ there exist integral curves $E_1,\ldots , E_n\subset X$ of geometric genus 1 so that $D=\sum _{i=1}^n a_i E_i$ for $a_i\in {\mathbb {Q}}_{\geq 0}$ ?

Proof. Suppose that there exists a locally free subsheaf $E\subset \Omega ^1_X$ of rank r such that $\mu (E) \ge \mu (\Omega ^1_X)$ . Then $L = \wedge ^r E$ is a subsheaf of $\Omega _X^r$ , and hence, $H^0(\Omega _X^r(-L)) \ne 0$ .

Let $\xi \in G$ and $f_m: C_m\to X$ be the sequence of morphisms associated to $\xi $ . Since $f_m(C_m)$ passes through a general point of X, we see that

$$ \begin{align*}H^0(C_m, f_m^* \Omega_X^r(-L)) \ne 0. \end{align*} $$

Then we have

$$ \begin{align*}h^0(M_{f_m}^r (-f_m^* L) ) + h^0(M_{f_m}^{r-1}(-f_m^* L) \otimes K_{C_m}) \ge h^0(f_m^* \Omega_X^r(-L))> 0 \end{align*} $$

by the left exact sequence

where $M_{f_m}^a = \wedge ^a M_{f_m}$ . On the other hand, we know that

$$ \begin{align*}H^0(V(-B)) = 0 \text{ if } \deg B> \mu_{max}(V) \end{align*} $$

for a vector bundle V and a divisor B on a smooth projective curve. It follows that

$$ \begin{align*}\begin{aligned} L. (f_m)_* C_m = \deg f_m^* L &\le \max\big(\mu_{max}(M_{f_m}^r), \mu_{\max}(M_{f_m}^{r-1}) + \deg K_{C_m}\big) \\ &\le \max\big(r \mu_{\max}(M_{f_m}), (r-1)\mu_{\max}(M_{f_m}) + \deg K_{C_m}\big)\\ &\le r \max(\mu_{\max}(M_{f_m}), \deg K_{C_m}). \end{aligned} \end{align*} $$

Therefore,

$$ \begin{align*}\left(\frac{L}r - \frac{K_X}{n}\right) \frac{(f_m)_* C_m}{\deg (f_m)_* C_m} \le \frac{n\max (\mu_{\max}(M_{f_m}), \deg K_{C_m}) - \deg f_m^* K_X}{n \deg (f_m)_* C_m}. \end{align*} $$

By our definition of G, we conclude that

$$ \begin{align*}\left(\frac{L}r - \frac{K_X}{n}\right) \xi \le 0 \end{align*} $$

for all $\xi \in G$ . On the other hand, since $\mu (E)\ge \mu (\Omega ^1_X)$ ,

$$ \begin{align*}\left(\frac{L}r - \frac{K_X}{n}\right) A^{n-1} \ge 0. \end{align*} $$

Fixing $\xi \in G$ , since $\mathrm {Amp}_1(X)_{\mathbb {R}}$ is open in $N_1(X)_{\mathbb {R}}$ ,

$$ \begin{align*}A^{n-1} - t\xi\in \mathrm{Amp}_1(X)_{\mathbb{R}} \end{align*} $$

for some $t>0$ sufficiently small. Since $\mathrm {Amp}_1(X)_{\mathbb {R}}$ is asymptotically generated by G,

$$ \begin{align*}A^{n-1} - t\xi = \sum_{m=1}^\infty t_m \xi_m \end{align*} $$

for some $t_m> 0$ and $\xi _m\in G$ . Finally, from

$$ \begin{align*}(nL - rK_X)A^{n-1}\ge 0,\ (nL - rK_X)\xi \le 0 \text{ and } (nL - rK_X)\xi_m \le 0, \end{align*} $$

we conclude that $(nL - rK_X)\xi = 0$ . Therefore,

$$ \begin{align*}(nL - rK_X)\xi = 0 \end{align*} $$

for all $\xi \in G$ . This implies that $nL - rK_X$ is numerically trivial since G also generates $N_1(X)_{\mathbb {R}}$ .

For a complex K3 surface X, it is easy to see that $E/\deg E\in G$ for every elliptic curve E on X. Since by hypothesis the elliptic curves generate the ample cone $\mathrm {Amp}(X)$ of X, $\mathrm {Amp}(X)$ is generated by G. If $\Omega _X^1$ is destabilised by a line bundle L, then L is numerically trivial. For K3 surfaces, this implies that $L = {\mathcal O}_X$ so that $H^0(\Omega _X^1(-L)) = H^0(\Omega _X^1) \ne 0$ which is a contradiction.

Proof. Since stability persists if we pass to a larger algebraically closed field, we may assume k is uncountable. Let $Y={\mathbb {P}}(\Omega ^1_X)$ , and suppose for a contradiction that $L={\mathcal O}_Y(1)$ is pseudoeffective. Then there is a Nakayama–Zariski decomposition of $L=E+N$ , where E is an effective ${\mathbb {R}}$ -divisor and N is nef in codimension 1 (due to [Reference NakayamaNak04] in characteristic 0 and [Reference MustaţăMus13, Reference Fulger and LehmannFL17] otherwise).

From [Reference LangerLan10, Theorem 4.1] (or Flenner or Mehta–Ramanathan’s theorem in characteristic zero), we may pick a very ample smooth curve C on X so that $\Omega ^1_X|_C$ is strongly semistable (or just semistable in characteristic zero). Then on the ruled surface $R={\mathbb {P}}(\Omega ^1_X|_C)$ every pseudoeffective line bundle is nef (in fact for the projectivisation of a degree zero strongly semistable bundle on a curve, these cones agree). On the other hand, $L|_R$ is not ample, since $L^2|_R=c_1(\Omega ^1_X)\cdot C=0$ . Hence, $L|_R$ is on the boundary of the nef cone of R. Write $E=aL+\pi ^*E'$ . As the Picard number of R is two and $E|_R$ is also ${\mathbb {R}}$ -effective, it must be that $E'.C\geq 0$ . If $E'.C>0$ , then $E'$ is effective on X (as C can vary), so in particular, $E|_R$ is big and hence ample. This contradicts $L|_R=E|_R+N|_R$ being boundary on the nef cone though. In other words, $CE'=0$ and as C can vary, $E'=0$ , forcing $E=aL$ . Then $a=0$ since from the assumption L has no effective multiple. It follows that $E=0$ and L is nef in codimension 1. In particular, it fails to be nef on at most countably many curves $C_i$ . Taking a hyperplane section H of Y, we see then that $L|_H$ is nef. In particular, $L^2\cdot H\ge 0$ . In terms of Chern classes, this means that

$$\begin{align*}-c_2(T_X)\ge 0, \end{align*}$$

which contradicts $c_2(T_X)=24$ .