Proof of the Regeneration principle 1.3.

We can suppose that ${\mathcal X}_0$ has Picard rank at least two and that $D_0$ is not proportional to the polarization; otherwise, by [Reference Charles, Mongardi and PacienzaCMP24, Corollary 3.5], we can deform a curve ruling $D_0$ over all of B, and obtain in this way a regeneration of $D_0$ .

Let $C_0$ be the class of a minimal curve ruling $D_0$ . Let ${\mathcal H}\in \operatorname {Pic}({\mathcal X})$ be a relative polarization and $H_0$ its restriction to the central fibre ${\mathcal X}_0$ . Let $H_0^\vee $ be the (ample) class of a curve dual to $H_0$ . We can choose $m\in {\mathbb N}$ big enough so that $mH_0^\vee -C_0$ is ample, primitive and of square bigger than $d_0$ . Therefore, by Hypothesis 1.2, we have a rational curve $R_0\in [mH_0^\vee -C_0]$ which rules an ample divisor $F_0$ inside ${\mathcal X}_0$ .

As the divisor $F_0$ is ample, we have $C_0\cdot F_0>0$ . Hence, we can fix a point in $C_0\cap F_0$ and pick a curve $R_0$ in the ruling of $F_0$ passing through this point. Notice that $C_0$ cannot coincide with the ruling of $F_0$ , as $C_0$ and $R_0$ are not proportional (because the divisors they rule are not). In this way, we obtain a connected rational curve of class $[C_0+R_0]$ . By abuse of notation, we denote this curve by $C_0+R_0$ . By [Reference BertiniBer21, Corollary 6.3], which generalizes [Reference Charles, Mongardi and PacienzaCMP24, Corollary 3.5] to the reducible case, the curve $C_0+R_0$ deforms in its Hodge locus $\mathrm {Hdg_{[C_0+R_0]}}$ of the class $[C_0+R_0]=[mH_0^\vee ]$ and keeps ruling a divisor on each point of $\mathrm {Hdg_{[C_0+R_0]}}$ . By construction, this Hodge locus coincides with B, as $C_0+R_0$ is a multiple of $H_0^\vee $ , and the result follows. Notice that here we might have to take a base change to a finite cover of B, in order to ensure the existence of a global family of divisors, whence the base change appearing in Definition 1.1.

Proof. Let $(S,h_S)$ be a polarized K3 of genus p and $(A,h_A)$ a polarized abelian surface of type $(1,p-1)$ . We denote by $r_n$ the class of an exceptional rational curve which is the general fiber of the Hilbert-Chow morphism $S^{[n]}\to S^{(n)}$ (resp. $K_n(A)\subset A^{[n+1]}\to A^{(n+1)}$ ) and by $h_S\in H_2(S^{[n]}, \mathbb Z)$ (resp. $h_A\in H_2(K_n(A), \mathbb Z)$ ) the image of the class $h_S\in H_2(S, \mathbb Z)$ (resp. $h_A\in H_2(A, \mathbb Z)$ ) under the inclusion $H_2(S, \mathbb Z)\hookrightarrow H_2(S^{[n]}, \mathbb Z)$ (resp. $H_2(A, \mathbb Z)\hookrightarrow H_2(K_n(A), \mathbb Z)$ ). Recall that $q(h_S)=2p-2=q(h_A)$ and $q(r_n)$ equals $1/(2n-2)$ in the $K3^{[n]}$ case and $1/(2n+2)$ in the Kummer case.

We take a primitive ample curve class $C\in H_2(X,{\mathbb Z})$ such that $q(C)>n-1$ (resp. $n+1$ for Kummer type). By [Reference Charles, Mongardi and PacienzaCMP24, Corollary 2.8] and [Reference Mongardi and PacienzaMP18, Theorem 4.2], the pair $(X,C)$ is deformation equivalent to the pair $(S^{[n]},h_S-2gr_n)$ with $2g\leq n-1$ or $(S^{[n]},h_S-(2g-1)r_n)$ with $2g\leq n$ (resp. $(K_n(A),h_A-2gr_n)$ or $(K_n(A),h_a-(2g-1)r_n)$ with $2g\leq n-1$ ).

If $p\leq g$ , we would get a contradiction since

$$ \begin{align*}n-1 \leq q(C)= q(h_S) -4g^2 \frac{1}{2(n-1)} = 2(p-1) -4g^2 \frac{1}{2(n-1)}\leq 2(g-1)-4g^2 \frac{1}{2(n-1)}\leq n -2. \end{align*} $$

Notice that the class of C equals $\frac {D^\vee }{m}$ with $m\leq 2n-2$ ; hence, the condition $q(D)\geq (2n-2)^2(n\pm 1)$ ensures $q(C)\geq n\pm 1$ (where the sign is according to the deformation type).

Therefore, $p\geq g$ , and by [Reference Charles, Mongardi and PacienzaCMP24, Section 4.1] and [Reference Mongardi and PacienzaMP21, Proof of Proposition 2.1], the curves we obtain in $S^{[n]}$ (resp. in $K_n(A)$ ) have a rational representative which covers a divisor by [Reference Charles, Mongardi and PacienzaCMP24, Proposition 4.1] and [Reference Mongardi and PacienzaMP21, Proposition 1.1]. Such rational curves then deform in its Hodge locus by [Reference Charles, Mongardi and PacienzaCMP24, Corollary 3.5], while still covering a divisor, and the proposition follows.

Proof of Theorem 1.4.

The result follows immediately from the combination of Proposition 2.1 and the Regeneration principle 1.3.

Proof of Theorem 1.5.

Again, the result follows from the combination of Proposition 2.1 and the Regeneration principle 1.3. Indeed, suppose that $(X,H)$ is a polarized IHS manifold of $K3^{[n]}$ -type, and let us consider a connected component ${\mathcal M}$ of the moduli space of polarized IHS manifolds containing $(X,H)$ . By [Reference Charles, Mongardi and PacienzaCMP24, Theorem 2.5], there exists a point in ${\mathcal M}$ which parametrizes the Hilbert scheme over a very general projective $K3 (S,H_S)$ . Let us choose any rational curve C in S, whose existence is guaranteed by Bogomolov-Mumford [Reference Mori and MukaiMM06] (see also [Reference Barth, Hulek, Peters and Van de VenBHPVdV04, Section VIII.23]), and let us consider the uniruled divisor $D_C=\{Z\in S^{[n]}$ such that $supp(Z)\cap C\neq \emptyset \}$ . We then apply the Regeneration principle 1.3 to $D_C$ and obtain a regeneration of it on all IHS manifolds corresponding to points of ${\mathcal M}$ . As the very general element of ${\mathcal M}$ has Picard rank one, the class of this regeneration is proportional to this unique class; hence, our regeneration has class $mH$ on X, for some m. For the generalized Kummer type, we proceed the same way, by using [Reference Mongardi and PacienzaMP18, Theorem 4.2] and [Reference Knutsen, Lelli-Chiesa and MongardiKLCM19b, Theorem 1.1] instead of the analogous results in the $K3^{[n]}$ -type case.

Proof. The proof is analogous to Theorem 1.5, with an extension to the case of square zero classes. If D has positive square, instead of the moduli space of polarized IHS manifolds, we consider the moduli space ${\mathcal M}$ of lattice polarized IHS manifolds such that $\operatorname {Pic}(X)$ contains a divisor of square $q(D)$ , and pick the connected component containing $(X,D)$ . Let us choose a parallel transport operator $\gamma $ on ${\mathcal M}$ such that $\gamma (X)$ has Picard rank $1$ . Therefore, $\gamma (D)$ is ample on $\gamma (X)$ . By Theorem 1.5, a multiple of $\gamma (D)$ is uniruled by a rational curve $\gamma (C)$ , which has class proportional to $\gamma (D)^\vee $ . Therefore, by [Reference Charles, Mongardi and PacienzaCMP24, Proposition 3.1], $\gamma (C)$ deforms in its Hodge locus, which by construction contains $(X,D)$ , and we obtain a rational curve C covering a multiple of D. If $q(D)=0$ , we can suppose that D is nef by [Reference MarkmanMar11, Proposition 5.6]; otherwise, we follow the same reasoning as above to reduce to the nef case. As X is projective, we have an ample divisor $H\in \operatorname {Pic}(X)$ . Let L be the saturated lattice generated by D and H, and let us consider the component ${\mathcal M}$ of the moduli space of L lattice polarized IHS manifolds containing $(X,L)$ . Inside of ${\mathcal M}$ , by [Reference Mongardi and PacienzaMP23, Theorem 3.13], we can pick a point $\gamma (X)$ such that $\gamma (D)$ stays nef and there exists a prime exceptional divisor E on $\gamma (X)$ such that $q(\gamma (D),E)>0$ .Footnote ² Let R be a curve ruling E. As $\gamma (D)$ is nef, there exists an $m\in {\mathbb N}$ such that $m\gamma (D)^\vee -R$ is an ample curve. Therefore, by Proposition 2.1, we produce a rational curve C of class $m\gamma (D)^\vee -R$ which rules an ample divisor and attach to it a rational tail R, so that the connected curve $C+R$ of class $m\gamma (D)^\vee $ rules a divisor and deforms in its Hodge locus by [Reference BertiniBer21, Corollary 6.3]. By construction, this Hodge locus contains $(X,D^\vee )$ , and the result follows.

Proof. Without loss of generality, we can suppose that the two points $\xi _i,\, i\in \{1,2\}$ correspond to reduced subschemes and that $\operatorname {supp}(\xi _1)\cap \operatorname {supp}(\xi _2)=\emptyset $ ; otherwise, we can take arbitrarily close approximations by reduced subschemes with such property. Therefore, we write

$$ \begin{align*}\xi_i=p^i_1+\dots p^i_n,\end{align*} $$

with $p^i_1,\ldots , p^i_n$ distinct points on S for $i=1,2$ . By Theorem 3.2, we have two ample irreducible curves $R^1_1, R^2_1$ arbitrarily near $p_1^1$ and $p_1^2$ , respectively. As these curves are ample, the rational curve $R_1=R^1_1\cup R^2_1$ is connected. Let us consider the rational curve $R_1+p^1_2+\dots p^1_n$ inside $S^{[n]}$ : this can be used to approximate the subschemes $p^1_1+p^1_2+\dots p^1_n$ and $p^2_1+p^1_2+\dots p^1_n$ . Iterating the argument, one obtains a rational curve (union of two irreducible ample curves) $R_j$ for all $j\in \{1,\dots n\}$ which approximates the two points $p^1_j$ and $p^2_j$ . Considering the curve $p^2_1+\dots +p^2_{j-1}+R_j+p^1_{j+1}+\dots + p^1_{n}$ , one can approximate the points $p^2_1+\dots +p^2_{j-1}+p^1_j+p^1_{j+1}+\dots + p^1_{n}$ and $p^2_1+\dots +p^2_{j-1}+p^2_j+p^1_{j+1}+\dots + p^1_{n}$ . Therefore, by taking the union of these curves, we obtain a chain of $2n$ rational irreducible curves which approximate the two points $\xi _1$ and $\xi _2$ . By construction, each of these rational curves C deforms in a family which covers the divisor $\{Z\in S^{[n]},\text { such that } \operatorname {supp}(Z)\cap C\neq \emptyset \}$ , and the corollary follows.

Proof of Theorem 1.6.

1) Let X be a very general IHS manifold in ${\mathcal M}$ . Let $x_1,x_2 \in X$ be two points on it. Thanks to [Reference Mongardi and PacienzaMP23, Corollary 1.2], we can pick a point in ${\mathcal M}$ which parametrizes the punctual Hilbert scheme of a very general projective $K3 (S,H)$ arbitrarily close to X and two points $\xi _1, \xi _2\in S^{[n]}$ approximating $x_1$ and $x_2$ , respectively. We take the chain R of $2n$ rational curves approximating $\xi _1$ and $\xi _2$ given by Corollary 3.3. We can now apply the Regeneration principle 1.3 to regenerate the union of the divisors ruled by the deformations of the irreducible components of R to obtain a chain of rational curves on X satisfying the statement.

2) Let A be an abelian surface, $\iota $ the $-(1)$ -involution and $S\to A/\iota $ the associated projective Kummer K3 surface. On S, we have infinitely many (ample) rational curves by [Reference Chen, Gounelas and LiedtkeCGL22]. These rational curves yield infinitely many (singular) hyperelliptic curves on the abelian surface A. By taking the fibers of the degree two map onto $\mathbb P^1$ to these infinitely many hyperelliptic curves, we associate infinitely many rational curves in $Kum_2 (A)$ . Notice that, by translation, deformations of each of these rational curves cover a uniruled divisor. By adding $(n-2)$ arbitrary points, we obtain the same conclusion on $Kum_n(A)$ , for all $n\geq 2$ . These divisors have positive square. Regenerating these infinitely many uniruled divisors, we obtain the conclusion on the very general deformation of $Kum_n(A)$ .

Proof of Theorem 1.7.

To prove the theorem, we will show the existence of an ample uniruled divisor with infinite ${\operatorname {Bir}}(X)$ -orbit. By [Reference OguisoOgu06, Theorem 1.1], as ${\operatorname {Bir}}(X)$ is infinite, there exists an element $g\in {\operatorname {Bir}}(X)$ of infinite order. Let D be an ample uniruled divisor, whose existence is granted by Theorem 1.5. We claim that the orbit of D via g is infinite, as otherwise, a multiple of g would give an isometry of the lattice $D^\perp \subset \operatorname {NS}(X)$ . The latter is negative definite as D is ample and has therefore finite isometry group. Hence, g would act with finite order on both D and $D^\perp $ , which is absurd, and the claim follows.

Proof. First of all, recall that a birational map between two IHS manifolds sending an ample class into an ample class can be extended to an isomorphism. As such, when $\rho (X)=1$ , we have $\operatorname {Aut}(X)={\operatorname {Bir}}(X)$ . By [Reference FujikiFuj78, Theorem 4.8] the group of automorphisms of a compact Kähler manifold that fix a Kähler class has only finitely many connected components. On the other hand the group of automorphisms of an IHS manifold X is discrete, since $h^0(X,T_X)=h^0(X,\Omega^1_X)=0$ . Hence $\mathrm{Aut}(X)$ must be finite.