Proof. Let $U \subseteq Z$ be the open subset on which the naive composition $f \circ g$ exists, and let $V \subseteq X$ be the maximal open on which f is defined. Then, by assumption, U is a big open of Z. The morphism $f \circ g \colon U \to Y$ is the composition of the morphism of varieties $U \to V$ with the orbifold morphism $V \to (Y, \Delta _Y)$ . As the image of the composition is not contained in $\operatorname {\mathrm {supp}} \Delta _Y$ , the composition $f \circ g \colon U \to (Y, \Delta _Y)$ is thus a morphism of orbifolds. This concludes the proof.

Proof. Let $T_1\subset T$ be the set of t in T such that $F_t$ factors over Z. We consider $T_1 = \sqcup _{t\in T_1} \operatorname {\mathrm {Spec}} \kappa (t)$ as an S-scheme. To show that $T_1$ is closed in T, it suffices to show that $\overline {T_1} = T_1$ . We endow $\overline {T_1}\subset T$ with the reduced closed subscheme structure. The natural morphism $T_1\subset \overline {T_1}$ is dominant. Since $X\to S$ is flat, the basechange $X\times _S T_1\to X\times _S \overline {T_1}$ is (also) dominant. Since $X\times _S T_1\to Y$ factors set-theoretically through Z and $X\times _S T_1\to X\times _S \overline {T_1}$ is dominant, we see that the restriction of F to $X\times _S \overline {T_1}$ (also) factors set-theoretically through Z. However, for any $t\in \overline {T_1}$ , the scheme $X\times _S \operatorname {\mathrm {Spec}} \kappa (t)$ is geometrically reduced over $\kappa (t)$ . In particular, $F_t$ factors (scheme-theoretically) through Z, as required.

Proof. Assume (a) holds. Let $Z \subseteq X$ be the locus of points where the rank of $df_x$ is less than $\dim Y$ . Since $K(Y)\subset K(X)$ is a separable field extension, the arguments used to prove [Reference HartshorneHar77, Lemma III.10.5] and [Reference HartshorneHar77, Proposition III.10.6] show that the dimension of $\overline {f(Z)}$ is less than $\dim Y$ . Thus, since f is dominant, this implies that $Z \neq X$ . In particular, there is a closed point $x \in X$ such that the rank of $df_x$ is equal to $\dim Y$ . Since Y is smooth, this shows that (b) holds.

Assume (b) holds. Then [Reference HartshorneHar77, Proposition III.10.4] shows that (c) holds.

Now assume that (c) holds. There is an open neighborhood $U \subseteq X$ of x such that $f\vert _U$ is smooth. Smooth maps are flat, and flat maps of varieties are open. Hence, $f(U)$ is a nonempty open of Y. Thus, f is dominant. In particular, f maps the generic point of X to the generic point of Y. The smoothness of $f\vert _U$ also implies that f is smooth at the generic point of X. Since generically smooth morphisms are separable, we see that (a) holds. This concludes the proof.

Proof. Assume (a) holds. Let $U \subseteq Y$ be the locus of smooth points of Y. Then U is a dense open of Y, and the restriction of $f \colon f^{-1}(U) \to U$ is still separably dominant. Thus, we may assume that Y is smooth. In particular, by Lemma 3.3, there is a closed point x in X such that $df_x \colon \textrm {T}_x X \to \textrm {T}_{f(x)} Y$ is surjective. Since X and Y are smooth of the same dimension, it follows that $df_x$ is an isomorphism.

Assume (b) holds. Then, the point $f(x)$ is a regular point of Y. We can thus replace Y by an open neighborhood of $f(x)$ and replace X by the preimage of that. The assumption on the dimensions continues to hold, so we may assume that Y is smooth. Consequently, by Lemma 3.3, the morphism f is separably dominant.

Proof. Replacing X by an alteration if necessary, we may assume that X is smooth [Reference de JongdJ96, Theorem 4.1]. Let $U \subseteq X \times T$ be the maximal open subset on which F is defined. The map F then induces a morphism of T-schemes $G \colon U \to Y \times T$ .

We claim that if $(x,t) \in U$ is any closed point, the differential $dG_{(x,t)}$ is an isomorphism if and only if the differential of the rational map is an isomorphism at $x \in X$ . To see this, note that the tangent spaces of $X \times T$ and $Y \times T$ are the products of the tangent spaces of the factors. Furthermore, the component of $dG_{(x,t)}$ which maps $\textrm {T}_t T \to \textrm {T}_{F(x,t)} X$ is the zero map, and the component which maps $\textrm {T}_t T \to \textrm {T}_t T$ is just the identity. Lastly, the component of $dG_{(x,t)}$ mapping $\textrm {T}_x X \to \textrm {T}_{F_t(x)} Y$ is just $dF_t$ . This implies the claim.

Let $t \in T$ be a closed point. By Lemma 3.4, $F_t$ is separably dominant if and only if there is a closed point $x \in X$ such that the differential $dF_{t,x}$ is an isomorphism. By the previous claim, this happens if and only if there is a closed point $x \in X$ such that $dG_{(x,t)}$ is an isomorphism.

Let $V \subseteq U$ be the set of all points at which the differential of G is an isomorphism. The set V is open in U, and hence open in $X \times T$ . The map $X \times T \to T$ is flat, and hence open. Thus, the projection of V to T is an open subset of T. By the previous paragraph, this is exactly the set of $t \in T$ for which $F_t$ is separably dominant, so we are done.

Proof. See [Reference MatsumuraMat80, §20F Corollary 1].

Proof. The case $\omega = 0$ is clear, so suppose $\omega \neq 0$ . By replacing T with an alteration if necessary, we may assume that T is smooth.

Consider the pullback form $G^*\omega $ , and note that it defines a rational section of the vector bundle $(\Omega ^n_{X \times T})^{\otimes m}$ . For any closed point $t \in T$ , we pullback $G^*\omega $ along the inclusion $\iota _t \colon X := X\times \{t\} \subseteq X \times T$ to get the form $\iota _t^*G^*\omega = G_t^*\omega $ . Let E and F be the divisors of zeroes and poles of $G^*\omega $ , respectively. For t in T, we define $E_t := E \cap X_t$ and $F_t := F \cap X_t$ . Since $G_t$ is separably dominant and $\omega \neq 0$ , we have that, for every t in T, $E_t$ and $F_t$ are (possibly trivial) effective divisors in $X_t$ . Note that whenever $G_t$ is defined in codimension one, we have that $E_t$ (resp. $F_t$ ) is the divisor of zeroes (resp. poles) of $G_t^*\omega $ . However, if $G_t$ is not defined at all points of codimension $1$ , it may happen that $E_t$ and $F_t$ are strictly bigger than the divisor of zeroes and poles, respectively.

We now prove the result by induction on $\dim (T)$ . The case $\dim (T)=0$ is clear. Consider the set S of all t in T such that $\dim (E_t \cap F_t) = n-1$ . By semicontinuity of fiber dimension, S is closed in T (since $\dim (E_t \cap F_t)> n-1$ cannot occur). The condition $t \in S$ implies that $G_t$ cannot be defined in codimension $1$ . Otherwise, the form $G_t^*\omega $ would have to have both a pole and a zero along the codimension $1$ subset $E_t \cap F_t$ , which is absurd. Since G is defined at all points of codimension $1$ , we see $\dim (S)<\dim (T)$ . By the inductive hypothesis, it follows that $S_\omega =S \cap T_\omega $ is closed.

We now show that $F\to T$ is flat using Lemma 3.8. First, note that it is locally cut out by the vanishing of a single equation given by the denominator of $G^*\omega $ . Furthermore, the morphism $X \times T \to T$ is flat and, for every closed point t in T, the scheme-theoretic fiber $F_t$ of $F \to T$ is a divisor of $X_t$ . Thus, we conclude that $F \to T$ is flat by Lemma 3.8.

In particular, there is a morphism $T \to \mathrm {Hilb}(X)$ representing the family $(F_t)_{t \in T}$ , where $\mathrm {Hilb}(X)$ is the Hilbert scheme of X over k. Now, as there are only finitely many effective divisors with the property of being $\leq D_X$ , the set of such divisors form a (finite) closed subscheme of $\mathrm {Hilb}(X)$ . It follows that $F_t \leq D_X$ is a closed condition on t.

For t in T, the condition $F_t\leq D_X$ implies that $G_t^* \omega \in \Gamma (X,\omega _X^{\otimes m}(D_X))$ . Moreover, outside the set S (defined above), the condition $G_t^*\omega \in \Gamma (X,\omega _X^{\otimes m}(D_X))$ is equivalent to $F_t \leq D_X$ . Thus, a point $t \in T$ lies in $T_\omega $ if and only if we have $t \in S \cap T_\omega $ or $F_t \leq D_X$ . As $S\cap T_\omega $ is closed in T and the set of t in T with $F_t\leq D_X$ is closed in T, this concludes the proof.

Proof. Campana shows this when $k=\mathbb {C}$ using computations in the analytic topology; see [Reference CampanaCam11, Proposition 2.11]. His arguments adapt to positive characteristic, as we show now.

We have a morphism of sheaves

$$\begin{align*}f^* \operatorname{\mathrm{Sym}}^n \Omega^p_Y(\log \left\lceil {\Delta_Y} \right\rceil ) \to \operatorname{\mathrm{Sym}}^n \Omega^p_X(\operatorname{\mathrm{supp}} (\Delta_X + f^* \Delta_Y)).\end{align*}$$

As $f^*S^n \Omega ^p_{(Y, \Delta _Y)}$ (resp. $S^n\Omega ^p_{(X,\Delta _X)}$ ) is a subsheaf of the source (resp. the target) of this morphism, we may argue étale-locally around a fixed point $\eta $ . As the sheaves involved are locally free, we may and do assume that $\eta \in X$ is of codimension $1$ (except for in the trivial situation in which X is zero-dimensional).

Locally around $\eta $ , the sheaf $f^* S^n \Omega ^p_{(Y,\Delta _Y)}$ is generated by the pullbacks of the local generators of $S^n \Omega ^p_{(Y,\Delta _Y)}$ around $f(\eta )$ (see Definition 4.1). Let $\omega \in S^n \Omega ^p_{(Y, \Delta _Y)}$ be such a generator. Let $\widetilde {Y} \to Y$ be a connected étale neighborhood of $f(\eta )$ such that

1. (i) $\widetilde {Y}$ is an étale open of $\mathbb {A}^d$ with $d = \dim Y$ ,

2. (ii) $\Delta _Y$ is in normal crossing form (i.e., $\Delta _Y$ is given by the pullback of $x_1\cdot \ldots \cdot x_\ell =0$ in $\mathbb {A}^d$ for some $\ell \geq 0$ ),

3. (iii) $f(\eta )$ specializes to the origin of $\mathbb {A}^d$ , and

4. (iv) $\omega $ has the form described in Definition 4.1.

There is a connected étale neighborhood $\widetilde {X} \to X$ of $\eta $ such that $f\vert _{\widetilde {X}}$ factors over $\widetilde {Y}$ and such that $\Delta _X$ is in normal crossings form. Since the induced morphism $f \colon (\widetilde {X}, \widetilde {X} \times _X \Delta _X) \to (\widetilde {Y}, \widetilde {Y} \times _Y \Delta _Y)$ is (still) orbifold, we may replace $(X, \Delta _X)$ by $(\widetilde {X}, \widetilde {X} \times _X \Delta _X)$ and $(Y, \Delta _Y)$ by $(\widetilde {Y}, \widetilde {Y} \times _Y \Delta _Y)$ .

As Y is an étale open of $\mathbb {A}^d$ , we obtain a map $X \to \mathbb {A}^d$ given by d maps $f_1,...,f_d \colon X \to \mathbb {A}^1$ . For $i=1,\ldots ,d$ , we let $m_i$ denote the multiplicity of the (pullback of the) prime divisor $\{y_i=0\}$ in $\Delta _Y$ . Since $f(X)$ is not contained in $\Delta _Y$ , we know that whenever $m_i> 1$ , the function $f_i$ is not identically zero. Viewing $f_i$ as an element of the DVR $\mathcal {O}_{X, \eta }$ , we decompose it as $f_i = t^{\nu _i} g_i$ with t a uniformizer and $g_i(\eta ) \neq 0$ . Since f is an orbifold morphism, for any i with $\nu _i \neq 0$ , we have $\nu _i \geq \left \lceil {m_i/m_\eta } \right \rceil $ , where $m_\eta $ is the multiplicity of the divisor $\eta $ in $\Delta _X$ .

Let $J \subseteq \{1,...,d\}$ be a p-element subset, and consider the rational p-form $dy_J/y_J$ on Y. If none of the functions $f_i$ with $i \in J$ vanish along $\eta $ , the pullback $f^*(dy_J/y_J)$ has no pole at $\eta $ . If such an $i \in J$ exists, then $f^*(dy_J/y_J)$ has a pole of order at most $1$ . Since we can always write $f^*(dy_J/y_J) = (dt/t) \wedge u + v$ , where u is a $(p-1)$ -form with no pole at $\eta $ and v a p-form with no pole at $\eta $ , the pullback of $\omega $ is given by

$$\begin{align*}f^*\omega = \prod_{i=1}^d (f_i)^{ \left\lceil {k_i/m_i} \right\rceil } \bigotimes_{\alpha=1}^n \left( (\frac{dt}{t} \wedge u_\alpha) + v_\alpha \right).\end{align*}$$

Here, as before, $u_\alpha $ and $v_\alpha $ are forms with no pole at $\eta $ . We can write the tensor product of sums as a sum of tensor products. When doing this, the order at $\eta $ of each summand occuring in such a rewriting is at least

$$ \begin{align*}\left(\sum_{i=1}^d \left\lceil {\frac{k_i}{m_i}} \right\rceil \nu_i\right) - k_t,\end{align*} $$

where $k_t$ counts the number of $((dt/t) \wedge u_\alpha )$ -factors occuring in that summand (as opposed to $v_\alpha $ -factors). Note that $k_t \leq \sum k_i$ , where the sum runs over those i for which $\nu _i \geq 0$ . Using our estimate $\nu _i \geq \left \lceil {m_i/m_\eta } \right \rceil $ from before, we obtain that the order at $\eta $ of each summand is at least

$$ \begin{align*}\left(\sum_{\substack{i=1 \\ \nu_i \neq 0}}^d \left\lceil {\frac{k_i}{m_\eta}} \right\rceil \right) - k_t \geq -k_t + \left\lceil {\frac{k_t}{m_\eta}} \right\rceil ,\end{align*} $$

so the pole at $\eta $ is at most of the order we allow for elements of $S^n \Omega ^p_{(X, \Delta _X)}$ . Hence, $f^*\omega \in S^n \Omega ^p_{(X, \Delta _X)}$ , which concludes the proof.

Proof. For the sheaf of log-differentials, we have $\Omega ^n_X(\log \left \lceil {\Delta } \right \rceil ) = \omega _X( \left \lceil {\Delta } \right \rceil )$ (see [Reference IitakaIit82, §11.1]). It follows that $\operatorname {\mathrm {Sym}}^N \Omega ^n_X(\log \left \lceil {\Delta } \right \rceil ) = \operatorname {\mathrm {Sym}}^N \omega _X( \left \lceil {\Delta } \right \rceil )$ . Since symmetric powers of line bundles agree with tensor powers, it follows that $\operatorname {\mathrm {Sym}}^N \omega _X( \left \lceil {\Delta } \right \rceil ) = \omega _X( \left \lceil {\Delta } \right \rceil )^{\otimes N}$ . Thus, $S^N \Omega ^n_{(X, \Delta )}$ is by construction a locally free subsheaf of $\omega _X( \left \lceil {\Delta } \right \rceil )^{\otimes N}$ . More precisely, we see that locally around a point $p \in X$ , it is the subsheaf generated by the single element

$$ \begin{align*}x_1^{N/m_1}...x_n^{N/m_n} \bigotimes_{l=1}^N \frac{dx_1 \wedge dx_2 ... \wedge dx_n}{x_1x_2...x_n},\end{align*} $$

where $x_1,...,x_n$ are a set of normal crossing coordinates for $\Delta $ , and $m_i$ denotes the multiplicity of the coordinate $x_i$ in the orbifold divisor. The subsheaf generated by this element is equal to $\omega _X(\Delta )^{\otimes N}$ in some neighborhood of p. The claim follows since p was arbitrary.

Proof. By Lemma 4.4, we have $\omega _Y(\Delta _Y)^{\otimes N}= S^N \Omega ^n_{(Y, \Delta _Y)}$ . Similarly, as $n = \dim X = \dim Y$ , we have $\omega _X(\Delta _X)^{\otimes N} = S^N \Omega ^n_{(X, \Delta _X)}$ , as this equality holds over a big open and both sides are line bundles. By Lemma 4.3, we get a morphism $f^* S^N \Omega ^n_{(Y, \Delta _Y)} \to S^N \Omega ^n_{(U, \Delta _X \cap U)}$ of sheaves on U, where $U \subseteq X$ denotes the intersection of the domain of definition of f with the snc locus of $(X, \Delta _X)$ . Since $U \subseteq X$ is a big open, by Remark 2.3, the line bundle $f^*\omega _Y(\Delta _Y)^{\otimes N}$ on U extends to a line bundle on X. Furthermore, as the morphism of locally free sheaves extends as well by Hartogs, this concludes the proof.

Proof. First assume that the result is true in the case of the smooth orbifolds. Let $X^o \subseteq X$ and $T^o \subseteq T$ denote the snc loci, and note that these are big opens. Then, since forming the pushforward commutes with forming direct sums, we see that $\iota _{(X^o \times T^o)*}\pi _{X^o}^*S^N \Omega ^m_{(X^o, \Delta _{X^o})}$ is a direct summand of $S^N \Omega ^m_{(X, \Delta _X) \times (T, \Delta _T)}$ . Since $\iota _{(X^o \times T^o)*} \pi _{X^o}^* = \pi _X^* \iota _{X^o*}$ for locally free sheaves, the desired result follows.

By the above, we may assume that $(X, \Delta _X)$ and $(T, \Delta _T)$ are smooth. Assume first that $\Delta _X$ and $\Delta _T$ are $\mathbb {Z}$ -divisors (i.e., that all multiplicities are either $1$ or $\infty $ ). In this case, we have $S^N \Omega ^m_{(X,\Delta _X)} = \operatorname {\mathrm {Sym}}^N \Omega ^m_X(\log \Delta _X)$ . The latter is a genuine symmetric power of an exterior power of $\Omega ^1_X(\log \Delta _X)$ . Notice that the decomposition

$$ \begin{align*}\Omega^1_{X \times T}(\log (\Delta_X \times T + X \times \Delta_T)) = \pi_X^* \Omega^1_X(\log \Delta_X) \oplus \pi_T^* \Omega^1_T(\log \Delta_T)\end{align*} $$

is still valid. Thus, the discussion of the previous paragraph applies, proving the result in this case.

Finally, we treat the general case in which $\Delta _X$ and $\Delta _T$ are not $\mathbb {Z}$ -divisors. By definition, the sheaf $S^N \Omega ^n_{(X,\Delta _X)}$ is a subsheaf of

$$\begin{align*}\operatorname{\mathrm{Sym}}^N \Omega^m_{X}(\log \left\lceil {\Delta_X} \right\rceil ), \end{align*}$$

and, similarly, the sheaf $S^N \Omega ^m_{(X,\Delta _X)\times (T,\Delta _T)}$ is a subsheaf of

$$\begin{align*}\operatorname{\mathrm{Sym}}^N \Omega^m_{X\times T}(\log \left\lceil {\Delta_X\times T + X\times \Delta_T} \right\rceil ). \end{align*}$$

By the previous paragraph, we know that the morphism

$$\begin{align*}\pi_X^* \operatorname{\mathrm{Sym}}^N \Omega^m_{X}(\log \left\lceil {\Delta_X} \right\rceil ) \to \operatorname{\mathrm{Sym}}^N \Omega^m_{X\times T}(\log \left\lceil {\Delta_X \times T + X \times \Delta_T} \right\rceil ) \end{align*}$$

is injective and has a retract. Since the projection map $\pi _X$ is orbifold, it follows from Lemma 4.3 that the above injection sends the subsheaf $\pi _X^*S^N\Omega ^m_{(X,\Delta _X)}$ to the subsheaf $S^N\Omega ^m_{(X,\Delta _X)\times (T,\Delta _T)}$ . To prove the claim, it thus suffices to show that the retraction also respects these subsheaves. This can be checked locally, and it suffices to consider the generators. This can be done very explicitly.

Indeed, let $(x,t) \in X \times T$ be any closed point, let $dx_1,...,dx_n$ be local coordinates for X around x which exhibit $\Delta _X$ in normal crossings form, and let $dt_1,...,dt_r$ be local coordinates for T around t which exhibit $\Delta _T$ in normal crossings form. Then $dx_1,...,dx_n, dt_1,..., dt_r$ are local coordinates for $X \times T$ exhibiting its orbifold divisor in normal crossings form. Let $\omega $ be a local generator of $S^N \Omega ^m_{(X, \Delta _X) \times (T,\Delta _T)}$ around $(x,t)$ . If $\omega $ contains any factors containing a $dt_i$ , the pullback $\iota _t^* \omega $ will be identically zero. Thus, it remains to consider the case where the only differentials appearing in $\omega $ are products of $dx_i$ terms. Pulling back such a generator of $S^N\Omega ^m_{(X,\Delta _X)\times (T,\Delta _T)}$ along $\iota _t$ yields a (formally identical) generator of $S^N \Omega ^m_{(X,\Delta _X)}$ . This finishes the proof.

Proof. We first show that the pullback map considered in the statement is well defined. To do so, let $X^o\subseteq X$ be the snc locus of $(X,\Delta _X)$ , and note that $X^o$ is a big open of X. The morphism $\iota _t \colon X \to X\times T$ induces an orbifold morphism $X^o\to X^o\times T$ (i.e., $\iota _t$ is orbifold over the snc locus). In particular, by Lemma 4.3, there is an induced morphism $\iota _t^* S^N \Omega ^m_{(X^o, \Delta _{X^o}) \times (T, \Delta _T)} \to S^N \Omega ^m_{(X^o, \Delta _{X^o})}$ . Since the sheaves involved are reflexive, this pullback map extends to a pullback map $\iota _t^* S^N \Omega ^m_{(X, \Delta _X) \times (T, \Delta _T)} \to S^N \Omega ^m_{(X, \Delta _X)}$ .

To see that the pullback map factors as claimed, note that $\pi _X \circ \iota _t = \operatorname {\mathrm {id}}_X$ is the identity morphism, so that, in fact, $\iota _t^* \pi _X^* S^N \Omega ^m_{(X,\Delta _X)} = S^N \Omega ^m_{(X,\Delta _X)}$ . This concludes the proof.

Proof. Before starting the proof, we note that the set S is well defined by Remark 2.3.

Let $(f, \phi )$ and $(g, \psi )$ be elements of S which induce the same rational map

By our assumptions on the line bundles, the space $\mathbb {P}(\mathcal {L}_X(X))$ contains a birational copy $\overline {X}$ of X and $\mathbb {P}(\mathcal {L}_Y(Y))$ contains Y. The following square commutes whenever the compositions are defined:

Here, the upper horizontal arrow can be either f or g. Note that the indeterminacy locus of $\gamma $ is a linear subspace and that $\overline {X}$ is not contained in any proper linear subspace of $\mathbb {P}(\mathcal {L}_X(X))$ . Hence, $\gamma $ is defined on some open of $\overline {X}$ , and the commutativity of the diagram above implies that $\gamma $ sends $\overline {X}$ to Y. So we get a rational map . The composition is equal to both f and g whenever it is defined, showing that $f=g$ on a dense open subset. As Y is separated, this implies that $f=g$ everywhere.

Proof. If there are no separably dominant near-maps from $(X,\Delta _X)$ to $(Y,\Delta _Y)$ , then we are done. Otherwise, let be a separably dominant near-map. By Corollary 4.5, for $N \in \mathbb {N}$ sufficiently divisible, we get an induced morphism of line bundles $f^* (\omega _Y(\Delta _Y))^{\otimes N} \to \omega _X(\Delta _X)^{\otimes N}$ . Since f is separably dominant, the morphism of line bundles $f^* (\omega _Y(\Delta _Y))^{\otimes N} \to \omega _X(\Delta _X)^{\otimes N}$ is non-zero, and hence injective. This implies that $\omega _X(\Delta _X)^{\otimes N}$ is a big line bundle. Increasing N if necessary, we can thus assume that all of the following hold:

• • $\omega _X(\Delta _X)^{\otimes N}$ and $\omega _Y(\Delta _Y)^{\otimes N}$ are well-defined line bundles.

• • The line bundle $\omega _X(\Delta _X)^{\otimes N}$ is very big.

• • There is an effective divisor $C \subseteq Y$ such that $(\omega _Y(\Delta _Y)^{\otimes N})(-C)$ is very ample.

We fix the integer N and the effective divisor $C \subseteq Y$ from the last bullet point. We define $V_X := \Gamma (X, \omega _X(\Delta _X)^{\otimes N})$ and $V_Y := \Gamma (Y, \omega _Y(\Delta _Y)^{\otimes N}(-C))$ . Note that we obtain a closed immersion $\iota _Y \colon Y \to \mathbb {P}(V_Y)$ and a rational map which is birational onto its scheme-theoretic image $\overline {X}$ . For every dominant near-map f, we get an induced vector space morphism $f^* \colon V_Y \to V_X$ . By Lemma 5.1, we can recover f from $f^*$ and even from $\mathbb {P}(f^*)$ . Thus, we are led to studying linear maps $V_Y \to V_X$ .

Let $H := \operatorname {\mathrm {Hom}}(V_Y, V_X)^\vee $ , with $^\vee $ denoting the dual. Composition of functions is a canonical bilinear map $V_X^\vee \times \operatorname {\mathrm {Hom}}(V_Y, V_X) \to V_Y^\vee $ , and after identifying $\operatorname {\mathrm {Hom}}(V_Y,V_X)$ with its double dual, we get a bilinear morphism $V_X^\vee \times H^\vee \to V_Y^\vee $ . It induces a relative (over $\mathbb {P}(H)$ ) rational map . For every closed point $h \in \mathbb {P}(H)$ , we denote by $F_h$ the rational map .

To prove the proposition, we are first going to construct a “small” locally closed subset $H_3$ of $\mathbb {P}(H)$ such that the set of separably dominant near-maps (still) injects into $H_3$ via $f\mapsto \mathbb {P}(f^\ast )$ .

Let $H_1 \subseteq \mathbb {P}(H)$ be the subset for which $F_h$ maps $\overline {X} \subseteq \mathbb {P}(V_X)$ to $Y \subseteq \mathbb {P}(V_Y)$ . To see that this is a meaningful condition, note that the indeterminacy locus of $F_h$ is a linear subspace and that $\overline {X} \subseteq \mathbb {P}(V_X)$ is contained in no proper linear subspace. Let $\eta _{\overline {X}}$ be the generic point of the scheme $\overline {X}$ . Since $H_1$ is the set of h in $\mathbb {P}(H)$ such that the morphism

$$\begin{align*}\{\eta_{\overline{X}}\}\times \mathbb{P}(H)\to \mathbb{P}(V_Y) \end{align*}$$

factors over Y, it follows from Lemma 3.1 that it is closed in $\mathbb {P}(H)$ .

Note that we obtain a relative rational map . Let $H_2\subset H_1$ be the subset of elements of $H_1$ for which the induced rational map is separably dominant. By Lemma 3.6, the set $H_2$ is open in $H_1$ .

Let $H_3\subset H_2$ be the subset of rational maps such that, for every global section $\omega $ of $\omega _Y(\Delta _Y)^{\otimes N}$ , the pullback $(g\circ \iota _X)^*\omega $ is a global section of $\omega _X(\Delta _X)^{\otimes N}$ . By applying Lemma 3.9 to every single $\omega $ and taking the intersection over all closed sets obtained this way, we see that $H_3$ is closed in $H_2$ . Hence, $H_3$ is locally closed in $\mathbb {P}(H)$ , and we give it the reduced scheme structure.

If is a separably dominant orbifold near-map, then the induced map $\mathbb {P}(f^*)$ lies in $H_3$ . As we mentioned before, by Lemma 5.1, different separably dominant orbifold near-maps induce different elements of $H_3$ . Therefore, to prove the proposition, it suffices to show that $H_3$ is finite. To do so, let $H_4$ be an irreducible component of $H_3$ , so that $H_4$ is a quasi-projective variety. Since $H_3$ is quasi-projective, it has only finitely many irreducible components. Therefore, to conclude the proof, it suffices to show that $H_4$ is a point.

Let $\overline {H_4}$ be the closure of $H_4$ in $\mathbb {P}(H)$ , and note that $\overline {H_4}$ is a projective variety. Since $H_1$ is closed in $\mathbb {P}(H)$ , we see that $\overline {H_4}$ is contained in $H_1$ . In particular, we can interpret every closed point of $\overline {H_4}$ as a (possibly non-dominant) rational map . Let $\widetilde {H_4}$ be a smooth projective variety, and let $\widetilde {H_4} \to \overline {H_4}$ be an alteration such that the preimage of $\overline {H_4} \setminus H_4$ in $\widetilde {H_4}$ is a strict normal crossings divisor $D_H$ (this exists by [Reference de JongdJ96, Theorem 4.1]). We let be the relative rational map induced by the above map .

By Lemma 4.6, the sheaf $S^N \Omega ^n_{(X, \Delta _X) \times (\widetilde {H_4}, D_H)}$ has $\pi _X^* S^N \Omega ^n_{(X,\Delta _X)}$ as a direct summand. We denote by

$$\begin{align*}\pi \colon S^N \Omega^n_{(X, \Delta_X) \times (\widetilde{H_4}, D_H)} \to \pi_X^* S^N \Omega^n_{(X,\Delta_X)}\end{align*}$$

the projection. We now define the morphism $\Psi \colon \widetilde {H_4} \to \operatorname {\mathrm {Hom}}(V_Y,V_X) = H^\vee $ by $\Psi (h) = \left [ \omega \mapsto \iota _h^*\pi (G^*\omega ) \right ]$ , where $\iota _h\colon X\to X\times \widetilde {H_4}$ is the inclusion map $x\mapsto (x,h)$ . We now show that $\Psi $ is a well-defined morphism of varieties.

To do so, fix a closed point $h \in \widetilde {H_4}$ and $\omega \in V_Y$ . Then $\omega \in \Gamma (Y, \omega _Y(\Delta _Y)^{\otimes N})$ . We now analyze the possible poles the pulled back form $G^* \omega $ on $X \times \widetilde {H_4}$ can have. First, we note that all occurring poles of $G^* \omega $ are logarithmic, as $\omega $ has only logarithmic poles. Furthermore, on the open $X \times (\widetilde {H_4} \setminus D_H)$ (which lies over $X \times H_3$ ), the form $G^* \omega $ can only have poles along $ \left \lceil {\Delta _X} \right \rceil \times \widetilde {H_4}$ , and these poles have orders bounded by the coefficients of $N\Delta _X$ by definition of $H_3$ . It follows that $G^* \omega $ defines a global section of $S^N \Omega ^n_{(X, \Delta _X) \times (\widetilde {H_4}, D_H)}$ . Consequently, $\pi (G^*\omega )$ is a global section of $\pi _X^* S^N \Omega ^n_{(X, \Delta _X)}$ . Global sections of $\pi _X^* S^N \Omega ^n_{(X, \Delta _X)}$ only have poles along subsets of $ \left \lceil {\Delta _X} \right \rceil \times \widetilde {H_4}$ , with these poles being logarithmic and of order bounded by the coefficients of $N\Delta _X$ . In particular, they can be pulled back along $\iota _h$ , and then give global sections of $S^N \Omega ^n_{(X, \Delta _X)} = \omega _X(\Delta _X)^{\otimes N}$ . In particular, we have $\iota _h^* \pi (G^*\omega ) \in V_X$ , as required.

On the open subset $\widetilde {H_4} \setminus D_H$ , which consists of the elements lying over $H_4$ , the morphism $\Psi $ has the simpler description $\Psi (h) = \iota _h^*G^*\omega $ . The two descriptions agree by Lemma 4.7. As elements of $H_4$ represent dominant rational maps , we could even slightly abuse notation and write $\Psi (h) = h^*\omega $ .

Let $h_1$ and $h_2$ be elements of $\widetilde {H_4}$ lying over $H_4$ such that $\Psi (h_1) = \Psi (h_2)$ . Since $\Psi (h_1) \colon V_Y \to V_X$ is the injective map $\omega \mapsto (G\circ \iota _{h_1})^\ast \omega $ and $\Psi (h_2) \colon V_Y\to V_X$ is the injective map $\omega \mapsto (G\circ \iota _{h_2})^\ast \omega $ , it follows from Lemma 5.1 that the dominant near-maps $G\circ \iota _{h_1}$ and $G\circ \iota _{h_2}$ are equal. This implies that $h_1$ and $h_2$ lie over the same element of $H_4$ (via the alteration $\widetilde {H_4}\to \overline {H_4}$ ).

However, since $\widetilde {H_4}$ is a projective variety and $H^\vee $ is affine, the morphism $\Psi $ is constant. Since $\Psi $ separates elements lying over distinct points of $H_4$ (see previous paragraph), we conclude that $H_4$ is a singleton. This concludes the proof.

Proof. Let $U \subseteq X \setminus \operatorname {\mathrm {supp}} \Delta _X$ be a smooth open subset. Let $V \to U$ be an alteration with a smooth compactification $V \subseteq \overline {V}$ (see [Reference de JongdJ96, Theorem 4.1]). Shrinking U if necessary, we may assume that $V \to U$ is quasi-finite and dominant. Then, by Lemma 2.5, the set of separably dominant near-maps injects into the set of separably dominant near-maps . Shrinking V if necessary, we may assume that the boundary $\overline {V} \setminus V$ is a (reduced) divisor $D \subset \overline {V}$ . Then, the set of separably dominant near-maps equals the set of separably dominant near-maps . The latter is finite by Proposition 5.2.

Proof. We may assume that the base field k is uncountable. As before, we may replace $(X, \Delta _X)$ by a dense open subset of $X \setminus \operatorname {\mathrm {supp}} \Delta _X$ , so we may assume that X is smooth and quasi-projective and that $\Delta _X$ is empty.

We argue by contradiction. Assume that is an infinite sequence of pairwise distinct separably dominant near-maps. If $\dim (X)> \dim (Y)$ , we will construct an $(\dim (X)-1)$ -dimensional subvariety $Z \subset X$ which still admits infinitely many pairwise distinct separably dominant orbifold near-maps to $(Y, \Delta _Y)$ . By iterating this process, we obtain a contradiction to Corollary 5.3. As the case $\dim (Y)=0$ is trivial, we may assume $\dim (X) \geq 2$ .

Let $X \subseteq \mathbb {P}^n$ be an immersion, and let $Z \subset X$ be a very general hyperplane section. Then Z is a smooth variety. Since Z is not contained in the indeterminacy locus of any $f_i$ , all of the $f_i$ induce rational maps . Moreover, being very general, Z does not contain any irreducible component of the indeterminacy locus of any $f_i$ . This implies that all are defined in codimension $1$ . For fixed $i, j$ , the locus where $f_i$ and $f_j$ are equal is contained in a proper closed subset of X, so that $f_i$ and $f_j$ are distinct after restricting to a generic hyperplane section. Thus, the near-maps $(f_i\vert _Z)_{i \in \mathbb {N}}$ are pairwise distinct. By the implication $(a) \implies (b)$ of Lemma 3.3, for every $i \in \mathbb {N}$ , there is a point $x \in X$ such that $df_{i,x} \colon \textrm {T}_x X \to \textrm {T}_{f_i(x)} Y$ is surjective. As this is an open condition on x, we see that the map on tangent spaces is surjective at a general point of X. Moreover, if $V \subset \textrm {T}_x X$ is a generic subspace of codimension $1$ , the composed map $V \to \textrm {T}_{f_i(x)} Y$ will still be surjective, as $\dim \textrm {T}_x X = \dim X> \dim Y = \dim \textrm {T}_{f_i(x)} Y$ . Consequently, we see that the restricted maps still have a surjective differential map at some point. Hence, they are separably dominant by the implication $(b) \implies (a)$ of Lemma 3.3. By Corollary 2.7, the separably dominant near-maps are orbifold near-maps. This concludes the proof.

Proof. Let $f \colon (X, \Delta ) \to (X, \Delta )$ be a separably dominant orbifold morphism. Let $f^n$ be the n-fold composition of f. Since every $f^n$ is a separably dominant orbifold morphism, by the finiteness of the set of separably dominant orbifold morphisms $(X,\Delta )\to (X,\Delta )$ , there are distinct positive integers m and n with $m>n$ such that $f^m = f^n$ . As f is dominant, it follows that $f^{m-n} = \operatorname {\mathrm {id}}_X$ , so that f is an automorphism of finite order. The finiteness of the automorphism group follows immediately from Proposition 5.2.