Proof. We prove p is firm along f if and only if such a diagram (2) exists with firm along $W \to S'$ . Parts (i)-(iv) then follow directly. By [Sta25, Tag 0487], all geometric points $\bar s \to S$ lift to $S'$ . Consider the map between the fibers

$$\begin{align*}W_{\bar{s}} \to X_{\bar{s}}.\end{align*}$$

As the identity is firm along $W \to S'$ , there are a strict geometric point $\bar w \to W_{\bar s}$ and a dashed retraction fitting in the sequence

$$\begin{align*}\bar M_{S, \bar s} \to \bar M_{X\times_Y^{\mathsf{fs}} S, \bar w} \to \bar M_{W, \bar w} \dashrightarrow \bar M_{S, \bar s}. \end{align*}$$

Such a retraction witnesses firmness of $\bar s \to S' \to S \to Y$ along $X \to Y$ . The converse implication holds by properties of the fiber product.

Part (vi) follows from the universal property of the pushout and the fact that $\bar M_{X\times _Y^{\mathsf {fs}}S,x}$ is the pushout of $\bar M_{Y,y}\to \bar M_{S,s}$ and $\bar M_{Y,y}\to \bar M_{X,x}$ . For (vii), let $\overline s'\to S'$ be a geometric point and $\overline s\to S$ be the induced point. Then firmness of p along f gives

The dashed arrow exists because the solid square is a pushout coming from the fiber product of fs log schemes. The proof of (viii) is similar. For (ix) it suffices to argue that if a composite $\bar s \to \bar t \to S$ is firm, so is $\bar t \to S$ . For this, one can use the same geometric point $\bar w \to X_{\bar s} \to X_{\bar t}$ for the requisite splitting in each case.

Proof. The strict étale site of an Artin cone $\Theta _{P}$ is trivial in that every strict étale cover admits a section [Reference Abramovich and WiseAW18, Corollary 2.2.8]. Pull back the cover of $\mathscr {C}$ and find a section of the cover of $\Theta _{P}$ .

Proof of Proposition 2.2

By definition, firmness is equivalent to finding a geometric point $\bar x \to X$ and a retraction

$$\begin{align*}\bar M_Y \to \bar M_{X, \bar x} \dashrightarrow \bar M_Y. \end{align*}$$

We can replace X by a strict étale cover using Lemma 2.1, in particular by a disjoint union of atomics $X = \bigsqcup X_i$ [Reference Molcho and WiseMW22, Proposition 2.2.2.5]. The Artin fan of X is then a disjoint union of the Artin cones of each connected component $\Theta _{X_i} = \Theta _{P_{i}}$ . Write $Q = \Gamma (Y, \bar M_Y)$ .

Sections of the map $\Theta _{X} = \bigsqcup \Theta _{P_{i}} \to \Theta _{Y} = \Theta _{Q}$ are the disjoint union of the sections of each map $\Theta _{P_{i}} \to \Theta _{Y}$ as $\Theta _{Y}$ is a connected Artin cone. A section $\Theta _{Y} \dashrightarrow \Theta _{P_{i}}$ is dual to a morphism

$$\begin{align*}P_i \to Q\end{align*}$$

such that the composite $Q \to P_i \to Q$ is the identity. As $\Theta _{P_{i}}$ is the Artin fan of $X_i$ , there exists a geometric point $\bar x \to X_i$ such that $\bar M_{X_i, \bar x} = P_i$ . This geometric point $\bar x$ necessarily maps to Y, so the proof is complete.

Proof. The question is local in X, so assume X is atomic. Then $\Theta _{X} = \Theta _{P}$ is an Artin cone. We can assume $X \neq \varnothing $ , so $Z \neq \varnothing $ and Z must contain the origin $v \in \Theta _{P}$ of Remark 2.5.

The point $v \in \Theta _{P}$ is always in the image of $X \to \Theta _{X}$ as in Remark 2.5. We assumed the set-theoretic image $\mathrm {Im}(X \to Z)$ is an open and it contains v. The only open of $\Theta _{P}$ containing v is the whole stack $\Theta _{P}$ , so $\mathrm {Im}(X \to Z)$ must likewise be Z.

Proof of Theorem 1.3

A log lift clearly implies firmness. For the converse, f.s. base change to assume $s = Y$ . Étale localize to assume the map $f : X \to Y$ has a compatible map on Artin fans $\Theta _{X} \to \Theta _{Y}$ by Lemma A.19. Remark 2.5 shows the map $Y \to \Theta _{Y}$ factors through the origin $v \in \Theta _{Y}$ . Write $v'$ for the pullback of $v \in \Theta _{Y}$ to $\Theta _{X}$ :

The map $X \to v'$ is log flat and strict, hence flat. It is also finitely presented, so open [Sta25, Tag 01UA]. Lemma 2.6 ensures $X \to v'$ is surjective.

The point is f-firm if and only if there is a section $\Theta _{Y} \dashrightarrow \Theta _{X}$ by Proposition 2.2. Then by the universal property of the fiber product there is a logarithmic section

To come up with a logarithmic lift, it remains to find a section

or equivalently a section of the morphism of algebraic spaces with log structures $X \times _{v'} s \to s$ . This morphism is strict, surjective, and finitely presented, so finding such a section is equivalent to finding a lift on the level of underlying algebraic spaces. This is possible after a finite field extension $s' \to s$ [Sta25, Tag 0487].

If f was further assumed log smooth, then $X \to v'$ is smooth. Firmness ensures a lift $s \to v'$ as above. Log lifts $s \dashrightarrow X$ correspond to k-points of the variety $X' = X \times _{v'} s$ . Because $X'$ is smooth and nonempty, it has a $k'$ -point for a finite separable extension $k'/k$ .

Proof. We can assume S is affine, and then $\widehat {S}$ is a cofiltered limit of affine log schemes given by the strict étale neighborhoods $\bar s \to U \to S$ [Sta25, Tag 04HX]. We are reduced to [Reference Molcho and WiseMW22, Lemma 2.2.3.4], which states that maps from cofiltered systems $\{S_i\}$ of affine log schemes to locally finite presentation morphisms $X \to Y$ are compatible with limits/colimits:

$$\begin{align*}{\mathrm{colim}}_i \mathrm{Hom}_{\mathrm{log sch}/Y}(S_i, X) = \mathrm{Hom}_{\mathrm{log sch}/Y}(\lim S_i, X). \end{align*}$$

Proof. Replace S by the strict henselization at $s \to S$ , to which all generizations lift. Assume $Y = S$ by Lemma 2.1(iii). As $s \to Y$ is f-firm, there are a strict log geometric point $x \to X = X \times _Y^{\mathrm { fs}} s$ and a factorization

$$\begin{align*}\bar M_{S, s} \to \bar M_{X, x} \to \bar M_{S, s} \end{align*}$$

of the identity map $id_{\bar M_{S, s}}$ . We can replace X by its strict henselization at x as well. Write

$$\begin{align*}P := \bar M_{S, s}, \qquad Q := \bar M_{X, x},\end{align*}$$

and $Q'$ for the sharpening of the factorization $Q \to P$ , so that there is a factorization

$$\begin{align*}P \to Q \to Q' \to P \end{align*}$$

of the identity. Locally near x and s, the morphism $X \to S$ is charted by the monoid map $P \to Q$ , that is, there exist strict étale neighborhoods U of x in X and V of s in S and a commutative square

with strict horizontal maps. Since X and S are local and strictly henselian with closed points $x,s$ , we may pick $U=X$ and $V=S$ . Write $W := S \times _{{\mathbb {A}}_P} {\mathbb {A}}_Q$ and expand the above chart:

By [Reference OgusOgu18, Theorem IV.4.1.7], the map $X \to W$ is flat.

Write

$$\begin{align*}v \in {\mathbb{A}}_Q(\mathrm{{Spec}\:} \mathbb{Z}), \qquad v' \in {\mathbb{A}}_{Q'}(\mathrm{{Spec}\:} \mathbb{Z}), \qquad w \in {\mathbb{A}}_P(\mathrm{{Spec}\:} \mathbb{Z})\end{align*}$$

for the closed points, with underlying schemes $ {\mathrm{Spec}} \mathbb {Z}$ . These are the origins of Remark 2.5. The point $(v, s) \in {\mathbb {A}}_Q \times s$ is a specialization of the point $(v', s) \in {\mathbb {A}}_{Q'} \times s \subseteq {\mathbb {A}}_Q \times s$ . Both points map to the image of $s \in {\mathbb {A}}_P \times S$ , so they lie in the closed subscheme $W_s = W \times _S s$ . The point $x \to X$ maps to $(v, s) \in {\mathbb {A}}_Q$ . We can lift the specialization $(v', s) \rightsquigarrow (v, s) \in W_s$ to a specialization $y \rightsquigarrow x$ in X by flatness of $X \to W$ [Sta25, Tag 03HV].

Replace X by its strict henselization at $y \in X$ and x by y. The point $s \to S$ remains f-firm, as the same section

$$\begin{align*}P \to Q \to Q' \to P\end{align*}$$

witnesses firmness of the point $y \in X$ with characteristic monoid $\bar M_{X, y} = Q'$ . After replacing X, our retraction $Q \to P$ is sharp and we have $Q = Q'$ and $v = v'$ .

Continue to use notation as above. Consider any geometric point $t \to S$ , necessarily a generization of the closed point $s \to S$ . The image $t'$ of the composite $t \to {\mathbb {A}}_P \to {\mathbb {A}}_Q$ with the section is a generization of the unique closed “point” $v \in {\mathbb {A}}_Q$ . So we have a specialization $(t', t) \rightsquigarrow (v, s)$ in ${\mathbb {A}}_Q \times S$ . This specialization lies inside the closed subscheme W. As $x \to X$ maps to $(v, s) \in {\mathbb {A}}_Q$ , we can again lift this specialization $(t', t) \rightsquigarrow (v, s)$ to a specialization $z \rightsquigarrow x$ in X by flatness of $X \to W$ [Sta25, Tag 03HV].

We claim $z \to X$ witnesses the firmness of $t \to S$ . Write

$$\begin{align*}\bar P := \bar M_{S, t}, \qquad \bar Q := \bar M_{X, z}.\end{align*}$$

As $t' \in {\mathbb {A}}_Q$ was the image of t under the section ${\mathbb {A}}_P \to {\mathbb {A}}_Q$ , we have a commutative diagram

It follows that the bottom row composes to the identity, so $t \to S$ is firm.

Proof. We can assume $\mathscr {B} = \sigma $ is an Artin cone because Lemma 2.4 guarantees maps $\tau \to \mathscr {B}$ factor through Artin cones $\sigma \to \mathscr {B}$ in a strict étale cover. Write $P = \Gamma (\tau , \bar M_\tau ), Q = \Gamma (\sigma , \bar M_\sigma )$ so that $\sigma \to \tau $ comes from a map of sharp monoids $\theta : P \to Q$ . The map $\theta $ is sharp if and only if the map $\sigma \to \tau $ does not factor through $\partial \tau \subseteq \tau $ .

Assuming $\theta $ is sharp, we need to show p admits a section by producing a map $Q \dashrightarrow P$ such that the composite $P \to Q \to P$ is the identity. Note that $P, Q$ are “toric” monoids as they are sharp, f.s., with free associated groups $P^{\mathrm {gp}}, Q^{\mathrm {gp}}$ and that the integral map $P \to Q$ is locally exact by [Reference OgusOgu18, Theorem I.4.7.7]. So Theorem I.4.8.14 of [Reference OgusOgu18] provides an isomorphism $P \times (Q \setminus \sqrt {K_\theta }) \simeq Q$ for a certain monoid ideal $\sqrt {K_\theta } \subseteq Q$ . Projecting away from this ideal gives us our retraction $Q \dashrightarrow P$ .

Proof of Theorem 1.4

Consider a strict log geometric point $\bar {y} \to Y$ . Replace Y by its strict henselization at $\bar {y}$ and X by a strict étale cover by a disjoint union of atomics using Lemma 2.1.

Let be the (f.s. and scheme-theoretic) fiber over $\bar {y}$ . There are compatible maps of Artin fans

by Lemmas A.18, A.19 because $X, Y$ are disjoint unions of atomics and $X' \to X, \bar {y} \to Y$ are strict. Because the rest of the morphisms in the square are compatible with maps of log schemes, there is a commutative square

The point $\bar {y} \to Y$ is in the set-theoretic image of f if and only if $X'$ is nonempty. In that case, the map $\pi : \Theta _{X'} \to \Theta _{\bar {y}}$ admits a section if and only if $\bar {y} \to Y$ is f-firm by Proposition 2.2. It is clear that firm points are in the set-theoretic image of f. So it suffices to show that if $\pi $ does not admit a section, then $X'$ is empty.

By Lemma 3.1, $\pi $ factors through $\partial \Theta _{\bar {y}} \subseteq \Theta _{\bar {y}}$ . Then we have a commutative rectangle

where $v \in \Theta _{\bar {y}}$ is the closed point as in Remark 2.5. But $\partial \Theta _{\bar {y}}$ is the complement $\Theta _{\bar {y}} \setminus v$ , so $X' = \varnothing $ .

Proof. Suppose given a map $S \to Y'$ . The composite $S \to Y' \to Y$ factors through $Y(f)$ if and only if, for all log geometric points $\bar s \to S$ , the identity is firm along the pullback $X_{\bar s} \to \bar s$ by Lemma 2.1(iii)(iv). But this is the same as being firm along the pullback $X^{\prime }_{\bar s}=X_{\bar s} \to \bar s$ , so S equivalently factors through $Y'(f')$ .

Proof. F. Kato’s results [Reference KatoKat22, Theorem 1.1, Corollary 1.2] show that such a base change is possible after passing to a strict étale neighborhood of any point of Y. Instead of a log alteration, he takes a finite Kummer map. Molcho [Reference MolchoMol21, Theorem 1.0.1] instead provides a log alteration. To conclude, we need to find a global log blowup/alteration $Y' \to Y$ refining any local one.

One can define a special family $Y^{\prime }_n \to Y$ of log blowups/alterations indexed by $n \in {\mathbb {N}}$ . First, take a log blowup of $\Theta _{Y}$ to assume its cones are free as in the proof of [Reference Kempf, Knudsen, Mumford and Saint-DonatKKMSD73, Chapter 1, §2, Theorem 11]. Subdivide further to also assume $\Theta _{Y}$ is without monodromy in the sense that it admits a Zariski cover by free Artin cones $\Theta ^r = \left [{{\mathbb {A}}^1/\mathbb {G}_m}\right ]^r$ .

For each cone $\sigma = \Theta ^r$ of $\Theta _{Y}$ , we define a system of log blowups and log alterations and leave the reader to check that they glue along faces. See Figure 3.

Figure 3

The sequence $\Sigma _n$ of log blowups of $\Theta ^2$ for $n = 1, 2, 3$ . This sequence eventually refines all log blowups. The log alteration $\Sigma ^{\prime }_n$ also adds in an increasing sequence of root stacks to refine all log alterations.

Figure 3 Long description

The figure consists of three panels arranged horizontally, each showing a first-quadrant coordinate system with a vertical and horizontal axis.

* The first panel on the left shows the most refined state. It contains nine vectors originating from the bottom-left corner. Moving clockwise from the vertical axis, the vectors alternate in color between blue and black. There are five black vectors and four blue vectors interspersed between them, creating a dense fan of rays.

* The second panel in the middle shows a less refined state. It contains five vectors originating from the origin. Moving clockwise from the vertical axis, the sequence is blue, black, blue, black, and blue. The spacing between these vectors is wider than in the first panel.

* The third panel on the right shows the simplest state. It contains only one blue vector originating from the origin, bisecting the angle between the vertical and horizontal axes at exactly 45 degrees.

A log alteration $\mathscr {B} \to \sigma $ factors as

$$\begin{align*}\mathscr{B} \to \mathscr{B}' \to \sigma \end{align*}$$

with $\mathscr {B} \to \mathscr {B}'$ a root stack and $\mathscr {B}' \to \sigma $ a log modification. By quasicompactness, there are a natural number $N \in {\mathbb {N}}$ and a factorization

$$\begin{align*}[N] : \mathscr{B}' \dashrightarrow \mathscr{B} \to \mathscr{B}' \end{align*}$$

of the multiplication by N map $[N]$ on $\mathscr {B}'$ . The log modification $\mathscr {B}' \to \sigma $ is refined by a log blowup, which can be refined by a stellar subdivision. So it suffices to construct a family of subdivisions that refines all stellar subdivisions and all roots to finite order $N \in {\mathbb {N}}$ .

Write $e_1, \cdots , e_r \in {\mathbb {N}}^r$ for the standard basis vectors. For each set of integers $a_1, \cdots , a_r \in \{0, 1, 2, \cdots , n\}$ bounded by n, we get a map ${\mathbb {N}}^r \to {\mathbb {N}}$ defined by the $(1 \times r)$ -matrix $[a_1 \cdots a_r]$ .

Consider the star subdivision of $\sigma $ at the dual map of Artin cones $\Theta \to \Theta ^r$ . If we perform star subdivision at a finite set of vectors $\Theta \to \Theta ^r$ , the result depends on the order of the vectors. Nevertheless, there is a unique minimal subdivision $\Sigma _n \to \Theta ^r$ refining each of these star subdivisions given by taking their intersection as subfunctors of $\Theta ^r$ depicted in Figure 4.

Let $\Sigma ^{\prime }_n \to \Sigma _n$ be the root stack given by rescaling the lattice by $n!$ . For $n \gg 0$ , $\Sigma _n \to \sigma $ refines any log blowup and $\Sigma ^{\prime }_n \to \sigma $ refines any log alteration. As we assumed $\Theta _{Y}$ was free, we get an induced log alteration of $\Theta _{Y}$ which pulls back to define $Y^{\prime }_n \to Y$ . Check that it refines every possible log alteration of Y or any $Y_0 \to Y$ .

Proof. The claim is Zariski-local in Y, so we can assume Y is affine and $T \subseteq Y$ is the intersection $U \cap D$ of an open U and a closed subset D of $|Y|$ . It suffices to handle the cases $T = U$ and $T = D$ separately, and $T = U$ is an open subscheme of Y. The functor $T = D$ is representable by the formal spectrum $\mathrm {{Spf}\:} (\mathcal {O}_Y)^\wedge _{I_D}$ of the completion of $\mathcal {O}_Y$ at the ideal $I_D$ of D, an ind-scheme. See [Sta25, Tag 0AIZ] for more details.

Proof. Lemma 3.3 provides a log alteration $Y' \to Y$ such that the pullback

of f is integral and saturated. By Lemma 3.2, the firm locus pulls back:

$$\begin{align*}Y(f) \times_Y^{\mathrm{fs}} Y' = Y'(f').\end{align*}$$

Replace f by the integral, saturated $f'$ . Theorem 1.4 identifies the set-theoretic image $f(X) \subseteq Y$ as the firm locus. By Chevalley’s theorem [Sta25, Tag 054K], $f(X)$ is locally constructible. If f is assumed log flat, it is flat (since it is also integral) by [Reference OgusOgu18, Theorem IV.4.3.5] and $f(X) \subseteq Y$ is open [Sta25, Tag 01UA].

Proof. Let $S \to Y$ be a map firm along f. The problem is local in $S, Y, X$ . Assume S is strictly henselian by Lemma 2.7 and then base change to assume $Y = S$ . Replace X by a strict étale cover to assume it is a disjoint union of atomics.

As Y is strictly henselian, we have a (f.s. and ordinary) pullback square

with $\mathscr {B} \to \mathscr {C}$ a log alteration of Artin fans by Definition A.22. We need to show that $S \to Y$ factors through $Y'$ .

The Artin fan is then the same as that of its closed point $\bar s \to S$ as in Example A.16:

$$\begin{align*}\Theta_{S} = \Theta_{\bar s}.\end{align*}$$

By Lemma 2.1(iv), S being firm implies that the log geometric point $\bar s \to S \to Y$ is firm. By Proposition 2.2, the log geometric point $\bar s \to S \to Y$ is firm if and only if the map $\Theta _{X_{\bar s}} \to \Theta _{\bar s}$ admits a section. We may have $\Theta _{X_{\bar s}} \neq \Theta _{X}$ as in Example 2.3. The strict map $X_{\bar s} \to X$ nevertheless induces $\Theta _{X_{\bar s}} \to \Theta _{X}$ by Lemma A.18 and the factorization through $\Theta _{X_{\bar s}}$ induces a dashed arrow

where the map $\Theta _{Y} \to \mathscr {C}$ exists because $Y \to \mathscr {C}$ is strict, hence factors via $\Theta _{Y}$ . But then $S \to Y \to \mathscr {C}$ factors through $\mathscr {B}$ and so $S \to Y$ factors through $Y' \to Y$ .

Proof. Any finitely presented, flat morphism $W \to Z$ with reduced geometric fibers admits an initial factorization through an étale Z-algebraic space by [Reference RomagnyRom11, Theorem 2.5.2]. The morphisms $X_{\bar {y}} \to \mathsf {{Log}}_{\bar {y}}$ , $X \to \mathsf {{Log}}_Y$ meet these criteria. Write $\mathscr {B}, \mathscr {C}$ for their initial factorizations, which are the relative Artin fans [Reference Herr, Molcho, Pandharipande and WiseHMPW27] of $X_{\bar {y}} \to \bar {y}$ and $X \to Y$ .

The maps $X_{\bar {y}} \to \mathscr {B}$ , $X \to \mathscr {C}$ have connected geometric fibers by loc. cit. We have a diagram

where the dashed arrow exists by the initialness of $\mathscr {B}$ and the fact that $\mathscr {C}_{\bar {y}} := \mathscr {C} \times _{\mathsf {{Log}}_Y} \mathsf {{Log}}_{\bar {y}}$ is étale and representable over $\mathsf {{Log}}_{\bar {y}}$ .

The morphisms $X_{\bar {y}} \to \mathscr {B}$ and $X \to \mathscr {C}$ are flat of finite presentation since $X_{\bar {y}} \to \mathsf {{Log}}_{\bar {y}}$ and $X \to \mathsf {{Log}}_Y$ are. In particular, they are open and we may write $U \subseteq \mathscr {B}, W \subseteq \mathscr {C}$ for their open, nonempty images. Since $X \to \mathsf {{Log}}_{\bar {y}}$ factors through the étale map $U \to \mathsf {{Log}}_{\bar {y}}$ , we have $U=\mathscr {B}$ by the initialness of $\mathscr {B}$ , so $X_{\bar {y}} \to \mathscr {B}$ is surjective. Likewise, $X \to \mathscr {C}$ is surjective. The maps

$$\begin{align*}X_{\bar{y}} \to \mathscr{B}, \qquad X_{\bar{y}} \to \mathscr{B} \to \mathscr{C}_{\bar{y}}\end{align*}$$

are surjective morphisms with connected geometric fibers. By [Reference Hablicsek, Herr and LeonardiHHL26, Proposition 3.13], the map $\mathscr {B} \to \mathscr {C}_{\bar {y}}$ is then an isomorphism.