Proof. By a result of Zariski [Reference Kollár and Mori21, Lemma 2.45], there is a finite sequence of birational proper morphisms

$$ \begin{align*}X_n\overset{f_n}{\longrightarrow} X_{n-1} \overset{f_{n-1}}{\longrightarrow} \dots \overset{f_1}{\longrightarrow} X_0=X\end{align*} $$

such that each $f_i$ is the blow-up of $X_{i-1}$ along the center $\mathfrak {c}_{i-1}$ of G on $X_{i-1}$ , and G appears on $X_n$ as an $f_n$ -exceptional divisor dominating $\mathfrak {c}_{n-1}$ . To prove the statement we may localize each $X_i$ in a neighbourhood of the generic point of $\mathfrak {c}_{i}$ , and so we can assume that each $\mathfrak {c}_i$ is a regular subvariety and that each $X_i$ is regular. Let $E_i\subset X_i$ be the unique $f_i$ -exceptional divisor: since $E_i=f_i^{-1}(\mathfrak {c}_{i-1})$ it holds that $\mathfrak {c}_i\subseteq E_i$ for each i, with equality if and only if $i=n$ . By abuse of notation, the strict transform of $E_i$ on each $X_{i+1},\dots ,X_n$ will also be denoted by $E_i$ .

Let $f\colon X_n\to X$ be the composition of all blow-ups. We write

$$ \begin{align*}f^*K_{X}=K_{X_n}-\sum_{i=1}^{n}a_iE_i.\end{align*} $$

We claim that $a_{i+1}>a_i$ for all i: since $a_1=\operatorname {\mathrm {codim}}(\mathfrak {c})-1$ and $E_n=G$ , the result will follow. These inequalities are easily proved by induction: we have

$$ \begin{align*}f_i^*\left(K_{X_{i-1}}-\sum_{j=1}^{i-1}a_jE_j\right) = K_{X_i}-\sum_{j=1}^{i-1}a_jE_j -\left(\operatorname{\mathrm{codim}}_{X_{i-1}}(\mathfrak{c}_{i-1})-1 +\operatorname{\mathrm{mult}}_{\mathfrak{c}_{i-1}}\sum_{j}a_jE_j\right) \cdot E_i.\end{align*} $$

Since $\mathfrak {c}_{i-1}\subsetneq E_{i-1}$ we have

$$ \begin{align*}a_i=\operatorname{\mathrm{codim}}_{X_{i-1}}(\mathfrak{c}_{i-1}) -1+\operatorname{\mathrm{mult}}_{\mathfrak{c}_{i-1}}\sum_{j}a_jE_j \geq 1+a_{i-1}.\end{align*} $$

This proves the lemma.

Proof. This is proved in [Reference Posva27, Theorem 4.2.5], under the additional assumption that $\Delta $ is effective. However, that assumption is not needed in the proof.

Proof. This can be checked Zariski-locally on $X, B$ , and $B'$ , so we may assume that $X=\operatorname {\mathrm {Spec}}(R)$ , $B=\operatorname {\mathrm {Spec}}(A)$ , $B'=\operatorname {\mathrm {Spec}}(A')$ and that $\mathcal {F}=\tilde {F}$ is generated by $\partial _1,\dots ,\partial _n\in \operatorname {\mathrm {Der}}_A(R)$ . Let $R'=R\otimes _A A'$ . We have a natural map

$$ \begin{align*}\mathfrak{g}'\colon \operatorname{\mathrm{Der}}_R(A)\to \operatorname{\mathrm{Der}}_{A'}(R'), \quad \mathfrak{g}'(\psi)\colon r\otimes a'\mapsto \psi(r)\otimes a',\end{align*} $$

and the $A'$ -sub-module $F'$ of $\operatorname {\mathrm {Der}}_{A'}(R')$ generated by $\mathfrak {g}'(F)$ sheafifies into $\mathcal {F}'$ . We check below that $F'$ is closed under Lie brackets and p-powers. As the $\partial _i'=\mathfrak {g}'(\partial _i)$ generate $F'$ we see that

$$ \begin{align*}(R')^{F'}= \ker\left((\partial_1',\dots,\partial_n')\colon R'\to (R')^{\oplus n}\right).\end{align*} $$

Similarly we have

$$ \begin{align*}R^F=\ker\left((\partial_1,\dots,\partial_n)\colon R\to R^{\oplus n}\right).\end{align*} $$

Since $A'$ is flat over A we therefore obtain that $R^F\otimes _A A'=(R')^{F'}$ , and the lemma follows.

It remains to check that $F'$ is closed under brackets and p-powers. Let $\xi ,\psi \in \mathfrak {g}'(F)$ and $r,s\in R'$ . As $\mathfrak {g}'$ commutes with Lie brackets we have $[r\xi , s\psi ]=rs\cdot [\xi ,\psi ]\in F'$ . By Hochschild’s formula [Reference Posva27, (2.3.0.a)] we see that $(r\xi )^{[p]}\in F'$ . Finally, a formula of Jacobson [Reference Jacobson13, p. 209] shows that $(r\xi +s\psi )^{[p]}$ and $(r\xi )^{[p]}+(s\psi )^{[p]}$ differ by an $R'$ -linear combination of multifold Lie brackets of $r\xi $ and $s\psi $ . Thus $(r\xi +s\psi )^{[p]}\in F'$ , and we are done.

Claim 4.1. $\mathcal {F}_m$ is a family of $1$ -foliations over $\mathbb {A}^1_t$ (Definition 2.11) if and only if $n\geq 2$ .

Proof. Clearly $\partial _m\in T_{(\mathbb {A}\times \mathbb {A}^1)/\mathbb {A}^1}$ , and we see that $\partial _m^{[p]}=0$ as in Claim 3.1 (and this survives along any specialization of t). Thus $\mathcal {F}_m$ is a relative $1$ -foliation over $\mathbb {A}^1_t$ , for any $n\geq 1$ . It is easily seen that the cokernel of $\mathcal {F}_m\hookrightarrow T_{\mathbb {A}\times \mathbb {A}^1/\mathbb {A}^1}$ is torsion-free, hence flat, over $\mathbb {A}^1$ .

The restriction of $\mathcal {F}_m$ to the generic fiber of the projection $\mathbb {A}\times \mathbb {A}^1_t\to \mathbb {A}^1_t$ is the restriction to $\mathbb {A}\times \operatorname {\mathrm {Spec}}(k(t))$ of the module generated by

$$ \begin{align*}\sum_{i=1}^nx_i^2\partial_{x_i}+t^m\mu(y)\partial_y\in \operatorname{\mathrm{Der}}_{k(t)}k(t)[x_1,\dots,x_n,y],\end{align*} $$

which is clearly a $1$ -foliation for every $n\geq 1$ . A similar computation shows that $\mathcal {F}_m|_{(t=\alpha )}$ is a $1$ -foliation for every $0\neq \alpha \in k$ and $n\geq 1$ . The only condition is given by the restriction to the central fiber $(t=0)$ : indeed, $\mathcal {F}_m|_{(t=0)}$ is generated by

$$ \begin{align*}\sum_{i=1}^nx_i^2\partial_{x_i}\in \operatorname{\mathrm{Der}}_{k}k[x_1,\dots,x_n,y],\end{align*} $$

so $\mathcal {F}_m|_{(t=0)}$ is a $1$ -foliation if and only $n\geq 2$ . (For $n=1$ , observe that $x_1^2\partial _{x_1}$ does not generate a saturated sub-module of $\operatorname {\mathrm {Der}}_{k}k[x_1,\dots ,x_n,y]$ .) The statement follows.

Proof. The statement about the singular locus is clear. On the $x_1$ -patch $U_1$ the blow-up b is given by

$$ \begin{align*} b^*\colon & k[x_1,\dots,x_n,y,t][\mu(y)^{-1}]\longrightarrow k[u_1,\dots,u_n,y,s][\mu(y)^{-1}], \\& x_1\mapsto u_1, \quad x_i\mapsto u_1u_i \ (1<i\leq n), \quad y\mapsto y,\quad t\mapsto u_1s. \end{align*} $$

Therefore we have

$$ \begin{align*} b^*\partial_m|_{U_1} &= u_1\cdot \left(u_1\partial_{u_1} -\sum_{i=2}^{n} u_i\partial_{u_i} - s\partial_s\right) + \sum_{i=2}^{n} u_1^2u_i^2 \cdot \frac{1}{u_1}\partial_{u_i} + u_1^ms^m \mu(y) \cdot \partial_{y} \\&= u_1 \cdot \underbrace{\left[ u_1\partial_{u_1} +\sum_{i=2}^{n} (-u_i+u_i^2)\partial_{u_i} +u_1^{m-1}s^m\mu(y)\partial_{y} -s\partial_s \right]}_{\psi} \end{align*} $$

Since $b^*\partial _m|_{U_1}$ is p-closed, so is $\psi $ . As $\psi (u_1)=u_1$ , by Lemma 2.3 we obtain that $\psi ^{[p]}=\psi $ . Thus $b^*\mathcal {F}_m|_{U_1}=\mathcal {O}_{U_1}\cdot \psi $ is lc by Proposition 2.6. Moreover, the above computations show that E is $b^*\mathcal {F}_m$ -invariant and that $a(E;\mathcal {F}_m)=-1$ . The situation on the other $x_i$ -patches $U_i$ is similar.

On the t-patch V the blow-up b is given by

$$ \begin{align*} b^*\colon & k[x_1,\dots,x_n,y,t][\mu(y)^{-1}]\longrightarrow k[v_1,\dots,v_n,y,s][\mu(y)^{-1}], \\& x_i\mapsto sv_i \ (1\leq i\leq n), \quad y\mapsto y, \quad t\mapsto s. \end{align*} $$

Therefore we have:

$$ \begin{align*} b^*\partial_m|_V &= \sum_{i=1}^n v_i^2s \partial_{v_i} + s^m\mu(y)\partial_{y} \\&= s\cdot \underbrace{\left[ \sum_{i=1}^{n}v_i^2\partial_{v_i}+s^{m-1}\mu(y)\partial_{y} \right]}_{\xi} \end{align*} $$

Under the description $V\cong D_{\mathbb {A}^{n+1}_{v_1,\dots ,v_n,y}}(\mu (y))\times \mathbb {A}^1_s$ , we see that $\xi $ corresponds to $\partial _{m-1}$ and thus $b^*\mathcal {F}_m|_V\cong \mathcal {F}_{m-1}$ . To finish the proof, it remains to observe that $\operatorname {\mathrm {Sing}}(\xi )=V(v_1,\dots ,v_n,s)\subset V$ is in fact a closed subset of X, and that the strict transform of $(t=0)$ is disjoint from the patch V.

Proof. Since $\partial _m(t)=0$ we have $k[t]\subset k[x_1,\dots ,x_n,y,t]^{\partial _m}$ : this proves the first point. On the principal open set $D(t)\subset \mathbb {A}\times \mathbb {A}^1$ the $1$ -foliation $\mathcal {F}_m|_{D(t)}$ is regular. Thus for $0\neq \alpha \in k$ , by Theorem 2.12 we have

$$ \begin{align*}Y_\alpha=\mathbb{A}/\mathcal{F}_{m,\alpha} \quad\text{for}\quad \mathcal{F}_{m,\alpha} = \mathcal{O}_{\mathbb{A}}\cdot \left( \sum_{i=1}^{n}x_i^2\partial_{x_i} + \alpha^m\mu(\alpha)\partial_{y} \right).\end{align*} $$

As $\mathcal {F}_{m,\alpha }$ is regular for $\alpha \neq 0$ we obtain that $Y_\alpha $ is smooth [Reference Posva27, Lemma 2.5.10].

Let us prove that $Y_0$ is not $S_2$ . For this we reduce to $m=1$ . Indeed, consider the cartesian diagram

where the vertical arrows are the projections and $u\colon \mathbb {A}^1_t\to \mathbb {A}^1_{t'}$ is given by $u^*(t')=t^m$ . By Lemma 2.13 we have $v^*\mathcal {F}_1=\mathcal {F}_m$ (where $\mathcal {F}_1$ is defined as in (4.1) using $t'$ instead of t) and an $\mathbb {A}^1_t$ -isomorphism

$$ \begin{align*}(\mathbb{A}\times \mathbb{A}^1_{t'})/\mathcal{F}_1 \times_{\mathbb{A}^1_{t'}}\mathbb{A}^1_t \cong Y.\end{align*} $$

Hence it suffices to show that $Y_0$ is non- $S_2$ in the case $m=1$ . As $\mathcal {F}_1$ is a family of $1$ -foliations over $\mathbb {A}^1_t$ , by Theorem 2.12 it is sufficient to prove that

$$ \begin{align*}\mathcal{O}_{\mathbb{A}\times \mathbb{A}^1,\textbf{0}}^{\partial_1} \otimes_{k[t]} k[t]/(t) \subsetneqq \left(\mathcal{O}_{\mathbb{A},\textbf{0}}/(t)\right)^{\bar{\partial}} \quad \text{with}\quad \bar{\partial}=\sum_{i=1}^{n}x_i^2\partial_{x_i},\end{align*} $$

where $\textbf {0}=V(x_1,\dots ,x_n,y,t)$ . Clearly $\bar {\partial }(y)=0$ , so it suffices to show that there is no $f\in \mathcal {O}_{\mathbb {A}\times \mathbb {A}^1,\textbf {0}}$ such that $\partial _1(y+tf)=0$ . If there was such an f, we would have

$$ \begin{align*}0=\partial_1(y+tf)=t\mu +\sum_{i=1}^n tx_i^2\partial_{x_i}(f)+t^2\mu \partial_y(f).\end{align*} $$

Taking into account that t is not a zero-divisor in $\mathcal {O}_{\mathbb {A}\times \mathbb {A}^1,\textbf {0}}$ , we would obtain

$$ \begin{align*}-\mu = \sum_{i=1}^nx_i^2\partial_{x_i}(f)+t\mu \partial_y(f).\end{align*} $$

However, the right-hand side belongs to the prime ideal $(x_1,\dots ,x_{n},t)$ , so this equality cannot hold in the local ring $\mathcal {O}_{\mathbb {A}\times \mathbb {A}^1,\textbf {0}}$ by our choice of $\mu $ . We have reached a contradiction, and hence $Y_0$ is not $S_2$ at the image of $\textbf {0}$ .

On the other hand, I claim that $Y_0$ is (geometrically) reduced. Since Y is $S_2$ , the Cartier divisor $Y_0$ is $S_1$ and it is sufficient to show that $Y_0$ is generically reduced. Observe that $Y_0$ is irreducible. The $1$ -foliation $\mathcal {F}_m$ is regular on the open set $U=D(x_1)$ , which intersects $Y_0$ nontrivially. By Theorem 2.12 it follows that the open subset $(U/{\mathcal {F}_m}|_U)_0$ of $Y_0$ is the quotient of $U_0$ by (the restriction of) $\mathcal {F}_{m,0}$ . In particular it is normal, so $Y_0$ is generically reduced.

It remains to show that $(Y,Y_0)$ is lc (in the general case $m\geq 1$ ). For this, as in the proof of Theorem 3.2, we blow-up $\mathbb {A}\times \mathbb {A}^1$ until the pullback of $\mathcal {F}_m$ has only lc singularities. By Lemma 4.2, this produces a sequence of m smooth blow-ups

$$ \begin{align*}X_m\longrightarrow X_{m-1}\longrightarrow \cdots \longrightarrow X_1\longrightarrow W=\mathbb{A}\times\mathbb{A}^1\end{align*} $$

such that, if $b_m\colon X_m\to W$ is the composition of the blow-ups, the $1$ -foliation $b_m^*\mathcal {F}_m$ is lc. Let $E_j\subset X_m$ be the strict transform of the exceptional divisor of $X_j\to X_{j-1}$ (for $1\leq j\leq m$ ), and denote $(t=0)\subset W$ by $W_0$ . We have $a(E_1;W,W_0)=n-1$ , and using Lemma 4.2 we compute

$$ \begin{align*} a(E_2;W,W_0)=&a(E_2;X_1, (X_1\to W)^{-1}_*W_0 + (1-n)E_1)\\ =& 2n-1. \end{align*} $$

Continuing by induction, we find

$$ \begin{align*}a(E_j;W,W_0)=jn-1, \quad j=1,\dots,m.\end{align*} $$

Now let $Z=X_m/b_m^*\mathcal {F}_m$ and consider the induced commutative diagram

where the horizontal arrows are the quotient morphisms and where $\mathfrak {b}\colon Z\to Y$ is birational. Let $F_j\subset Z$ be the prime divisor which is the image of $E_j$ . Using Lemma 4.2 we get

$$ \begin{align*}a(E_j;\mathcal{F}_m)=-j, \quad j=1,\dots,m.\end{align*} $$

Now we use Theorem 2.10: since $W_0$ is $\mathcal {F}_m$ -invariant we find

$$ \begin{align*} a(F_j;Y,Y_0) =& a(E_j; W,W_0)+(p-1)\cdot a(E_j; \mathcal{F}_m) \\ =& j(n-p+1)-1. \end{align*} $$

Therefore we have the crepant equation

$$ \begin{align*}K_Z+\mathfrak{b}^{-1}_*Y_0-\sum_{j=1}^m(j(n-p+1)-1)F_j =\mathfrak{b}^*(K_Y+Y_0).\end{align*} $$

To show that $(Y,Y_0)$ is lc it suffices to show that

$$ \begin{align*}\operatorname{\mathrm{discrep}}\left(Z,\mathfrak{b}^{-1}_*Y_0-\sum_{j=1}^m(j(n-p+1)-1)F_j\right)\geq -1.\end{align*} $$

Since $b_m^*\mathcal {F}_m$ is lc, by Theorem 2.10 it suffices to show that

$$ \begin{align*}\operatorname{\mathrm{discrep}}\left(X_m,(b_m)^{-1}_*W_0-\sum_{j=1}^m(j(n-p+1)-1)E_j\right)\geq -1.\end{align*} $$

As $(X_m,(b_m^{-1})_*W_0+\sum _{j=1}^mE_j)$ is log smooth, this holds by [Reference Kollár16, Corollary 2.11] as soon as ${1-j(n-p+1)\leq 1}$ for every $j\geq 1$ . This is the case when $n+1\geq p$ . The proof is complete.

Proof. The computations in the proof of Theorem 4.3 show that the central fiber $Y_0$ of $Y\to \mathbb {A}^1_t$ is not $S_2$ along the image of $V(x_1,\dots ,x_n,t)$ . Thus Y is non- $S_3$ along that curve. To show that Y is canonical, as the open set $Y\setminus Y_0$ is regular and thus terminal, it suffices to show that $a(E;Y)\geq 0$ for every divisor E over Y with $c_Y(E)\subseteq Y_0$ . As $Y_0$ is an effective Cartier divisor and $(Y,Y_0)$ is lc we have

$$ \begin{align*}a(E;Y)=a(E;Y,Y_0)+\operatorname{\mathrm{mult}}_{E}Y_0\geq -1+1=0\end{align*} $$

and we are done.

Proof. Since E is supersingular we have $\omega ^{[p]}=0$ [Reference Katz and Mazur14, 12.4.1.3]. An argument similar to Claim 3.1 shows that $\partial _{(0)}$ is p-closed. Therefore it induces a $1$ -foliation $\mathcal {G}$ of rank $1$ on $\mathbb {P}^{N-1}\times E\times \operatorname {\mathrm {Spec}}(A)$ . We compute the expression of $\partial _{(0)}$ on the other charts. Let $X_0,\dots ,X_{N-1}$ be homogeneous coordinates of $\mathbb {P}^{N-1}$ : we may assume that $x_i=X_i/X_0$ for $1\leq i\leq N-1$ . If $u_i=X_i/X_{N-1}$ for $0\leq i<N-1$ , then we have

$$ \begin{align*} x_i&=\frac{u_i}{u_0} \ (1\leq i<N-1), \quad x_{N-1}=\frac{1}{u_0},\\u_i&=\frac{x_i}{x_{N-1}} \ (1\leq i< N-1), \quad u_0=\frac{1}{x_{N-1}}. \end{align*} $$

Therefore

$$ \begin{align*}\partial_{x_i}=u_0\partial_{u_i} \ (1\leq i<N-1),\quad \partial_{x_{N-1}}=-u_0^2\partial_{u_0}-\sum_{i=1}^{N-2}u_0u_i\partial_{u_i}\end{align*} $$

and so

(4.2) $$ \begin{align} \partial_{(0)}=\frac{-1}{u_0}\cdot \underbrace{\left[ u_0\partial_{u_0}+\sum_{i=1}^{N-2}(u_i-u_i^2)\partial_{u_i} - au_0\omega \right]}_{\partial_{(N-1)}}. \end{align} $$

This shows that $\mathcal {G}|_{D_+(X_{N-1})\times E\times \operatorname {\mathrm {Spec}}(A)}$ is generated by $\partial _{(N-1)}$ and that $K_{\mathcal {G}}^{-1}=\mathcal {G}\cong \mathcal {O}_{\mathbb {P}^{N-1}}(-1)\boxtimes \mathcal {O}_{E\times \operatorname {\mathrm {Spec}}(A)}$ . This proves the first assertion.

On $D_+(X_{N-1})\times E\times \mathbb {A}^1$ , notice that $\partial _{(N-1)}$ is p-closed and that $\partial _{(N-1)}(u_0)=u_0$ . Therefore $\partial _{(N-1)}^{[p]}=\partial _{(N-1)}$ by Lemma 2.3, which shows that $\mathcal {G}$ has only multiplicative singularities on $D_+(X_{N-1})\times E\times \mathbb {A}^1$ . The computations on the patches $D_+(X_i)\times E\times \mathbb {A}^1$ for $1\leq i<N-1$ are similar. On $D_+(X_0)\times E\times \mathbb {A}^1$ , we see that the singular locus of $\mathcal {G}$ is $\{[1:0:\dots : 0]\}\times E\times V(a)$ . This proves the second assertion. In case $A=k[t]$ and $a=t^m$ , we proceed as in Claim 4.1 over each chart to see that $\mathcal {G}$ is a family of $1$ -foliations if and only if $N\geq 3$ , which holds by assumption.

Proof of Theorem 4.7

For clarity, we divide the proof into several steps. If $p=2$ we can use a slightly different construction, which we describe in Step 5.

Step 1: Construction of $\mathcal {Y}\to \mathbb {A}^1$ . Let E be a supersingular elliptic curve and $\omega \in H^0(E,T_E)$ be a global generator. If $X_0,\dots ,X_{N-1}$ are the homogeneous coordinates on $\mathbb {P}^{N-1}$ , consider on $D_+(X_0)\times E\times \mathbb {A}^1_t$ the derivation

$$ \begin{align*}\partial_{(0)}=\sum_{i=1}^{N-1}x_i^2\partial_{x_i}+t\omega, \quad x_i=X_i/X_0.\end{align*} $$

By Lemma 4.8 it induces a $1$ -foliation $\mathcal {G}$ on $W=\mathbb {P}^{N-1}\times E\times \mathbb {A}^1$ . We let $\mathcal {Y}=(\mathbb {P}^{N-1}\times E\times \mathbb {A}^1)/\mathcal {G}$ , with quotient morphism $q\colon W\to \mathcal {Y}$ . By construction the projection $W\to \mathbb {A}^1$ factors through a flat projective morphism $\mathcal {Y}\to \mathbb {A}^1$ .

Step 2: Construction of the divisor $\mathcal {H}$ . The divisor $\mathcal {H}$ is chosen as follows. Take a $\mathbb {Q}$ -divisor H on W such that:

• • $H=H_0\times \mathbb {A}^1$ for some $\mathbb {Q}$ -divisor $H_0$ on $\mathbb {P}^{N-1}\times E$ ;

• • $K_W+(p-1)K_{\mathcal {G}}+H$ is ample over $\mathbb {A}^1$ ,

• • $(W,W_\alpha +H)$ is lc for every $\alpha \in \mathbb {A}^1\setminus \{0\}$ ,

• • the support of H lies in general position with respect to the curve $\{[1:0:\dots : 0]\}\times E\times \{0\}$ , and

• • H is not $\mathcal {G}$ -invariant.

The last three properties are satisfied for a general choice of divisor of the form $H_0\times \mathbb {A}^1$ , thus by choosing $H_0$ general in a sufficiently ample linear system on $\mathbb {P}^{N-1}$ , we see there exists an H with all these properties. Notice that $K_W+(p-1)K_{\mathcal {G}}\cong \mathcal {O}_{\mathbb {P}^{N-1}}(-N+p-1)\boxtimes \mathcal {O}_{E\times \mathbb {A}^1}$ by Lemma 4.8, so because of the second property and on the assumption $N\geq \max \{p,3\}$ we cannot take $H=0$ .

We let $\mathcal {H}\subset \mathcal {Y}$ be the prime divisor with support $q(H)$ .

Step 3: Behaviour under base-change. Let us immediately observe what happens when we base-change $\mathcal {Y}\to \mathbb {A}^1_t$ along a flat finite morphism $g\colon C\to \mathbb {A}^1_t$ . By Lemma 2.13 we have a natural isomorphism

$$ \begin{align*}\mathcal{Y}\times_{\mathbb{A}^1}C\cong \left( \mathbb{P}^{N-1}\times E\times C\right)/f^*\mathcal{G}\end{align*} $$

where $f\colon \mathbb {P}^{N-1}\times E\times C\to \mathbb {P}^{N-1}\times E\times \mathbb {A}^1$ is the base-change of g. This shows that $\mathcal {Y}_C$ is normal. Observe that $H_C=f^*H$ still satisfies the five properties listed in Step 2. The pullback of $K_{\mathcal {Y}}+\frac {1}{p}\mathcal {H}$ along $\mathcal {Y}_C\to \mathcal {Y}$ is equal to $K_{\mathcal {Y}_C}+\frac {1}{p}\mathcal {H}_C$ : therefore the latter is ample over C as soon as $K_{\mathcal {Y}}+\frac {1}{p}\mathcal {H}$ is ample over $\mathbb {A}^1$ . By Proposition 2.9 and the non- $\mathcal {G}$ -invariance of H we have

(4.3) $$ \begin{align} q^*\left(K_{\mathcal{Y}}+\frac{1}{p}\mathcal{H}\right)=K_W+(p-1)K_{\mathcal{G}}+H \end{align} $$

which, by choice of H, is ample over $\mathbb {A}^1$ .

To complete the proof, it remains to check properties (2) and (3), and that $(\mathcal {Y}_C, \frac {1}{p}\mathcal {H}_C)\to C$ is locally stable. Properties (2) and (3) can be checked after an arbitrary surjective base-change. For local stability, by [Reference Kollár18, 2.15.5] and [Reference Hu and Zong12] it is enough to check it in case $g=F^r_{\mathbb {A}^1/k}$ is an iterate of the k-linear Frobenius morphism of $\mathbb {A}^1$ (Footnote ⁵ ).

Step 4: Local stability and singularities of fibers. To summarize, we have reduced the proof to the following: let $q'\colon W=\mathbb {P}^{N-1}\times E\times \mathbb {A}^1_t\to \mathcal {Y}'$ be the quotient by the $1$ -foliation $\mathcal {G}'$ induced by the derivation

$$ \begin{align*}\partial_{(0)}'=\sum_{i=1}^{N-1}x_i^2\partial_{x_i}+t^{p^r}\omega, \quad \text{on}\quad D_+(X_0)\times E\times \mathbb{A}^1_t.\end{align*} $$

It is endowed with a flat proper morphism $\mathcal {Y}'\to \mathbb {A}^1_t$ . Let $H'$ be a $\mathbb {Q}$ -divisor on W satisfying the properties listed in Step 2: we let $\mathcal {H}'=q(H')$ . We must show that $(\mathcal {Y}',\frac {1}{p}\mathcal {H}')\to \mathbb {A}^1$ is locally stable, that $\mathcal {Y}^{\prime }_t$ has only $\mu _p$ -quotient singularities for $t\neq 0$ , and that $\mathcal {Y}_0'$ is not $S_2$ . As we will see, everything is a consequence of Theorem 4.3 and Lemma 4.8.

Step 4.1: Singularity of the central fiber. First, let us prove that $\mathcal {Y}_0'$ is not $S_2$ along the image through $q'$ of the proper curve $\{[1:0:\dots :0]\}\times E\times \{0\}$ . Indeed, let $z=\{[1:0:\dots :0]\}\times \{e\}\times \{0\}$ where e is an arbitrary closed point of E. As $\mathcal {Y}_0'$ is a Cartier divisor in $\mathcal {Y}$ , it is equivalent to showing that $\mathcal {O}_{\mathcal {Y}',q'(z)}$ is not $S_3$ . I claim that this is étale-local over z. For if $h\colon \operatorname {\mathrm {Spec}}(\mathcal {O}_{W,z}^h)\to \operatorname {\mathrm {Spec}}(\mathcal {O}_{W,z})$ is the henselization, by [Reference Posva29, Lemma 3.2.1] we have a cartesian diagram

where g is an étale morphism. Therefore it suffices to prove that $\operatorname {\mathrm {Spec}}(\mathcal {O}_{W,z}^h)/h^*\mathcal {G}'$ is not $S_3$ . If $y\in \mathcal {O}_{E,e}^h$ is an étale coordinate at $e\in E$ , then an étale-local generator for $\mathcal {G}'$ at z is given by

$$ \begin{align*}\partial^h=\sum_{i=1}^{p}x_i^2\partial_{x_i} + t^{p^r} \mu(y) \partial_y\end{align*} $$

where $\mu (y)\in \mathcal {O}_{E,e}^h$ is a unit such that $\omega \otimes \mathcal {O}_{E,e}^h=\mu (y)\partial _y$ . Then we apply Theorem 4.3 (or rather its formal-local version at the origin, which is proved similarly) to see that the $\partial ^h$ -invariant sub-ring of $\mathcal {O}_{W,z}^h$ is not $S_3$ .

Since $\mathcal {Y}'$ is $S_2$ , the irreducible Cartier divisor $\mathcal {Y}_0'$ is nonetheless $S_1$ . As it is generically reduced by Theorem 4.3, we obtain that $\mathcal {Y}_0'$ is (geometrically) reduced.

Step 4.2: Local stability. Now we prove that $(\mathcal {Y}',\frac {1}{p}\mathcal {H}')\to \mathbb {A}^1$ is locally stable. First, let $0\neq \alpha \in k$ . As $\mathcal {G}'|_{D(t)}$ is lc and $(W,W_\alpha +H')$ is lc, by Theorem 2.10 we have that $(\mathcal {Y}',\mathcal {Y}^{\prime }_\alpha +\frac {1}{p}\mathcal {H}')$ is lc along $\mathcal {Y}_\alpha $ . Moreover, as $\mathcal {G}'|_{D(t)}$ has only multiplicative singularities by Lemma 4.8, by Theorem 2.12 we obtain that

$$ \begin{align*}\mathcal{Y}^{\prime}_\alpha= (\mathbb{P}^{N-1}\times E)/\mathcal{G}^{\prime}_\alpha\end{align*} $$

where $\mathcal {G}^{\prime }_\alpha $ is the $1$ -foliation induced by $\sum _{i=1}^{N-1}x_i^2\partial _{x_i}+\alpha ^{p^r}\omega $ . By Lemma 4.8 we see that $\mathcal {G}^{\prime }_\alpha $ has only multiplicative singularities. Therefore $\mathcal {Y}_\alpha $ has only $\mu _p$ -quotient singularities by [Reference Posva27, Proposition 4.1.1].

It remains to prove that $(\mathcal {Y}',\mathcal {Y}^{\prime }_0+\frac {1}{p}\mathcal {H}')$ is lc. To this end, we blow-up W to simplify the singularities of $\mathcal {G}'$ . By Lemma 4.8 the non-lc singularities of $\mathcal {G}'$ are contained on the chart $D_+(X_0)\times E\times \mathbb {A}^1$ , where (étale-locally on the factor E) the $1$ -foliation $\mathcal {G}'$ is generated by a derivation of the form (4.1) with $m=p^r$ . Therefore, the proofs of Lemma 4.2 and Theorem 4.3 show that if we blow-up repeatedly the locus of non-lc singularities of $\mathcal {G}'$ , we obtain a birational proper morphism

$$ \begin{align*}b\colon X=X_{p^r}\longrightarrow W\end{align*} $$

with $p^r$ exceptional divisors $E_1,\dots ,E_{p^r}\subset X$ , such that $b^*\mathcal {G}'$ is lc, $(X,b_*^{-1}W_0+\sum _j E_j)$ is log smooth, each $E_j$ is $b^*\mathcal {G}'$ -invariant, $a(E_j;\mathcal {G}')=-1$ and $a(E_j;W,W_0)=j(N-1)-1$ . By our choice of $H'$ , we also have that $(X,b_*^{-1}(W_0+H')+\sum _jE_j)$ is log smooth and

$$ \begin{align*}a(E_j;W,W_0+H')=a(E_j; W,W_0) \quad \forall j=1,\dots,p^r.\end{align*} $$

The arguments are now the same as the ones at the end of the proof of Theorem 4.3. Consider the commutative diagram

If $F_1,\dots ,F_{p^r}\subset Z$ denote the $\mathfrak {b}$ -exceptional divisors, then by Theorem 2.10 we have

$$ \begin{align*} a\left(F_j; \mathcal{Y}', \mathcal{Y}_0'+\frac{1}{p}\mathcal{H}'\right) &= a(E_j; W, W_0+H')+(p-1)\cdot a(E_j;\mathcal{G}') \\&= a(E_j; W, W_0) +(p-1)(-j) \\&= j(N-p)-1 \end{align*} $$

Hence we have the crepant equation

$$ \begin{align*}K_Z+\mathfrak{b}^{-1}_*\left(\mathcal{Y}_0'+\frac{1}{p}\mathcal{H}'\right)+\sum_{j=1}^{p^r}(1-j(N-p))F_j =\mathfrak{b}^*\left(K_{\mathcal{Y}'}+\mathcal{Y}^{\prime}_0+\frac{1}{p}\mathcal{H}'\right).\end{align*} $$

Notice that $1-j(N-p)\leq 1$ for every $j\geq 1$ is equivalent to $N\geq p$ , which holds as $N=\max \{p,3\}$ . Therefore we have that

$$ \begin{align*}\operatorname{\mathrm{discrep}}\left(X,b^{-1}_*(W_0+H)+\sum_{j=1}^{p^r}(1-j(N-p))E_j\right)\geq -1.\end{align*} $$

Hence by Theorem 2.10 we obtain that

$$ \begin{align*}\operatorname{\mathrm{discrep}}\left(Z,\mathfrak{b}^{-1}_*(\mathcal{Y}_0+\frac{1}{p}\mathcal{H})+\sum_{j=1}^{p^r}(1-j(N-p))F_j\right)\geq -1,\end{align*} $$

and therefore $(\mathcal {Y}',\mathcal {Y}^{\prime }_0+\frac {1}{p}\mathcal {H}')$ is lc.

Step 5: the case $p=2$ . It follows from Remark 4.9 that there exists a smooth blow-up $Z\to \mathbb {P}^{N-1}$ whose (reducible) center does not contain $[1:0:\dots :0]$ , and such that if $b\colon Z\times E\times \mathbb {A}^1\to W$ is the induced morphism, the birational pullback $b^*\mathcal {G}$ is regular away from the preimage of $\{[1:0: \dots : 0]\}\times E\times \{0\}$ . We take

$$ \begin{align*}\mathcal{Z}=(Z\times E\times \mathbb{A}^1)/b^*\mathcal{G}\longrightarrow \mathbb{A}^1\end{align*} $$

as the example. By [Reference Posva27, Proposition 5.2.4, Lemma 2.5.10] we have that $\mathcal {Z}_\alpha $ is regular for $\alpha \neq 0$ . The rest of the properties are proved exactly as above. This completes the proof.

Claim 4.12. For $p>2$ , we have $((1-2x^2)\partial _x)^{[p]}=2^{(5p-3)/2} (1-2x^2)\partial _x$ .

Proof. After the change of variable $u=1+\sqrt {2}x$ , we reduce to consider the p-th power of the derivation $\partial =\sqrt {2}u(2-u)\partial _u$ on $k[u]$ . By a formula of Jacobson [Reference Jacobson13, p. 209], we have

$$ \begin{align*}\partial^{[p]}=2^{p/2}\left((-u^2\partial_x)^{[p]}+(2u\partial_u)^{[p]}+S\right)\end{align*} $$

where S is a k-linear combination of $(p-1)$ -fold commutators of $-u^2\partial _x$ and $2u\partial _u$ . Now we make several observations. First, by Hochschild’s formula $\partial $ is p-closed and so $\partial ^{[p]}$ is a $k[u]$ -scaling of $\partial $ . As $(-u^2\partial _u)^{[p]}=0$ and $(2u\partial _u)^{[p]}=2^{p}u\partial _u$ , it follows that $S\neq 0$ . As

$$ \begin{align*}[u^2\partial_u,\ 2u\partial_u]=-2u^2\partial_u\end{align*} $$

we see that $S\neq 0$ can only be of the form $-\alpha u^2\partial _u$ for some $\alpha \in k^\times $ . Hence $\partial ^{[p]}=2^{p/2}(2^pu\partial _u-\alpha u^2\partial _u)$ is a $k[u]$ -scaling of $\partial =2^{1/2}(2u\partial _u-u^2\partial _u)$ , which forces $\alpha =2^{(5p-3)/2}$ .

Proof. Let us first consider the blow-up $b_1$ . We have three blow-up charts, which we denote by $U_{A}$ , $U_B$ , and $U_C$ :

• • Chart $U_A$ , given by $x\mapsto u, \ y\mapsto uv, \ z\mapsto z, \ t\mapsto us$ . Then

  $$ \begin{align*}b_1^*\partial_m|_{U_A}= u^3v^3\partial_u+(1-u^2v^4)\partial_v-u^2v^3s\partial_s+u^ms^m\partial_z\end{align*} $$
  which shows that $b_1^*\mathcal {F}_m|_{U_A}$ is a regular $1$ -foliation. Notice that $b_1^*\partial _m(u)\in (u)$ , so E is $b_1^*\mathcal {F}_m$ -invariant. Since $b_1^*\partial _m|_{U_A}$ generates $b^*\mathcal {F}_m|_{U_A}$ , without having to saturate, we have $a(E;\mathcal {F}_m)=0$ .

• • Chart $U_B$ , given by $x\mapsto uv, \ y\mapsto v, \ z\mapsto z, \ t\mapsto vs$ . Then

  $$ \begin{align*}b_1^*\partial_m|_{U_B}= (v^2-u^2)\partial_u+uv\partial_v-us\partial_s+v^ms^m\partial_z\end{align*} $$
  which is singular along the plane $u=v=0$ . These singularities are non-lc.

• • Chart $U_C$ , given by $x\mapsto us, \ y\mapsto vs, \ z\mapsto z, \ t\mapsto s$ . Then

  $$ \begin{align*}b_1^*\partial_m|_{U_C}= v^3s^2\partial_u+u\partial_v+s^m\partial_z\end{align*} $$
  which is singular along the plane $u=s=0$ . These singularities are non-lc.

We see that the locus $S\subset Y$ of singularities of $b_1^*\mathcal {F}_m$ is smooth, closed, and irreducible of codimension $2$ . As can be checked on chart $U_B$ , that singular locus is not contained in the strict transform $b_*^{-1}(t=0)$ .

Next let us blow-up S. We have two blow-up charts above $U_B$ (resp. above $U_C$ ), which we denote by $U_{BA}$ and $U_{BB}$ (resp. by $U_{CA}$ and $U_{CB}$ ):

• • Chart $U_{BA}$ , given by $u\mapsto \bar {u},\ v\mapsto \bar {u}\bar {v}, \ z\mapsto z, \ s\mapsto s$ . Then

  $$ \begin{align*}b^*\partial_m|_{U_{BA}}= \bar{u} \cdot \underbrace{\left[ \bar{u}(\bar{v}^2-1)\partial_{\bar{u}} +\bar{v}(2-\bar{v}^2)\partial_{\bar{v}} -s\partial_s + \bar{u}^{m-1}\bar{v}^ms^m\partial_z\right]}_{\psi_{BA}}.\end{align*} $$
  As $\psi _{BA}(s)=-s$ , by Lemma 2.3 we see that $\psi _{BA}^{[p]}=-\psi _{BA}$ . Thus $b^*\mathcal {F}_m$ is lc on the chart $U_{BA}$ . As $\psi _{BA}(\bar {u})\in (\bar {u})$ we see that F is $b^*\mathcal {F}_m$ -invariant. As $b^*\partial _m=\bar {u}\cdot \psi _{BA}$ we find that $a(F;\mathcal {F}_m)=-1$ .

• • Chart $U_{BB}$ , given by $u\mapsto \bar {u}\bar {v},\ v\mapsto \bar {v}, \ z\mapsto z, \ s\mapsto s$ . Then

  $$ \begin{align*}b^*\partial_m|_{U_{BB}}= \bar{v} \cdot \underbrace{\left[ (1-2\bar{u}^2)\partial_{\bar{u}} +\bar{u}\bar{v}\partial_{\bar{v}} -\bar{u}s\partial_s + \bar{v}^{m-1}s^,\partial_z\right]}_{\psi_{BB}}.\end{align*} $$
  By Claim 4.12 we see that $\psi _{BB}(u)=2^{(5p-3)/2}\psi _{BB}(u)$ . Thus by Proposition 2.6 we see that $b^*\mathcal {F}_m$ is lc on the chart $U_{BB}$ .

• • Chart $U_{CA}$ , given by $u\mapsto \bar {u},\ v\mapsto v, \ z\mapsto z, \ s\mapsto \bar {u}\bar {s}$ . Then

  $$ \begin{align*}b^*\partial_m|_{U_{CA}}= \bar{u} \cdot \underbrace{\left[ \bar{u}v^3\bar{s}^2\partial_{\bar{u}} +\partial_{v} +\bar{u}^{m-1}\bar{s}^m\partial_z - v^3\bar{s}^3\partial_{\bar{s}}\right]}_{\psi_{CA}}.\end{align*} $$
  As $\psi _{CA}$ generates a regular $1$ -foliation, we see that $b^*\mathcal {F}_m$ is regular on the chart $U_{CA}$ .

• • Chart $U_{CB}$ , given by $u\mapsto \bar {u}\bar {s},\ v\mapsto v, \ z\mapsto z, \ s\mapsto \bar {s}$ . Then

  $$ \begin{align*}b^*\partial_m|_{U_{CB}}= \bar{s} \cdot \underbrace{\left[ v^3\partial_{\bar{u}} +\bar{u}\partial_{v} +\bar{s}^{m-1}\partial_z\right]}_{\psi_{CB}}.\end{align*} $$
  Under the $\mathbb {A}^1_t$ -isomorphism $U_{CB}\cong \mathbb {A}^3_{x,y,z}\times \mathbb {A}^1_t$ given by $(\bar {u},\bar {v},z,\bar {s})\mapsto (x,y,z,t)$ , we see that $\psi _{CB}$ corresponds to $\partial _{m-1}$ , and thus $b^*\mathcal {F}_m|_{U_{CB}}$ corresponds to $\mathcal {F}_{m-1}$ .

The above computations show that the non-lc locus of $b^*\mathcal {F}_m$ is contained in $U_{CB}$ . It is elementary (but tedious) to check that this locus does not intersect the other patches $U_{CA}$ , $U_{BA}$ , and $U_{BB}$ . It remains to observe that the strict transforms of $E\subset Y$ and of $(t=0)\subset \mathbb {A}^3\times \mathbb {A}^1_t$ are disjoint from the chart $U_{CB}$ .

Proof. The first item is clear. For the remaining ones, as in the proof of Corollary 4.5 and Theorem 4.7, we reduce the proof to the following: if $Y'=(\mathbb {A}^3\times \mathbb {A}^1)/\mathcal {F}_{p^r}$ then $Y'\to \mathbb {A}^1_t$ is locally stable, has smooth fibers above $t\neq 0$ and a non- $S_2$ reduced central fiber $Y^{\prime }_0$ . The statements about the singularities of the fibers are proved as in Theorem 4.3 (and are true whenever $p>2$ ). It remains to check that $Y'\to \mathbb {A}^1$ is locally stable above $t=0$ .

As usual by now, we blow-up $W=\mathbb {A}^3\times \mathbb {A}^1_t$ . By Lemma 4.13, there is a sequence of smooth blow-ups

$$ \begin{align*}b\colon X=X_{2p^r}\longrightarrow W=\mathbb{A}^3\times\mathbb{A}^1_t\end{align*} $$

with exceptional divisors $E_i,F_i\subset X$ for $i=1,\dots ,p^r$ such that: $(X,\sum _i E_i+F_i)$ is log smooth, $b^*\mathcal {F}_{p^r}$ has only lc singularities, the $E_i$ ’s and the $F_i$ ’s are $b^*\mathcal {F}_{p^r}$ -invariant, and

$$ \begin{align*}a(E_i;\mathcal{F}_{p^r})=-i+1, \quad a(F_i;\mathcal{F}_{p^r})=-i\quad \forall i=1,\dots,p^r.\end{align*} $$

Using Lemma 4.13 and proceeding by induction, one computes

$$ \begin{align*}a(E_i; W,W_0)=3i-2,\quad a(F_i;W,W_0)=3i-1.\end{align*} $$

Now let $Z=X/b^*\mathcal {F}_{p^r}$ and denote by $E^{\prime }_i\subset Z$ (resp. by $F^{\prime }_i\subset Z$ ) the image of $E_i$ (resp. of $F_i$ ). Then by Lemma 4.13 and Theorem 2.10 we have

$$ \begin{align*}a(E_i';Y',Y^{\prime}_0)=a(E_i;W,W_0)+(p-1)\cdot a(E_i;\mathcal{F}_{p^r})=i(4-p)+p-3\end{align*} $$

and

$$ \begin{align*}a(F_i';Y',Y^{\prime}_0)=a(F_i;W,W_0)+(p-1)\cdot a(F_i;\mathcal{F}_{p^r})=i(4-p)-1.\end{align*} $$

As $b^*\mathcal {F}_{p^r}$ is lc, by Theorem 2.10 it suffices to ensure that

$$ \begin{align*}\operatorname{\mathrm{discrep}}\left(Z, -\sum_i a(E_i';Y',Y_0')E_i'-\sum_i a(F_i';Y',Y_0')F_i'\right)\geq -1\end{align*} $$

to obtain that $(Y',Y_0')$ is lc. This holds as soon as $i(4-p)\geq 0$ for every $i=1,\dots ,p^r$ , which happens if (and only if) $p=3$ , so the proof is complete.