Proof. Write $n=m+\delta$ and $\rho$ for $\rho ^{(m)}$. We have

\[ \rho_\ast{\mathcal {O}}(\delta)={\mathcal {O}}^{\delta+1}\oplus{\mathcal {O}}(-1)^{m-\delta-1}. \]

This is shown for instance in the introduction to [Reference AchingerAch12] when $\rho$ is a power of the Frobenius, but the proof given applies unchanged in our situation. In terms of graded modules over $k[s,t]$, the corresponding graded module

\[ M_\bullet=\bigoplus_n\operatorname{H}^{0}({\boldsymbol{\rm {P}}}^{1},(\rho_\ast{\mathcal {O}}(\delta))(n)) \]

admits the following description: $M_n$ is the free $k$-vector space spanned by $s^{i}t^{j}$ with $i+j=\delta +mn$. Given positive integers $i$ and $j$ such that $i+j=\delta$, let $N^{i,j}$ be the graded submodule of $M_\bullet$ with $N^{i,j}_n$ the submodule spanned by $s^{a}t^{b}$ such that $a+b=\delta +mn$, $a\equiv i\pmod {m}$, and $b\equiv j\pmod {m}$. There is a graded splitting

\[ M_\bullet\to N^{i,j}_\bullet \]

defined as follows: given $\alpha$ and $\beta$ such that $\alpha +\beta =\delta +mn$, send $s^{\alpha } t^{\beta }$ to $0$ if $\alpha \not \equiv i\pmod m$ or $\beta \not \equiv j\pmod m$, and to $x^{\alpha } y^{\beta }$ otherwise. There results a free summand

\[ N_\bullet=\bigoplus_{i+j=a}N^{i,j}_\bullet\subset M_\bullet. \]

Since the rank of $N$ is $\delta$, we see that this gives the full free summand of $\rho _\ast {\mathcal {O}}(\delta )$.

To compute the sections of $\rho _\ast {\mathcal {O}}(m+\delta )=\rho _\ast {\mathcal {O}}(n)$, we shift the decomposition $M_\bullet =N_\bullet \oplus K_\bullet$ by $1$. That is, the non-split divisors will correspond to the elements of $N_\bullet$ of degree $1$. These are precisely the forms of degree $n$ in $s$ and $t$ each of whose terms are divisible by $s^{m}$ or $t^{m}$. This gives the desired result.

Proof. First, we note that the coefficients of $P_{\lambda _1,\ldots,\lambda _n}$ are homogeneous in $z_1,\ldots,z_n$, so it suffices to prove the statement for the image of $A$ in the projective space ${\boldsymbol{\rm {P}}}^{n-1}$ with homogeneous coordinates $z_1,\ldots,z_n$. Let $j$ be minimal such that $\tau _j\in [\delta +1,m-1]$. Consider the projective subspace $L\subset {\boldsymbol{\rm {P}}}^{n-1}$ given by the vanishing of $z_1,\ldots,z_j$. This is a codimension $j$ subspace, corresponding to the polynomials

\[ P_{\lambda_1+\cdots+\lambda_j,\lambda_{j+1},\ldots,\lambda_n}(s,t)=t^{\tau_j}(t-z_{j+1}s)^{\lambda_{j+1}}\cdots(t-z_ns)^{\lambda_n} \]

in the variables $z_{j+1},\ldots,z_n$. The lowest $t$-degree term of this polynomial is

\[ z_{j+1}^{\lambda_{j+1}}\cdots z_n^{\lambda_n}t^{\tau_j}s^{p+\delta-\tau_j}. \]

Since $\tau _n<2m$, we see that this term must vanish in order for the polynomial not to be $m$-split. This means that one of $z_{j+1},\ldots,z_n$ must vanish, giving a union of hyperplanes in $L$. The restriction of $P$ to each of these hyperplanes has the form

\[ P_{\tau_j+\lambda_q,\lambda_{j+1},\ldots,\lambda_n} \]

(with $\lambda _q$ taken out of the sequence). Since the $\lambda _i$ are non-increasing, the shortest sequence of such polynomials whose lowest $t$-degree term lies in $[\delta +1,m-1]$ has length $B:=B_m(\lambda _1,\ldots,\lambda _m)$. It follows that $A\cap L$ has codimension at least $B$ in $L$. Since the codimension can only go down upon intersection, we conclude that $A$ has codimension at least $B$, as desired.

Proof. Write $\sum \lambda _i=13+\delta$ and $\sum q\lambda _i=12q+1+\delta '$. We have $13q+\delta q=12q+1+\delta '$, so that $q(1+\delta )=\delta '+1$. Note that the partial sums $\tau _j(q\lambda _1,\ldots,q\lambda _n)$ satisfy

\[ \tau_j(q\lambda_1,\ldots,q\lambda_n)=q\tau_j(\lambda_1,\ldots\lambda_n). \]

The bounds on the partial sums $\tau _j(q\lambda _1,\ldots,q\lambda _n)$ to compute $B_{12q+1}(q\lambda _1,\ldots,q\lambda _n)$ are $\delta '+1$ and $12q$. On the other hand, by the above calculation we have $\delta '+1=q(1+\delta )$. The set of $\tau _j(q\lambda _1,\ldots,q\lambda _n)$ lying between $\delta '+1$ and $12q$ is thus the same as the set of $\tau _j(\lambda _1,\ldots,\lambda _n)$ lying between $\delta +1$ and $12$. This gives the desired equality.

Proof. This follows immediately from Corollary 3.6, once we note that $p^{2e}\equiv 1\pmod {12}$ for all positive integers $e$ and all primes $p>3$.

Proof. The $e$th absolute Frobenius $F_X^{e}\colon X\to X$ factors through $X^{(e)}$. By minimality of $Y_e$, it also factors through the map $g_e$, giving a map $h:X\to Y_e$, and hence a map of sheaves ${\mathcal {O}}_{Y_e}\to h_*{\mathcal {O}}_X$. Applying $F^{e}_*{\boldsymbol{\rm {R}}}^{1}(\varphi _e)_*$, we find a map

\[ F^{e}_*{\boldsymbol{\rm {R}}}^{1}(\varphi_e)_*{\mathcal {O}}_{Y_e}\to F^{e}_*{\boldsymbol{\rm {R}}}^{1}(\varphi_e)_*h_*{\mathcal {O}}_X\to{\boldsymbol{\rm {R}}}^{1}f_*F^{e}_{X*}{\mathcal {O}}_X \]

where the second map is induced by the appropriate Leray spectral sequence. We obtain the following diagram.

(4.2.1)

Note that the vertical arrow is identified with the canonical map ${\mathcal {O}}_C\to F^{e}_*{\mathcal {O}}_C$ tensored with the invertible sheaf ${\boldsymbol{\rm {R}}}^{1} f_*{\mathcal {O}}_X$. Now, a splitting of ${\mathcal {O}}_X\to F^{e}_*{\mathcal {O}}_X$ gives a splitting of the long diagonal arrow of (4.2.1), which induces a splitting of the diagonal arrow (4.0.2), and a splitting of (4.0.2) induces a splitting of the vertical arrow.

Proof. Using the universal property of the minimal resolution, we find a factorization

\[ {\boldsymbol{\rm {R}}}^{1}f_*{\mathcal {O}}_X\to{\boldsymbol{\rm {R}}}^{1}(\varphi_{e'})_*{\mathcal {O}}_{Y_{e'}}\to{\boldsymbol{\rm {R}}}^{1}(\varphi_e)_*{\mathcal {O}}_{Y_e}. \]

Thus, a splitting of (4.0.2) for $e$ gives rise to a splitting for $e'$.

Proof. We have a commutative diagram

where the vertical arrows are the isomorphisms induced by the Leray spectral sequence, and the upper horizontal arrow is the map on $\operatorname {H}^{1}$ induced by the map (4.0.2). A split injection of sheaves induces an injection on cohomology, which gives the result.

Proof. It is immediate from the Artin–Tate isomorphism (6.0.1) that $\alpha =0$ if and only if the torsor $X_{\alpha }^{\circ }\to C$ admits a section. If this is the case, then $f_{\alpha }\colon X_{\alpha }\to C$ is Jacobian. Conversely, any section of $f_{\alpha }$ must be contained in the smooth locus of $f_{\alpha }$, and hence gives rise to a section of the torsor, so (i) is equivalent to (ii). Clearly (ii) implies (iii), and conversely taking the closure of a rational point shows (iii) implies (ii).

Proof. Note that it suffices to prove the result for any resolution, since the Brauer group is a birational invariant for regular schemes of dimension $2$. (Birational invariance of the $p$-primary part is subtle in higher dimension.) Given any such $C'$ and $Y$, functoriality of the Leray spectral sequence yields the following diagram.

Here, the right vertical arrow is deduced from the isomorphism $Y_{k(C')}\xrightarrow {\sim } X_{k(C')}$ of generic fibers, the injectivity of the left horizontal arrows follows from the regularity of $X$ and $Y$, and the isomorphy of the right horizontal arrows follows from the Leray spectral sequence. It follows from Lemma 6.1 that $\alpha$ is in the kernel of the pullback map if and only if there exists a map $\Sigma$ fitting into the following diagram.

Letting $C'\to C$ range over all purely inseparable covers gives the result.

Proof. This is [Reference MirandaMir89, Lemma VII.1.2]. Note that while Miranda seems to make a uniform (often unstated) assumption in [Reference MirandaMir89] that the base field is ${\boldsymbol{\rm {C}}}$, the results we quote in this paper do not depend upon that assumption, as one can see from the proofs. We do freely use the assumption that the base field has characteristic at least $5$.

Proof. As explained in [Reference MirandaMir89, Lemma II.5.7], the degree is at most $0$, and it is $0$ if and only if the family has all smooth fibers. (The proof there, while it may be implicitly stated over a field of characteristic $0$, is true over any field of characteristic at least $5$.) On the other hand, we can compute the discriminant of $Y_e\to {\boldsymbol{\rm {P}}}^{1}$ around a point $x\in {\boldsymbol{\rm {P}}}^{1}$ (with local uniformizer $t$) as follows. If $\Delta$ is the discriminant of a minimal Weierstrass model for $X\to {\boldsymbol{\rm {P}}}^{1}$, pullback by $F^{e}$ gives a discriminant of $\Delta ^{p^{e}}$ for the naïve pullback of the Weierstrass equation. As in the proof of Lemma 4.8, to make the pulled back curve minimal near $x$, one scales the coordinates $x$ and $y$ by powers of $t$ so that the discriminant gets scaled by a power of $t^{12}$. Since $p$ is prime to $12$, it is impossible to make $\Delta ^{p^{e}}t^{12n}$ invertible at $x$, whence $Y_e$ still has a singular fiber at $x$.

Proof. Suppose that ${\mathfrak{X}}_{\infty }$ admits a purely inseparable multisection. By Proposition 6.2, there exist a smooth proper curve $C_{\infty }$ and a finite purely inseparable morphism $\pi \colon C_{\infty }\to {\boldsymbol{\rm {P}}}^{1}_{\overline {k(\!\hspace {0.03em}( t)\!\hspace {0.03em}) }}$ such that, for any resolution of the pullback of $X\operatorname {\otimes }\overline {k(\!\hspace {0.03em}( t)\!\hspace {0.03em}) }$ by $\pi$, the class $\alpha _{\infty }$ is in the kernel of the pullback map. The curve $C_{\infty }$ has genus 0, so by Tsen's theorem is isomorphic to ${\boldsymbol{\rm {P}}}^{1}_{\overline {k(\!\hspace {0.03em}( t)\!\hspace {0.03em}) }}$. By [Sta17, Tag 0CCZ], we can find isomorphisms identifying $\pi$ with the $n$th relative Frobenius of ${\boldsymbol{\rm {P}}}^{1}_{\overline {k(\!\hspace {0.03em}( t)\!\hspace {0.03em}) }}$ over $\overline {k(\!\hspace {0.03em}( t)\!\hspace {0.03em}) }$.

Let $Y$ be a smooth resolution of the pullback of $X$ by the $n$th relative Frobenius of ${\boldsymbol{\rm {P}}}^{1}$ over $k$. Consider the following diagram.

By Lemmas 6.4 and 6.3, we have that $\operatorname {Pic}_X$ and $\operatorname {Pic}_{Y}$ are both finitely generated free constant group schemes. Thus,

\[ \operatorname{H}^{1}(\operatorname{Spec} k(\!\hspace{0.03em}( t)\!\hspace{0.03em}) ,\operatorname{Pic}_X)=0=\operatorname{H}^{1}(\operatorname{Spec} k(\!\hspace{0.03em}( t)\!\hspace{0.03em}) , \operatorname{Pic}_{Y}), \]

and it follows from the Leray spectral sequence that the vertical maps are injective. Note that $Y\operatorname {\otimes }\overline {k(\!\hspace {0.03em}( t)\!\hspace {0.03em}) }$ is a resolution of the pullback of $X\operatorname {\otimes }\overline {k(\!\hspace {0.03em}( t)\!\hspace {0.03em}) }$ along $\pi$ (since $Y$ and $X$ are defined over the algebraically closed field $k$, where regular and of finite type implies smooth), and therefore $\alpha _{\infty }$ is in the kernel of the upper horizontal map. It follows that $\alpha _{\eta }$ is in the kernel of the lower horizontal map, as desired.

Proof. Suppose $C\to {\boldsymbol{\rm {P}}}^{1}$ is a purely inseparable cover. By [Sta17, Tag 0CCZ], we may assume that $C={\boldsymbol{\rm {P}}}^{1}$ and the map is the $e$th power of the Frobenius map. Let $Y=Y_e\to {\boldsymbol{\rm {P}}}^{1}$ be the resolution studied in § 4. Consider the kernel of the pullback map

\[ \operatorname{\operatorname{Br}}(X\operatorname{\otimes} k[\!\hspace{0.03em}[ t]\!\hspace{0.03em}])\to\operatorname{\operatorname{Br}}(Y\operatorname{\otimes} k[\!\hspace{0.03em}[ t]\!\hspace{0.03em}]). \]

Restricting to $k[\varepsilon ]:=k[t]/(t^{2})$, we get a diagram

in which the vertical arrows are the usual pullback maps. It follows from the hypotheses on $\alpha$ that the restriction

\[ \alpha_{k[\varepsilon]}\in\operatorname{\operatorname{Br}}(X\operatorname{\otimes} k[\varepsilon]) \]

of $\alpha$ is the image of a generator for $\operatorname {H}^{2}(X,{\mathcal {O}})$. Since each $\gamma _e(f)$ is injective, the left vertical map is injective, and we conclude that $\alpha _{Y\operatorname {\otimes } k[\!\hspace {0.03em}[ t]\!\hspace {0.03em}]}$ is non-zero. Since $Y$ is smooth over $k$, we have that $Y\operatorname {\otimes } k[\!\hspace {0.03em}[ t]\!\hspace {0.03em}]$ is regular, and thus

\[ \operatorname{\operatorname{Br}}(Y\operatorname{\otimes} k[\!\hspace{0.03em}[ t]\!\hspace{0.03em}])\to\operatorname{\operatorname{Br}}(Y\operatorname{\otimes} k(\!\hspace{0.03em}( t)\!\hspace{0.03em}) ) \]

is injective.

Putting everything together, we conclude that $\alpha |_{Y_e\operatorname {\otimes } k(\!\hspace {0.03em}( t)\!\hspace {0.03em}) }\neq 0$ for all $e$. By Proposition 6.5, we see that the original fibration ${\mathfrak{X}}_\infty \to {\boldsymbol{\rm {P}}}^{1}_{\overline {k(\!\hspace {0.03em}( t)\!\hspace {0.03em}) }}$ cannot have any purely inseparable multisections, as claimed.

Proof. This follows immediately from Theorem 6.6, using the fact that

\[ \alpha_{X\operatorname{\otimes} k[t]/(t^{2})}\in\widehat{\operatorname{\operatorname{Br}}}(k[t]/(t^{2}))=\operatorname{H}^{2}(X,{\mathcal {O}}_X) \]

is a generator.

Proof. Since ${\boldsymbol{\rm {P}}}^{1}$ is separated, the condition that the composition ${\boldsymbol{\rm {P}}}^{1}\to X\to {\boldsymbol{\rm {P}}}^{1}$ equal $f$ is closed. Thus it suffices to show that the scheme $\operatorname {\operatorname {Hom}}_e({\boldsymbol{\rm {P}}}^{1}_U,X)$ parametrizing maps of degree $e$ is of finite type. Since $\operatorname {\operatorname {Hom}}({\boldsymbol{\rm {P}}}^{1}_U,X)\subset \operatorname {\operatorname {Hom}}({\boldsymbol{\rm {P}}}^{1}_U,{\boldsymbol{\rm {P}}}^{n}_U)$ is a closed immersion, it suffices to show the same thing for

\[ \operatorname{\operatorname{Hom}}_e({\boldsymbol{\rm {P}}}^{1}_U,{\boldsymbol{\rm {P}}}^{n}_U)=\operatorname{\operatorname{Hom}}_e({\boldsymbol{\rm {P}}}^{1},{\boldsymbol{\rm {P}}}^{n})\times U, \]

whence it suffices to show it for the absolute scheme $\operatorname {\operatorname {Hom}}_e({\boldsymbol{\rm {P}}}^{1},{\boldsymbol{\rm {P}}}^{n})$. By the universal property of projective space, this is given by the scheme of quotients ${\mathcal {O}}^{n+1}\to {\mathcal {O}}(e)$ on ${\boldsymbol{\rm {P}}}^{1}$. This $\operatorname {Quot}$-scheme is of finite type by [Sta17, Tag 0DPA].

Proof. Suppose that $f:X\to {\boldsymbol{\rm {P}}}^{1}$ is an elliptic supersingular K3 surface such that $f$ is $\infty$-Frobenius split (see, for instance, Examples 5.1 and 5.2). As shown in § 4.3 of [Reference Bragg and LieblichBL18], there exists a class $\alpha '\in \operatorname {H}^{2}(X\times {\boldsymbol{\rm {A}}}^{1},\boldsymbol {\mu }_p)$ which restricts to a class over $X\otimes k[[t]]$ whose associated Brauer class is the universal Brauer class. Moreover, there is a corresponding elliptic surface $\widetilde {{\mathfrak{X}}}\to {\boldsymbol{\rm {P}}}^{1}_{{\boldsymbol{\rm {A}}}^{1}}$ over ${\boldsymbol{\rm {A}}}^{1}$.

Let ${\mathcal {O}}(1)$ be a relative polarization of $\widetilde {{\mathfrak{X}}}$ over ${\boldsymbol{\rm {A}}}^{1}$. Given a positive integer $e$, let $H_{e,b}\subset \operatorname {\operatorname {Hom}}_{{\boldsymbol{\rm {A}}}^{1}}({\boldsymbol{\rm {P}}}^{1},\widetilde {{\mathfrak{X}}})$ be the Hom scheme parametrizing morphisms whose composition ${\boldsymbol{\rm {P}}}^{1}\to \widetilde {{\mathfrak{X}}}\to {\boldsymbol{\rm {P}}}^{1}$ is the $e$th relative Frobenius and whose image has ${\mathcal {O}}(1)$-degree at most $b$ in each fiber. By Lemma 6.8, $H_{e,b}\to {\boldsymbol{\rm {A}}}^{1}$ is of finite type. By Corollary 6.7, the image of $H_{e,b}$ does not contain the generic point of ${\boldsymbol{\rm {A}}}^{1}$, and hence the image is finite. Applying this for all $e$ and $b$, we see that the fiber of $\widetilde {{\mathfrak{X}}}$ over a very general point $k$-point of ${\boldsymbol{\rm {A}}}^{1}$ is a non-Jacobian elliptic supersingular K3 surface that does not admit a purely inseparable multisection.

Proof. Let $g:X'\to {\boldsymbol{\rm {P}}}^{1}$ be an elliptic K3 surface that is a form of $f$. The Brauer group of a supersingular K3 surface is $p$-torsion [Reference ArtinArt74, Theorem 4.3], so $g$ admits a section after pullback along $X[p]\to {\boldsymbol{\rm {P}}}^{1}$, where $X[p]$ is the $p$-torsion group scheme of the smooth locus of $f$. The generic fiber of $f$ is supersingular, so $X[p]\to {\boldsymbol{\rm {P}}}^{1}$ is purely inseparable of degree $p^{2}$. It follows that the map $Y_{2}\to X$ induces the zero map on formal Brauer groups, and therefore the pullback $\operatorname {H}^{2}(X,{\mathcal {O}}_X)\to \operatorname {H}^{2}(Y_2,{\mathcal {O}}_{Y_2})$ vanishes. By Lemma 4.6, the fibration $f$ is not $e$-Frobenius split for any $e\geq 2$.

Notation 7.2 We will write $\mathcal {W\!F}_Y$ for the category of Weierstrass fibrations over $Y$. Given a fixed $k$-scheme $Z$, we will write $\mathcal {W\!F}(Z)$ for the fppf $k$-stack whose fiber over a scheme $T$ is the groupoid $\mathcal {W\!F}_{Z\times _k T}$. There is an open substack $\mathcal {W\!F}(Y)^{\circ }$ whose objects over $T$ are Weierstrass fibrations over $Y$ parametrized by $T$ such that for each geometric point $t\to T$, the induced fibration $X_t\to Y_t$ has a smooth fiber.

Proof. It follows from the various definitions that there is a canonical isomorphism $\mathcal {W\!F}(Z)=\operatorname {\operatorname {Hom}}(Z,\mathcal {W\!F}(\operatorname {Spec} k))$. Applying [Reference OlssonOls06, Theorem 1.1], we see that it suffices to establish the result for $Z=\operatorname {Spec} k$. We will write ${\mathcal {M}}:=\mathcal {W\!F}(\operatorname {Spec} k)$ for the sake of notational simplicity.

Given a family $\pi :X\to T$, $\sigma :T\to X$ in ${\mathcal {M}}$, cohomology and base change tells us that $\pi _\ast {\mathcal {O}}(3\sigma (T))$ is a locally free sheaf of rank $3$, and that the natural morphism $X\to \operatorname {Proj}(\operatorname {Sym}^{\ast }\pi _\ast {\mathcal {O}}(3\sigma (T)))$ is a closed immersion. Let $G\subset \operatorname {GL}_3$ be the subgroup whose image in $\operatorname {PGL}_2$ is the stabilizer of the point $(0:1:0)$. We can construct a natural $G$-torsor $M\to {\mathcal {M}}$ whose fiber over $(\pi,\sigma )$ is the scheme of isomorphisms $\pi _\ast {\mathcal {O}}(3S)\to {\mathcal {O}}^{\oplus 3}$ with the property that the composition

\[ {\mathcal {O}}\to\pi_\ast{\mathcal {O}}(3S)\to{\mathcal {O}}^{\oplus 3} \]

lands in the span of $(0,1,0)$, where the first map above is the adjoint of the map ${\mathcal {O}}_X\to {\mathcal {O}}_X(3S)$ corresponding to the divisor $3S$. The $G$-torsor $M$ admits an open immersion into the space of cubic curves in ${\boldsymbol{\rm {P}}}^{2}$ passing through $(0:1:0)$.

Consider the diagonal of ${\mathcal {M}}$. By well-known results (for example, [Reference GrothendieckGro95, paragraph 4.c]), we know that $\Delta :{\mathcal {M}}\to {\mathcal {M}}\times {\mathcal {M}}$ is representable by schemes of finite presentation. We claim that $\Delta$ is finite. To see this, it suffices to work with two families $\pi :X\to T,\ \sigma :T\to X$ and $\pi ':X'\to T,\ \sigma ':T\to X$ with $T$ the spectrum of a dvr. The divisors ${\mathcal {O}}(3S)$ and ${\mathcal {O}}(3S')$ define embeddings $X,X'\hookrightarrow {\boldsymbol{\rm {P}}}^{2}_T$. An isomorphism of the generic fibers over $T$ gives a pointed isomorphism, which gives a change of coordinates that conjugates the Weierstrass equations of $X_K$ and $X'_K$. Since the minimal Weierstrass models over $T$ are unique, this shows that the isomorphism extends uniquely to an isomorphism $X\to X'$, giving properness of $\Delta$. On the other hand, we know that the automorphism group of any fiber has size at most $6$, when we conclude that $\Delta$ is finite.

It follows that we can identify ${\mathcal {M}}$ with the quotient stack $[M/G]$, which has finite diagonal. This shows the desired result. (One could also prove this result using Artin's representability theorem, and checking various deformation-theoretic conditions, but the proof given here is more concrete and informative.)

Notation 7.5 We will write $\mathcal {WD}_T$ for the groupoid of Weierstrass data parametrized by $T$. Given an integral $k$-scheme $Y$ of finite type, we will write $\mathcal {WD}(Y)$ for the fppf stack whose fiber over a $k$-scheme $T$ is the subgroupoid $\mathcal {WD}_{Y\times _k T}^{\circ }\subset \mathcal {WD}_{Y\times _k T}$ consisting of those Weierstrass data such that the discriminant $\Delta$ is not identically zero in any geometric fiber over $T$.

Proof. There is a representable forgetful morphism $\mathcal {WD}(Y)\to \operatorname {Pic}_{Y/\operatorname {Spec} k}$ whose fiber over an invertible sheaf $L$ on $Y\times T$ is the space of pairs $(a,b)$ of sections of $L^{\operatorname {\otimes } -4}$ and $L^{\operatorname {\otimes } -6}$ such that the discriminant does not identically vanish, which is itself open in the space of pairs. The scheme of sections of an invertible sheaf is representable by a geometric line bundle. Since $\operatorname {Pic}_{Y/\operatorname {Spec} k}$ is an Artin stack locally of finite type over $k$, we conclude that the same is true for $\mathcal {WD}(Y)$, as desired.

Proof. To prove the statement it suffices to prove that the functor

\[ \mathcal{WD}(Y)_T\to\mathcal{W\!F}(Y)_T \]

is an equivalence (respecting the indicated subcategories), where $T$ is a strictly local ring. This then reduces to [Reference MirandaMir81, Theorem 2.1], whose proof invokes [Reference Mumford and SuominenMS72, Theorem 1$^{\prime }$], whose proof is, in turn, an unrecorded relativization of [Reference Mumford and SuominenMS72, Theorem 1]). Furthermore, one can see that the proof of [Reference Mumford and SuominenMS72, Theorem 1] is a consequence of cohomology and base change for the fibers of Weierstrass fibrations, so it applies more generally over non-reduced bases (which is not a priori allowed by the hypotheses of [Reference Mumford and SuominenMS72, Theorem 1] and [Reference MirandaMir81, Theorem 2.1]).

Proof. This is a topological condition, so it suffices to show that the locus $R$ of points $t\in T$ such that $(L,a,b)_t$ is minimal is open. First, note that any Weierstrass datum is locally induced by a Weierstrass datum over a Noetherian scheme. Thus, we may assume $T$ is Noetherian. The locus $R$ is closed under generization, since the order of vanishing of a section of an invertible sheaf can only increase under specialization. To show the desired openness it suffices to show that $R$ is constructible. To do this, we may assume $T$ is integral. There is an open subscheme $U\subset T$ over which the vanishing loci of $a$ and $b$ are flat. Shrinking if necessary and making a finite flat extension of $U$, we may assume that in fact we can write the vanishing locus of $a$ as $\sum _{i\in I} a_i s_i$ and the vanishing locus of $b$ as $\sum _{j\in J} b_j t_j$, where $s_i$ and $t_j$ are sections of the projection $Y_U\to U$ and $a_i, b_j\in {\boldsymbol{\rm {N}}}$. The points in $R_U$ parametrizing minimal Weierstrass data can be described as follows. Given $i\in I$ and $j\in J$, let $Z_{ij}\subset U$ be the image of $s_i\cap t_j$, which is a closed subset. If $Z_{ij}\neq \emptyset$, then $R$ contains $Z_{ij}$ if and only if $a_i<4$ or $b_j<6$. We can thus write $R_U$ as the complement of finitely many $Z_{ij}$, depending upon the values of $a_i$ and $b_j$. This defines an open subset of $U$. By Noetherian induction, we see that $R$ is constructible, as desired.

Notation 7.10 We will write $\mathcal {W\!F}$ for $\mathcal {W\!F}({\boldsymbol{\rm {P}}}^{1})^{\circ }$ and $\mathcal {WD}$ for $\mathcal {WD}({\boldsymbol{\rm {P}}}^{1})$ in what follows. We will write $\mathcal {WD}^{n}$ for the substack consisting of minimal Weierstrass data $(L,a,b)$ where $L$ has degree $n$ (that is, $L\cong {\mathcal {O}}_{{\boldsymbol{\rm {P}}}^{1}}(n)$).

Notation 8.1 Suppose that $\underline {\Phi }=(\Phi _1,\dots,\Phi _n)$ is a configuration of additive fibers. Let $\lambda _i=\delta (\Phi _i)$, and suppose that the $\Phi _i$ are ordered such that the sequence $(\lambda _1,\dots,\lambda _n)$ is non-increasing. We write $B=B_{13}(\lambda _1,\dots,\lambda _n)$ for the integer defined by (3.4.1).

Proof. Let $\operatorname {\mathcal {P}}^{ab\neq 0}[\underline {\Phi }]$ be the preimage of $\mathcal {WD}^{-2,ab\neq 0}[\underline {\Phi }]$ under the flat cover of Lemma 7.14. As in the proof of Proposition 7.21, it is covered by copies of ${\boldsymbol{\rm {A}}}^{22-\zeta (\underline {\Phi })}$ with coordinates

\[ (a_0',a_1',\ldots,a'_{8+n-\sum\alpha_i},b_0',b'_1,\ldots,b'_{12+n-\sum\beta_i}). \]

To such a point we associate the polynomial

\[ \prod_{i=1}^{n}(t-a'_is)^{\lambda_i}. \]

This is a projection onto the affine space ${\boldsymbol{\rm {A}}}^{n}$ parametrizing polynomials of the form

\[ P(s,t)=\prod_{i=1}^{n}(t-z_is)^{\lambda_i}. \]

We can thus cover $\operatorname {\mathcal {P}}^{ab\neq 0}[\underline {\Phi }]$ by components that are flat over ${\boldsymbol{\rm {A}}}^{n}$. To prove the proposition, it thus suffices to prove a corresponding codimension statement for certain subsets of ${\boldsymbol{\rm {A}}}^{n}$.

By Corollary 3.7, the locus in ${\boldsymbol{\rm {A}}}^{n}$ parametrizing polynomials $P$ such that $P^{({p^{2e}-1})/{12}}$ is not $p^{2e}$-split for some $e$ is an ascending union of closed subschemes of codimension at least $B$. The preimage of this locus in $\operatorname {\mathcal {P}}^{ab\neq 0}[\underline {\Phi }]$ is exactly equal to the preimage of the locus in $\mathcal {WD}^{-2,ab\neq 0}[\underline {\Phi }]$ parametrizing fibrations that are not $2e$-Frobenius split for some $e$. By Lemma 4.5, a fibration is not $2e$-Frobenius split for some $e$ if and only if it is not $\infty$-Frobenius split.