Minimal models of L*

TL;DR

This paper classifies the minimal models of certain G_m-torsors associated with numerically trivial line bundles on abelian algebraic spaces over algebraic spaces, providing a structural understanding of these models.

Contribution

It introduces a classification of S-minimal models of G_m-torsors derived from numerically trivial line bundles on abelian algebraic spaces, a novel structural insight.

Findings

Classification of S-minimal models into two types

Structural understanding of G_m-torsors over abelian spaces

Framework for analyzing numerically trivial line bundles

Abstract

Let S be an algebraic space, A an S-abelian algebraic space, L an S-fiberwise numerically trivial invertible module on A, and L* the sheaf of regular sections of L considered as a G_m-torsor on A. We classify the S-minimal models of L* into two types.

Equations

$$1\to {\mathbf{G}}_m\to L^{\ast }\to A\to 1$$

$$(f{\times }_{S}s{)}^{!}k(s)\simeq \mathcal{O}[d]$$

$$C_{nL}:=L^{\ast }\stackrel{{\mathbf{G}}_m}{\wedge }{C}_{\infty }/n\mathbf{Z}$$

$$P_L:=L^{\ast }\stackrel{{\mathbf{G}}_m}{\wedge }{\mathbf{P}}^1$$

$${\rho }^1:A=0_1\to {\Sigma }_1,{\rho }^n:A={\infty }_n\to {\Sigma }_n$$

$$F:A\stackrel{\rho }{\to }\Sigma \stackrel{\epsilon }{\to }{A}^{\prime }$$

$$P_L:=L^{\ast }\stackrel{{\mathbf{G}}_m}{\wedge }P$$

$${\mathrm{Res}}_0(\lambda )+{\mathrm{Res}}_{\infty }(\lambda )=0$$

$${\phi }^{\prime }:X_t^{\prime }-Z_t^{\prime }=U_t\hookrightarrow L_t^{\ast }=P_{Lt}-\{0,\infty \}$$

$$I|D^{\prime }\simeq ({\phi }^{\prime }|D^{\prime }{)}^{!}J$$

$$-Z^{\prime }|D^{\prime }\sim -{\Sigma }_0^{\prime }|D^{\prime }-{\Sigma }_{\infty }^{\prime }|D^{\prime }+E^{\prime }$$

$$Z^{\prime }|D^{\prime }-{\Sigma }_0^{\prime }|D^{\prime }-{\Sigma }_{\infty }^{\prime }|D^{\prime }+E^{\prime }$$

$$H^0(D^{\prime },\mathcal{O})=H^0(X^{\prime },\mathcal{O})=k$$

$$\rho :{\Sigma }_0^{\prime }\cup {\Sigma }_{\infty }^{\prime }\to Z$$

$$\gamma :\tilde{\Gamma }\to \overline{\Gamma }$$

$$({\psi }^{!}J{)}_v={\mathcal{O}}_v(r_{v}v)$$

$$I{\mathcal{O}}_v\subset {\mathcal{O}}_v(-v)\subset {\mathcal{O}}_v(r_{v}v)=({\psi }^{!}J{)}_v=(h^{!}I{)}_v$$

$$c{l}_p:p^{\ast }{\Omega }_{A/k}^{d-1}\to p^{!}{\Omega }_{A/k}^{d-1}=H^{-d+1}{K}_{Z^{\prime }}|V^{\prime }$$

$$c{l}_p({\mathrm{Res}}_0(\lambda ))=c{\mathrm{Res}}_{Z^{\prime }|V^{\prime }}({\lambda }_{V^{\prime }})$$

$$c\in H^0(Z^{\prime }|V^{\prime },\mathcal{O})$$

$$Z^{\prime }|V^{\prime }=({\Sigma }_0^{\prime }+{\Sigma }_{\infty }^{\prime })|V^{\prime }$$

$$({\phi }^{\prime }{)}^{-1}=p_1\circ (p_2|{\Gamma }^o{)}^{-1}$$

$$D_X({\mathcal{O}}_{X^{\prime }})=I,D_X({\mathcal{O}}_{X^{\prime }}/{\mathcal{O}}_X)={\mathcal{O}}_Z[-1],D_X({\mathcal{O}}_Z)=({\mathcal{O}}_{X^{\prime }}/{\mathcal{O}}_X)[-1]$$

$$0\to {\mathcal{O}}_X\to {\mathcal{O}}_{X^{\prime }}\to {\mathcal{O}}_{X^{\prime }}/{\mathcal{O}}_X\to 0$$

$${\mathcal{O}}_Z={\mathcal{O}}_X/I\subset {\mathcal{O}}_{X^{\prime }}/I={\mathcal{O}}_{Z^{\prime }}$$

$$D_X({\mathcal{O}}_Z)=({\mathcal{O}}_{X^{\prime }}/{\mathcal{O}}_X)[-1],$$

$${\Sigma }_0^{\prime }\cup {\Sigma }_{\infty }^{\prime }\to Z$$

$$\rho :A={\Sigma }_i^{\prime }\to {\Sigma }_i:=\rho ({\Sigma }_i^{\prime })$$

$$k(A{)}^2\subset k(\Sigma )\subset k(A)$$

$$F:A\stackrel{\rho }{\to }\Sigma \stackrel{\epsilon }{\to }{A}^{\prime }$$

Full text

Minimal models of $L^{*}$

Ying Zong

zongying.upenn@gmail.com

1. Introduction

Let $S$ be an algebraic space, $A$ an $S$-abelian algebraic space, and $L$ an invertible module on $A$. We suppose that $L$ is fiberwise numerically trivial. This assumption, according to the theorem of square, is equivalent to that, on the $\mathbf{G}_{m}$-torsor $L^{*}:=\underline{\mathrm{Isom}}_{A}(\mathcal{O}_{A},L)$, there exists, locally for the étale topology of $S$, a compatible structure of extension of $A$ by $\mathbf{G}_{m}$:

[TABLE]

Definition. — *A minimal model of $L^{*}\times\mathbf{Z}/n\mathbf{Z}$, $n\geq 1$, consists of a pair $(f,\varphi)$, where *:

a) $f:X\to S$ is a proper flat finitely presented $S$-algebraic space, with geometrically connected fibers, such that, for each geometric point $s$ of $S$, there is an isomorphism

[TABLE]

in $D(f^{-1}(s),\mathcal{O})$, $d=\mathrm{dim}\ f^{-1}(s)$.

b) $\varphi$ is an $S$-pseudo-isomorphism of $X$ to $L^{*}\times\mathbf{Z}/n\mathbf{Z}$, which is defined and étale at all points $x$ of codimension $\leq 1$ in $f^{-1}(f(x))$ where $f$ is smooth.

This notion, relevant in the study of fibers of lagrangian fibrations on symplectic manifolds, cf. [6] Prop. 3, is motivated by Néron polygons of Deligne-Rapoport [4], the starting point for investigating boundaries of various moduli of elliptic curves. My purpose, more birational, was to see whether, even under the weakest possible assumption on the pseudo-isomorphism $\varphi$, the following example, which we call standard $n$-gon analogous to loc.cit., exhausts in fact all minimal models, at least fiberwise.

Example. — Let the projective line over $\mathrm{Spec}(\mathbf{Z})$ be $P$, let it be pinched with the coordinate sections $T=0,\infty$ identified, and let $c:C\to\mathrm{Spec}(\mathbf{Z})$ be the resulting nodal curve. One can make explicit, via the normalization $P\to C$, the dualizing module $H^{-1}c^{!}\mathcal{O}$; it has as $\mathcal{O}_{C}$-basis the logarithmic differential $d_{P/\mathbf{Z}}(T)/T$, which is of residue $1$ and $-1$ along $T=0$ and $T=\infty$ respectively. And one calculates the group $H^{1}(C,\mathbf{Z})$ of the isomorphism classes of $\mathbf{Z}$-torsors, in the sense of étale topology, on $C$; it is infinite cyclic, with a generator $C_{\infty}$ represented by an unending chain of projective lines, any two lines in succession glued along the section $\infty$ on the former and along the section [math] on the latter. The multiplicative group $\mathbf{G}_{m}$ acts canonically, compatibly on $C$ and $C_{\infty}$ as well as all intermediate quotients $C_{\infty}/n\mathbf{Z}$. The compactification

[TABLE]

of $L^{*}\times\mathbf{Z}/n\mathbf{Z}$ is a minimal model.

It turns out, when $A$ is non-trivial, models other than the standard do exist. We call them $n$-chains, all of which, fiber by fiber, are obtained from $C_{nL}$ by normalization and gluing and pinching (Artin [1] 6.1).

Theorem. — *Let $(f,\varphi)$ be a minimal model of $L^{*}\times\mathbf{Z}/n\mathbf{Z}$. Then *:

i) The $S$-pseudo-isomorphism $\varphi^{-1}$ is everywhere defined and an isomorphism onto the domain of definition of $\varphi$. The geometric number of connected components of $f^{-1}(s)-\mathrm{Im}(\varphi_{s}^{-1})$, $s\in S$, is $n$ or $n+1$, and locally constant on $S[1/2]$.

ii) Assume $S=\mathrm{Spec}(k)$, $k$ an algebraically closed field, and $X-\mathrm{Im}(\varphi^{-1})$ of $n$ connected components. Then $X$ is $C_{nL}$.

iii) Assume $S=\mathrm{Spec}(k)$, $k$ an algebraically closed field, and $X-\mathrm{Im}(\varphi^{-1})$ of $n+1$ connected components. Let

[TABLE]

and $\{0,\infty\}$ its distinguished pair of sections over $A$, and let $Q_{nL}$ be the chain obtained from $P_{L}\times\mathbf{Z}/n\mathbf{Z}$ by gluing the $i$-th section $\infty_{i}$ with the $(i+1)$-th section $0_{i+1}$, $i=1,\cdots,n-1$. Then $X$ is pinched from $Q_{nL}$ along finite $k$-morphisms

[TABLE]

both of generic degree $2$, with $\Sigma_{i}$ proper over $\mathrm{Spec}(k)$, normal, integral, Cohen-Macaulay, of dualizing module $K_{\Sigma_{i}}=\rho^{i}_{*}\mathcal{O}_{A}/\mathcal{O}_{\Sigma_{i}}$, $i=1,n$. Fix either $\rho:A\to\Sigma$ and a volume form $\mu$ on $A/k$.

When $k(A)/k(\Sigma)$ is inseparable, the relative Frobenius of $A/k$ factors through $\rho$ as

[TABLE]

with $\varepsilon$ finite flat, and $\mathrm{Tr}_{\rho}(\mu)=0$ in $H^{0}(\Sigma,K_{\Sigma})$.

When $k(A)/k(\Sigma)$ is separable, $\Sigma$ is the quotient $A/\sigma$ by an involutive automorphism $\sigma$ of $A$, and $\sigma^{*}\mu=-\mu$. In characteristic $2$, the fixed point set of $\sigma$ is either empty or purely of codimension $1$ in $A$.

The essential of the proof is arguing that, over the spectrum of an algebraically closed field, $X$ shares normalization with $C_{nL}$, see I) p) below. The fact that $\varphi^{-1}$ is everywhere defined and an isomorphism onto $\mathrm{Dom}(\varphi)$ is proved in III) d). That the geometric number of connected components of $f^{-1}(s)-\mathrm{Im}(\varphi^{-1}_{s})$, $s\in S[1/2]$, is locally constant is proved in III) h).

2. Proof of the theorem

I) Assume first $n=1$ and that $S$ is the spectrum of an algebraically closed field $k$.

Let $d=\mathrm{dim}\ X$. Since $X$ is pseudo-isomorphic to $L^{*}$, $X$ is integral, and $A$ is of dimension $d-1$. Let $t$ denote the generic point of $A$.

Let $U$ consist of all points of $X$ where $X$ is normal. Let $\rho:X^{\prime}\to X$ be the normalization of $X$, and let $U$ be via $\rho$ identified with $\rho^{-1}(U)$.

Write $I=H^{0}\rho^{!}\mathcal{O}$ for the conductor of $X^{\prime}/X$, coherent ideal in $\mathcal{O}_{X^{\prime}}$ and in $\mathcal{O}_{X}$, by which one defines the preferred closed sub-algebraic space structure of $Z^{\prime}=X^{\prime}-U$ (resp. $Z=X-U$) in $X^{\prime}$ (resp. $X$).

The hypothesis that $f^{!}k\simeq\mathcal{O}[d]$ in $D(X,\mathcal{O})$ says in particular that $X$ is Cohen-Macaulay. As $I\simeq H^{-d}(f\rho)^{!}k$, $I$ is dualizing at precisely those points of $X^{\prime}$ where $X^{\prime}$ is Cohen-Macaulay.

Let $P$ be the projective line over $\mathrm{Spec}(k)$, and

[TABLE]

and $\varphi^{\prime}$ the pseudo-isomorphism of $X^{\prime}$ to $P_{L}$ induced by $\varphi$.

Let $D^{\prime}$ consist of all points $x^{\prime}$ of $X^{\prime}$ where $X^{\prime}$ is $k$-smooth and $\varphi^{\prime}$ is defined. One has $\mathrm{codim}(X^{\prime}-D^{\prime},X^{\prime})\geq 2$.

Write $C_{L}$ for $C_{1L}$, and $\pi:C_{L}\to\mathrm{Spec}(k)$ its structural morphism. The normalization of $C_{L}$ is $P_{L}$. Let $J$ be the conductor of $P_{L}/C_{L}$. In $P_{L}$, $V(J)$ is formed of two disjoint $A$-sections $0,\infty$.

A basis of $H^{-d}\pi^{!}k$ can be thought of as a non-zero global $d$-form on $P_{L}/\mathrm{Spec}(k)$ with simple poles along $V(J)$. Such a $d$-form $\lambda$ has residue along either section $i=0,\infty$ a volume form $\mathrm{Res}_{i}(\lambda)\in H^{0}(A,\Omega^{d-1}_{A/k})$, and

[TABLE]

*a) The pseudo-morphism $\varphi|U$ is defined on all of $U$ *:

By hypothesis of minimal model, $\varphi$ is defined at all codimension $\leq 1$ points of $U$. Now apply Weil [2] 4.4/1, $L^{*}$ being a smooth $k$-algebraic group.

*b) The morphism $\varphi|U$ is étale *:

It is étale at every codimension $\leq 1$ point of $U$ by hypothesis, hence étale throughout by purity of branch locus [5] 2.4.

*c) The birational morphism $\varphi|U$ induces an open immersion of $U$ into $L^{*}$ *:

This follows by b) and Zariski’s Main Theorem. By c) we identify $U$ with its image in $L^{*}$.

*d) The pseudo-morphism of $X^{\prime}$ to $A$, composition of $\rho$, of $\varphi$ and of the projection $L^{*}\to A$, is everywhere defined *:

This follows by Weil [2] 4.4/1, since $X^{\prime}$ is normal and $A$ an abelian variety.

*e) The fiber of the morphism $X^{\prime}\to A$ in d) over the generic point $t$ of $A$ is a projective $t$-line, on which $Z^{\prime}_{t}$ consists of two distinct $t$-rational points *:

The fiber $X^{\prime}_{t}$ is proper over $t$, normal, in which $U_{t}$ is open dense. The open immersion

[TABLE]

uniquely extends to an isomorphism of $X^{\prime}_{t}$ with $P_{Lt}\simeq\mathbf{P}^{1}_{t}$, under which the image of $Z^{\prime}_{t}$ contains $\{0,\infty\}$. So $X^{\prime}_{t}$ is a projective $t$-line, $I_{t}$ the dualizing module, and $Z^{\prime}_{t}$ an anti-canonical divisor, on $X^{\prime}_{t}$. Being of degree $2$ over $t$, $Z^{\prime}_{t}$ consists of two distinct $t$-rational points.

Write $Z^{\prime}_{t}=\{0,\infty\}$, and let $\Sigma^{\prime}_{i}$ be the closed image of $i$, $i=0,\infty$, in $Z^{\prime}$.

Note that the formalism of duality implies

[TABLE]

which, in terms of $1$-codimensional cycles, amounts to a linear equivalence

[TABLE]

where $E^{\prime}$ is $(\varphi^{\prime}|D^{\prime})$-exceptional.

*f) One has $E^{\prime}\geq 0$ *:

For, $P_{L}$ is smooth over $\mathrm{Spec}(k)$, cf. the proof of [5] 4.5.

*g) One has $Z^{\prime}|D^{\prime}=(\Sigma^{\prime}_{0}+\Sigma^{\prime}_{\infty})|D^{\prime}$, $E^{\prime}=0$ *:

The cycle

[TABLE]

is $\geq 0$ and linearly equivalent to [math], therefore is [math], since

[TABLE]

h) The pseudo-isomorphism of $X^{\prime}-(\Sigma^{\prime}_{0}\cup\Sigma^{\prime}_{\infty})$ to $L^{*}$ induced by $\varphi^{\prime}$ is everywhere defined and an open immersion. In particular, the closed immersion $\Sigma^{\prime}_{0}\cup\Sigma^{\prime}_{\infty}\hookrightarrow Z^{\prime}$ is surjective. The restriction

[TABLE]

*is surjective, so $Z$ either is irreducible or consists of two irreducible components *:

By g) the complement of $U=X^{\prime}-Z^{\prime}$ in $X^{\prime}-(\Sigma^{\prime}_{0}\cup\Sigma^{\prime}_{\infty})$ is of codimension $\geq 2$. Since $U\hookrightarrow L^{*}$ is an open immersion, the claim is immediate by the combination of Weil [2] 4.4/1, purity of branch locus [5] 2.4 and Zariski’s Main Theorem.

*i) The birational morphism $\varphi^{\prime}|D^{\prime}$ induces an open immersion of $D^{\prime}$ into $P_{L}$ *:

By Zariski’s Main theorem and by purity of branch locus [5] 2.4, it suffices to show that $\varphi^{\prime}|D^{\prime}$ is étale at all codimension $\leq 1$ points of $D^{\prime}$. According to f), the $(\varphi^{\prime}|D^{\prime})$-exceptional cycle $E^{\prime}$ is [math], hence i).

*j) The domain of definition of $\varphi^{\prime}$ is $D^{\prime}$ *:

By definition, $D^{\prime}$ consists of all points of $\mathrm{Dom}(\varphi^{\prime})$ where $X^{\prime}$ is $k$-smooth. As $X^{\prime}-D^{\prime}$ is of codimension $\geq 2$ in $X^{\prime}$, and $\varphi^{\prime}|D^{\prime}$ an open immersion i), it follows by purity of branch locus [5] 2.4 that $\varphi^{\prime}$ is étale on $\mathrm{Dom}(\varphi^{\prime})$. A priori, $\mathrm{Dom}(\varphi^{\prime})$ is $k$-smooth, hence j).

Let $D^{\prime}$ be via $\varphi^{\prime}|D^{\prime}$ identified as an open sub-scheme of $P_{L}$ so that $I|D^{\prime}=J|D^{\prime}$.

*k) The fiber of $X^{\prime}\to A$ over every codimension $\leq 1$ point of $A$ is a projective line *:

This has been shown over the generic point $t$ of $A$. Assume $s$ is a codimension $1$ point of $A$. Clearly, $X^{\prime}\to A$ is flat around $X^{\prime}_{s}$. Since $\mathrm{codim}(X^{\prime}-D^{\prime},X^{\prime})\geq 2$, $D^{\prime}_{s}$ is schematically dense in $X^{\prime}_{s}$. So $X^{\prime}_{s}$ is geometrically integral and, as a flat specialization of $X^{\prime}_{t}$, is a projective $s$-line itself.

*l) Let $V^{\prime}$ consist of all points of $X^{\prime}$ where $X^{\prime}$ is smooth over $\mathrm{Spec}(k)$. Then $\mathrm{codim}(X^{\prime}-V^{\prime},X^{\prime})\geq 3$ *:

Otherwise, there would exist a maximal point $x^{\prime}$ of $X^{\prime}-V^{\prime}$, of codimension $2$ in $X^{\prime}$, with image $s$ of codimension $>1$ in $A$ by k).

Write $\Gamma$ (resp. $\overline{\Gamma}$) for the graph of $\varphi^{\prime}$ (resp. the closed image of $\Gamma$ in $X^{\prime}\times_{A}P_{L}$). Let

[TABLE]

be a proper morphism from a normal integral scheme $\widetilde{\Gamma}$ such that $\gamma$ is an isomorphism over $D^{\prime}\subset X^{\prime}$ and such that $\widetilde{\Gamma}$ is regular above $x^{\prime}$. Such partial resolution exists since excellent two dimensional singularities are canonically resolvable.

Call $h$ (resp. $\psi$) the projection $\widetilde{\Gamma}\to\overline{\Gamma}\to X^{\prime}$ (resp. $\widetilde{\Gamma}\to\overline{\Gamma}\to P_{L}$).

The ideal $I$ being dualizing at $x^{\prime}$, let $h^{!}I$ and $\psi^{!}J$ be along $h^{-1}(x^{\prime})$ identified via the unique isomorphism that is compatible with $I|D^{\prime}=J|D^{\prime}$.

Each maximal point $v$ of $h^{-1}(x^{\prime})$ is of codimension $1$ in $\widetilde{\Gamma}$ and, as $s$ is of codimension $>1$ in $A$, is $\psi$-exceptional. One has

[TABLE]

for an integer $r_{v}\geq 0$, since $P_{L}$ is regular (cf. the proof of [5] 4.5). And

[TABLE]

So $X^{\prime}$ presents at worst rational singularity at $x^{\prime}$ ($I\mathcal{O}_{v}\subset(h^{!}I)_{v}$) and this rational singularity is of multiplicty $1$ (the inclusion $I\mathcal{O}_{v}\subset(h^{!}I)_{v}$ is strict). That is, $X^{\prime}$ is regular, hence smooth over $\mathrm{Spec}(k)$, at $x^{\prime}$.

*m) Both morphisms $\Sigma^{\prime}_{i}\to A$, $i=0,\infty$, are isomorphisms *:

Fix a basis $\lambda$ of $J^{-1}\Omega^{d}_{P_{L}/k}$. The invertible module $(I|V^{\prime})^{-1}\Omega^{d}_{V^{\prime}/k}$ of $d$-forms on $V^{\prime}/\mathrm{Spec}(k)$ with simple poles along $Z^{\prime}|V^{\prime}$ has basis $\lambda_{V^{\prime}}$, the unique extension of $\lambda|D^{\prime}$ to $V^{\prime}$.

Denote by $p$ the composition $Z^{\prime}|V^{\prime}\hookrightarrow V^{\prime}\to A$; it is of local complete intersection and is étale at the two points of $Z^{\prime}_{t}$. Consider the fundamental class of $p$,

[TABLE]

where $K_{Z^{\prime}}=z^{{}^{\prime}!}k\in\mathrm{Ob}\ D(Z^{\prime},\mathcal{O})$, $z^{\prime}:Z^{\prime}\to\mathrm{Spec}(k)$ the structural morphism.

With respect to the basis $\mathrm{Res}_{Z^{\prime}|V^{\prime}}(\lambda_{V^{\prime}})$ of $H^{-d+1}K_{Z^{\prime}}|V^{\prime}$ and the volume form $\mathrm{Res}_{0}(\lambda)$ on $A$, one can write

[TABLE]

with coefficient

[TABLE]

This function $c$ takes value $1$ (resp. $-1$) on $\Sigma^{\prime}_{0}|V^{\prime}$ (resp. $\Sigma^{\prime}_{\infty}|V^{\prime}$), for $c|Z^{\prime}_{t}$ is $1$ (resp. $-1$) at [math] (resp. $\infty$).

Naturally, $\Sigma^{\prime}_{0}\cap\Sigma^{\prime}_{\infty}\cap V^{\prime}=\emptyset$, if $k$ is of characteristic $\neq 2$.

In all characteristics, $cl_{p}$ an isomorphism. Indeed,

[TABLE]

by g) and because $V^{\prime}-D^{\prime}$ is of codimension $\geq 2$ in $V^{\prime}$.

From that $cl_{p}$ is an isomorphism, it follows that $p$ is unramified, and that $p|(\Sigma^{\prime}_{i}|V^{\prime})$ is unramified thus étale (SGA 1 I 9.11).

By purity of branch locus [5] 2.4, if $\overline{\Sigma}^{\prime}_{i}$ denotes the normalization of $\Sigma^{\prime}_{i}$, the projection $\overline{\Sigma}^{\prime}_{i}\to A$ is étale ($\mathrm{codim}(X^{\prime}-V^{\prime},X^{\prime})\geq 3$) and hence an isomorphism by Zariski’s Main Theorem. So $\Sigma^{\prime}_{i}\to A$ is an isomorphism.

Consider henceforth $\Sigma^{\prime}_{i}$, $i=0,\infty$, as $A$-sections of $X^{\prime}$.

*n) The morphism $X^{\prime}\to A$ is equi-dimensional of relative dimension $1$ with $U$ dense in every fiber *:

Because it is smooth of relative dimension $1$ outside of $\Sigma^{\prime}_{0}\cup\Sigma^{\prime}_{\infty}$, h), and restricts to finite morphisms on $\Sigma^{\prime}_{i}$, $i=0,\infty$, m).

*o) Over each codimension $\leq 1$ point of $A$, the pseudo-isomorphism $\varphi^{\prime}$ is everywhere defined and an isomorphism *:

This is evident on k) and h): over any codimension $\leq 1$ point of $A$, the two $A$-sections $\Sigma^{\prime}_{0}$ and $\Sigma^{\prime}_{\infty}$ cannot intersect.

*p) The pseudo-isomorphism $\varphi^{\prime}$ is everywhere defined and an isomorphism *:

Let $\Gamma$ (resp. $\overline{\Gamma}$) be the graph of $\varphi^{\prime}$ (resp. the closed image of $\Gamma$ in $X^{\prime}\times_{A}P_{L}$). Let $\Gamma^{o}$ consist of every point of $\overline{\Gamma}$ where the projection $p_{2}|\overline{\Gamma}:\overline{\Gamma}\to P_{L}$ is étale. By Zariski’s Main Theorem, $p_{2}|\Gamma^{o}$ is an open immersion. By o) and by [3] Note (1) 2 (ii), $\Gamma^{o}$ is a neighborhood of the $A$-sections $\Sigma^{\prime}_{i}\times_{A}i$, $i=0,\infty$, so

[TABLE]

is defined in a neighborhood of $V(J)$. At each point of $V(J)$, $(\varphi^{\prime})^{-1}$ is obviously $A$-fiberwise quasi-finite in view of n), hence quasi-finite, hence étale by Zariski’s Main Theorem. Therefore, $X^{\prime}\to A$ is smooth also at the points of $\Sigma^{\prime}_{i}$, $i=0,\infty$, all whose fibers are projective lines. Then by h), $\Sigma^{\prime}_{0}$ and $\Sigma^{\prime}_{\infty}$ are disjoint. The assertion p) follows.

Identify from now on $X^{\prime}$ with $P_{L}$ via $\varphi^{\prime}$, so that $I=J$, $Z^{\prime}=V(J)$, $U=L^{*}$.

Let $D_{X}$ denote the duality auto-functor of $D^{b}_{\mathrm{Coh}}(X,\mathcal{O})$ with respect to its dualizing object $\mathcal{O}_{X}=H^{-d}f^{!}k$.

q) In $D^{b}_{\mathrm{Coh}}(X,\mathcal{O})$, one has

[TABLE]

This first identity follows by definition of $I$, for $X^{\prime}=P_{L}$ is smooth over $\mathrm{Spec}(k)$, in particular, Cohen-Macaulay. The other two are equivalent by biduality. One proves the second identity by applying $D_{X}$ to the exact sequence

[TABLE]

*r) The algebraic space $Z$ is normal, Cohen-Macaulay and of dualizing module $\mathcal{O}_{X^{\prime}}/\mathcal{O}_{X}=H^{-d+1}z^{!}k$, $z:Z\to\mathrm{Spec}(k)$ the structural morphism *:

Note first that $Z$ is reduced. In fact, $Z^{\prime}$ is reduced, and one has the inclusion

[TABLE]

If $\widetilde{Z}$ denotes the normalization of $Z$, then $\mathcal{O}_{\widetilde{Z}}/\mathcal{O}_{Z}$, which is a sub-$\mathcal{O}_{Z}$-module of the Cohen-Macaulay module $\mathcal{O}_{Z^{\prime}}/\mathcal{O}_{Z}=\mathcal{O}_{X^{\prime}}/\mathcal{O}_{X}$ and has support everywhere of codimension $\geq 1$ in $Z$, is [math]. So $Z$ is normal.

The other two statements together and the formula of q),

[TABLE]

are equivalent.

*s) Either $Z$ is integral with each projection $\Sigma^{\prime}_{i}\to Z$ biregular, or $Z$ has two irreducible components $\rho(\Sigma^{\prime}_{i})$ with each extension $k(i)/k(\zeta_{i})$ quadratic, where $\zeta_{i}$ denotes the generic point of $\rho(\Sigma^{\prime}_{i})$, $i=0,\infty$ *:

This is clear on h) and r): the projection $Z^{\prime}\to Z$ is finite, $Z$ normal, and the module $\mathcal{O}_{Z^{\prime}}/\mathcal{O}_{Z}=\mathcal{O}_{X^{\prime}}/\mathcal{O}_{X}$, being dualizing on $Z$, is invertible generically.

*t) When $Z$ is integral, $X$ is $C_{L}$ *:

That is, $X$ is pinched from $P_{L}$ along the finite morphism

[TABLE]

where each $A=\Sigma^{\prime}_{i}\to Z$, $i=0,\infty$, is an isomorphism. This is clear.

u) When $Z$ has two irreducible components, $X$ is pinched from $P_{L}$ along the finite morphisms

[TABLE]

*both of generic degree $2$, where $\Sigma_{i}$ is normal, integral, Cohen-Macaulay and of dualizing module $K_{i}=\mathcal{O}_{A}/\mathcal{O}_{\Sigma_{i}}$, $i=0,\infty$ *:

Again, this is clear.

To consider the case u) more closely, fix $i\in\{0,\infty\}$, put $\Sigma=\Sigma_{i}$, identify $\Sigma^{\prime}_{i}$ with $A$, let $K=\mathcal{O}_{A}/\mathcal{O}_{\Sigma}$, and choose a volume form on $A$, $\mu\in H^{0}(A,\Omega^{d-1}_{A/k})$.

— Assume, in u), $k$ of characteristic $2$, and the extension $k(A)/k(\Sigma)$ inseparable.

The inclusion of function fields

[TABLE]

corresponds, according to Weil [2] 4.4/1, to a factorization

[TABLE]

of the Frobenius of $A/k$, $A^{\prime}$ the base change of $A$ by the Frobenius

[TABLE]

Note that $\varepsilon:\Sigma\to A^{\prime}$ is finite. For, it is proper and, as $F$ is finite and $\rho$ surjective, quasi-finite. This finite morphism $\varepsilon$ is also flat, since $\Sigma$ is Cohen-Macaulay and $A$ regular (EGA O 17.3.5).

The module $K$, as it is dualizing on $\Sigma$, is flat over $A^{\prime}$. From the exact sequence

[TABLE]

one finds by duality relative to $K$ that

[TABLE]

That is, $\mathrm{Tr}_{\rho}(\mu)=0$, hence $\mathrm{Tr}_{F}(\mu)=C(\mu)=0$, where $C$ denotes the Cartier operation. This implies, for example, when $d=2$, that $A$ is a supersingular elliptic curve, $\Sigma=A^{\prime}$, and $K$ is $B\Omega^{1}_{A/k}$, the module of locally exact forms, which admits a basis over $\mathcal{O}_{A^{\prime}}$.

— Assume, in u), $k$ of characteristic $2$, and the extension $k(A)/k(\Sigma)$ separable.

Call $\sigma$ the generator of $\mathrm{Gal}(k(A)/k(\Sigma))\simeq\mathbf{Z}/2\mathbf{Z}$. By Weil [2] 4.4/1, $\sigma$ extends to an involution of $A$. Hence, $\Sigma$ is the quotient $A/\sigma$.

On $\mathcal{O}_{A}$, one has $(\sigma^{*}-1)^{2}=(\sigma^{*})^{2}-1=0$. And, $\sigma^{*}\mu=\mu$,

[TABLE]

When $\sigma$ has fixed points, one may by a translation assume that it fixes the origin of $A$, that is, assume that $\sigma$ is a $k$-group automorphism of $A$. Define

[TABLE]

two $\sigma$-invariant sub-abelian varieties of $A$. On $A_{1}$ and $A_{-1}$, $\sigma$ acts as $1$ and $-1$ respectively. The sum

[TABLE]

is an isogeny and $\sigma$-compatible, which by quotient gives

[TABLE]

Observe that the endomorphism $\mathrm{1-\sigma}:A\to A$ has kernel of (pure) codimension $1$ in $A$, so that $A_{-1}$ is an elliptic curve, and $A_{-1}/\sigma$ a projective line.

In fact, $\rho:A\to\Sigma$ is étale if it is étale at all codimension $\leq 1$ points of $A$. For, then $K=\mathcal{O}_{\Sigma}\mu$, the inclusion

[TABLE]

is bijective, the exact sequence

[TABLE]

shows that $\mathcal{O}_{A}$ is locally free over $\mathcal{O}_{\Sigma}$ of rank $2$, hence $A$ faithfully flat over $\Sigma$ and $\Sigma/k$ smooth, and $A$ is étale over $\Sigma$ by purity of branch locus.

— Assume, in u), $k$ of characteristic $\neq 2$.

Call $\sigma$ the generator of $\mathrm{Gal}(k(A)/k(\Sigma))\simeq\mathbf{Z}/2\mathbf{Z}$. By Weil [5] 4.4/1, $\sigma$ extends to an involution of $A$, and thus $\Sigma$ is the quotient $A/\sigma$.

We prove now that $\sigma^{*}\mu=-\mu$ :

*Case where $\sigma$ acts freely on $A$ *:

Then, $A\to\Sigma$ is étale, and $\Omega^{d-1}_{\Sigma/k}$ is the kernel of

[TABLE]

As $k$ is of characteristic $\neq 2$, the exact sequence

[TABLE]

splits. In particular, $H^{0}(\Sigma,K)=H^{0}(\Sigma,\mathcal{O}_{A}/\mathcal{O}_{\Sigma})=0.$

If $\sigma^{*}\mu$ were $\mu$ rather than $-\mu$, then $K$, as it is isomorphic to $\Omega^{d-1}_{\Sigma/k}$, would admit a basis. This proves the claim.

*Case where $\sigma$ acts on $A$ with fixed points *:

One may by a translation assume that $\sigma$ is a $k$-group automorphism of $A$. Consider the sub-abelian varieties of $A$,

[TABLE]

On $A_{1}$ and $A_{-1}$, $\sigma$ acts as $1$ and $-1$ respectively. The sum

[TABLE]

is an étale isogeny and $\sigma$-equivariant. Let $d_{-1}=\mathrm{dim}\ A_{-1}$. The set of points of $A$ where $A\to\Sigma$ is not étale is purely of codimension $d_{-1}$ in $A$.

Notice that the claim $\sigma^{*}\mu=-\mu$ is equivalent to that $A_{-1}$ is of odd dimension.

Assume on the contrary that $d_{-1}$ is even, in particular, $\geq 2$. Then $K$ is identical to the kernel of

[TABLE]

Since $k$ is of characteristic $\neq 2$, the exact sequence

[TABLE]

splits, by which $H^{0}(\Sigma,K)=0$. If $\sigma^{*}\mu$ were $\mu$ rather than $-\mu$, then $K$ would have a non-zero global section, a contradiction.

II) Assume $n\geq 1$ and that $S$ is the spectrum of an algebraically closed field $k$.

As $X$ is pseudo-isomorphic to $L^{*}\times\mathbf{Z}/n\mathbf{Z}$, $X$ is reduced. Moreover, $X$ is Cohen-Macaulay.

Let $U$ consist of every point of $X$ where $X$ is normal. Let $\rho:X^{\prime}\to X$ denote the normalization of $X$, and let $U$ be identified via $\rho$ with $\rho^{-1}(U)$. Write $I=H^{0}\rho^{!}\mathcal{O}$ for the conductor of $X^{\prime}/X$, coherent ideal in $X^{\prime}$ and in $X$, by which $Z^{\prime}=X^{\prime}-U^{\prime}$ (resp. $Z=X-U$) is given a closed sub-algebraic space structure in $X^{\prime}$ (resp. $X$).

As in I), one argues, on each connected component $X^{\prime}_{j}$ of $X^{\prime}$, that the pseudo-isomorphism $\varphi$ induces an $k$-isomorphism $\varphi^{\prime}_{j}$ of $X^{\prime}_{j}$ with $P_{L}$, that $Z^{\prime}\cap X^{\prime}_{j}$ is formed of two disjoint $A$-sections $\Sigma^{\prime}_{j0}$ and $\Sigma^{\prime}_{j\infty}$, $j\in\mathbf{Z}/n\mathbf{Z}$, and that $Z$ is normal, Cohen-Macaulay, of dualizing module $\mathcal{O}_{Z^{\prime}}/\mathcal{O}_{Z}$.

Also as in I), above each connected component $\Sigma$ of $Z$, either $\rho^{-1}(\Sigma)$ consists of two connected components $\Sigma^{\prime}_{j\alpha}$, $\Sigma^{\prime}_{l\beta}$, both biregular to $\Sigma$, $j,l\in\mathbf{Z}/n\mathbf{Z}$, $\alpha,\beta\in\{0,\infty\}$, or, $\rho^{-1}(\Sigma)=\Sigma^{\prime}_{j\alpha}$ is connected generically of degree $2$ over $\Sigma$, $j\in\mathbf{Z}/n\mathbf{Z}$, $\alpha\in\{0,\infty\}$.

These $\Sigma^{\prime}_{j\alpha}$ are $2n$ in number, and $X^{\prime}$ has $n$ connected components. So $X$, as it is connected by hypothesis, is pinched from $X^{\prime}$ by gluing along the $\Sigma^{\prime}_{j\alpha}$,

— either in a circle in which case $X=C_{nL}$,

— or in a chain on whose two ends the restrictions $\rho|\Sigma^{\prime}_{10}$, $\rho|\Sigma^{\prime}_{n\infty}$ are generically of degree $2$.

III) General case.

Let $U$ consist of all points of $X$ where $f$ is smooth. Its fiber over each geometric point $s$ of $S$ is formed exactly of all points of $f^{-1}(s)$ where $f\times_{S}s$ is smooth, since by assumption $f$ is flat of finite presentation.

Let $D=\mathrm{Dom}(\varphi)$.

*a) The $S$-algebraic space $U$ is parafactorial along $U-D$ universally. That is, for any $S$-algebraic space $S_{1}$, if $U_{1}$ (resp. $D_{1}$) denotes the base change of $U$ (resp. $D$) by $S_{1}\to S$, $U_{1}$ is parafactorial along $U_{1}-D_{1}$ *:

For, the complement of $D\cap U$ in $U$ is fiberwise of codimension $\geq 2$ in $U$ by hypothesis of minimal model, and $U$ smooth over $S$.

*b) The composition of the pseudo-isomorphism $\varphi$ with the projection $L^{*}\times\mathbf{Z}/n\mathbf{Z}\to A$ is defined on all of $U$ *:

In terms of the dual abelian algebraic space $A^{*}$ of $A$, the composition

[TABLE]

can be interpreted as the data of an invertible module on the abelian algebraic space over $D|U$, $A^{*}\times_{S}D|U$, which is fiberwise numerically trivial and trivial along the zero section.

Call this invertible module $N$. By a), an extension of $N$ to an invertible module on $A^{*}\times_{S}U$ exists, which is up to unique isomorphisms unique, trivial along the zero section, and, since $A$ is open and closed in $\mathrm{Pic}_{A^{*}/S}$, fiberwise numerically trivial, hence the claim.

*c) The pseudo-isomorphism $\varphi$ is defined on all of $U$ *:

In view of b), it suffices to observe that $L^{*}$ is affine over $A$, and that, by a), $\mathrm{prof}_{U-D}(U)\geq 2$.

*d) The morphism $\varphi|U$ is an isomorphism of $U$ with $L^{*}\times\mathbf{Z}/n\mathbf{Z}$ *:

The assertion can be verified fiber by fiber. Then it is immediate on the study made in I) and II).

*e) The geometric number $N(s)$ of connected components of $f^{-1}(s)-\mathrm{Im}(\varphi^{-1}_{s})$, $s\in S$, is a constructible function on $S$ *:

This follows from EGA IV 9.7.9.

*f) The function $s\mapsto N(s)$ is upper semi-continuous on $S[1/2]$ *:

By e), one may assume that $S$ is the spectrum of a discrete valuation ring with generic point $t$ and closed point $s$, $k(s)$ algebraically closed. One needs show that the case where $N(t)=n+1$, $N(s)=n$ does not happen.

Otherwise, replacing if necessary $S$ by a spectrum $S^{\prime}$ of discrete valuation ring faithfully flat over $S$, and $(f,\varphi)$ by $(f\times_{S}S^{\prime},\varphi\times_{S}S^{\prime})$, one would have $f^{-1}(s)=C_{nL}$, and that $f^{-1}(t)-\mathrm{Im}(\varphi^{-1}_{t})$ admit one component of the form $A_{t}/\sigma_{t}$, where $\sigma_{t}$ is an involutive automorphism of $A_{t}$. Recall from I) that $\sigma^{*}_{t}$ acts as $-1$ on the volume forms of $A_{t}$.

As $A$ is the $S$-Néron model of $A_{t}$, a unique involution $\sigma$ of $A$ extends $\sigma_{t}$. Write

[TABLE]

where $\alpha$ is an $S$-group automorphism of $A$, and $a\in A(S)$. That $\sigma^{2}=1$ amounts to that $\alpha^{2}=1$, $\alpha(a)=-a$.

Let $\Sigma$ denote the closed image of $A_{t}/\sigma_{t}$ in $X$. Its closed fiber $\Sigma_{s}$ is connected by Zariski and purely of dimension $d-1$, hence is a connected component of $f^{-1}(s)-\mathrm{Im}(\varphi^{-1}_{s})$. In particular, $(\Sigma_{s})_{\mathrm{red}}\simeq A_{s}$ and does not contain $k(s)$-rational curves, so the projection $A_{t}\to\Sigma_{t}$ extends to an $S$-morphism $A\to\Sigma$ ([5] 2.1).

This morphism $A\to\Sigma$ is proper, thus surjective, thus finite. For, $A_{s}\to(\Sigma_{s})_{\mathrm{red}}=A_{s}$, being surjective, is up to translation an isogeny, with say finite kernel $E_{s}$.

Therefore, $\Sigma$ is of normalization $A/\sigma$. Note that $(A/\sigma)_{s}=A_{s}/\sigma_{s}$, since $k(s)$ is of characteristic $\neq 2$. The factorization

[TABLE]

implies that $\sigma(x)-x$, $\forall\ x\in A_{s}(s)$, are contained in $E_{s}(s)$. Namely, the sub-abelian variety $\mathrm{Im}(\alpha_{s}-1)$ of $A_{s}$ is contained in $-a_{s}+E_{s}$.

But $E_{s}$ is finite over $s$. So $\mathrm{Im}(\alpha_{s}-1)=0$, $\alpha_{s}=1$, and so $\alpha=1$, since the functor of $S$-group automorphisms of $A$ is representable and locally unramified over $S$.

This shows that $\sigma$ is a translation. Hence, $\sigma^{*}_{t}\mu$ is $\mu$ rather than $-\mu$ for any volume form $\mu$ on $A_{t}$, a contradiction.

*g) The function $s\mapsto N(s)$ is lower semi-continuous on $S[1/2]$ *:

Since $s\mapsto h^{1}(s):=\mathrm{dim}_{k(s)}\ H^{1}(f^{-1}(s),\mathcal{O})$ is upper semi-continuous, it suffices to show that $h^{1}(s)$ is $d(s)$ and $<d(s)$, when $N(s)=n$ and $N(s)=n+1$ respectively, where $d(s)=\mathrm{dim}\ f^{-1}(s)$, $s\in S[1/2]$; note that $f$ is equi-dimensional, as $\mathrm{Im}(\varphi^{-1})$ is $S$-dense in $X$.

— *Calculation of $h^{i}(s):=\mathrm{dim}_{k(s)}\ H^{i}(f^{-1}(s),\mathcal{O})$, when $N(s)=n$ *:

One may assume $S=s=\mathrm{Spec}(k)$, $k$ an algebraically closed field. Then $X=C_{nL}$. Let $d=\mathrm{dim}\ X$. Consider the canonical exact sequence

[TABLE]

where $X^{\prime}=P_{L}\times\mathbf{Z}/n\mathbf{Z}$ is the normalization of $X$.

With the notations of II), write

[TABLE]

for the projection.

Both $H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}})$ and $H^{0}(X,\mathcal{O}_{X^{\prime}}/\mathcal{O}_{X})=H^{0}(Z,\mathcal{O}_{Z^{\prime}}/\mathcal{O}_{Z})$ are of dimension $n$ over $k$, and the exact sequence

[TABLE]

shows that $\mathrm{Ker}\ H^{0}(\Pi)$ is of dimension $1$. So $\mathrm{Coker}\ H^{0}(\Pi)$ is of dimension $1$ as well.

Observe that, for each integer $i$, $H^{i}(A,\mathcal{O})\simeq H^{i}(P_{L},\mathcal{O})$, via which the map

[TABLE]

can be identified with

[TABLE]

In particular, both kernel and cokernel of $H^{i}(\Pi)$ have the same dimension as $H^{i}(A,\mathcal{O})$. From the cohomology exact sequence

[TABLE]

it follows finally that $H^{i}(X,\mathcal{O})$ is of dimension

[TABLE]

— *Proof of $h^{1}(s)<d(s)$, when $N(s)=n+1$ *:

Similarly, one may assume $S=s=\mathrm{Spec}(k)$, $k$ an algebraically closed field of characteristic $\neq 2$. Let $d=\mathrm{dim}\ X$. Again, consider the exact sequence

[TABLE]

In the notations of II), $\mathcal{O}_{X^{\prime}}/\mathcal{O}_{X}=\mathcal{O}_{Z^{\prime}}/\mathcal{O}_{Z}$.

Recall that $X$ is pinched from $X^{\prime}=P_{L}\times\mathbf{Z}/n\mathbf{Z}$ in a chain. On its two ends, $\rho(\Sigma^{\prime}_{10})=\Sigma_{10}$, $\rho(\Sigma^{\prime}_{n\infty})=\Sigma_{n\infty}$, the dualizing module $K_{Z}$ has no non-zero global sections; in between, which consists of $n-1$ connected components, $K_{Z}$ is isomorphic to $\mathcal{O}_{Z}$. So $H^{0}(Z,\mathcal{O}_{Z^{\prime}}/\mathcal{O}_{Z})=H^{0}(Z,K_{Z})$ is of dimension $n-1$ over $k$, thus

[TABLE]

is exact, and thus

[TABLE]

which as one verifies easily is isomorphic to

[TABLE]

where $X^{\prime}_{1}=P_{L}\times\{1\}$ is the left most component of $X^{\prime}$. This finishes the proof, in view that $H^{1}(X^{\prime}_{1},\mathcal{O})\simeq H^{1}(A,\mathcal{O})$ is of dimension $d-1$ over $k$.

*h) The function $s\mapsto N(s)$ is locally constant on $S[1/2]$ *:

Combine f) and g).

The proof is now complete.