# Conic bundles that are not birational to
numerical Calabi–Yau pairs

János Kollár

###### Abstract

Let $X$ be a general conic bundle over ${\mathbb{P}}^{2}$ with branch curve of
degree at least 19. We prove that there is no normal projective variety $Y$
that is birational to $X$ and such that some multiple of its anticanonical
divisor is effective.

  *     * Keywords. Rationally connected variety, conic bundle, Calabi–Yau variety, birational equivalence.

2010 Mathematics Subject Classification. 14M22, 14J45, 14J20 (Primary); 14J32,
14E05 (Secondary).

[Français]

Titre. Fibrés en coniques qui ne sont pas birationnels à des paires de
Calabi–Yau numériques Résumé. Soit $X$ un fibré en coniques général sur
${\mathbb{P}}^{2}$ avec une courbe de branchement de degré au moins 19. Nous
montrons qu’il n’existe pas de variété projective normale $Y$ qui soit
birationnelle à $X$ et telle qu’un multiple de son diviseur anticanonique soit
effectif.

Understanding the difference between rationally connected varieties and Fano
varieties has long been a goal of birational geometry. In any dimension,
smooth Fano varieties come in finitely many families but rationally connected
varieties form infinitely many families. Thus one expects that not every
rationally connected variety is birational to a Fano variety, but actual
proofs of this fact have been quite subtle; see [Sar80] or [Cor95].

A more general form of this problem asks if every rationally connected variety
$X$ is birational to the underlying variety of a ${\mathbb{Q}}$-Fano pair
$(Y,\Delta)$. Since ${\mathbb{Q}}$-Fano pairs form infinitely many families
up-to birational equivalence [Oka09], this is a harder variant. This form of
the question was posed in [CG13] and a negative answer is established in
[Kry14]. Closely related results describing Calabi–Yau fiber-space structures
on certain Fano 3-folds are proved in [Che04, Ryd06, CK10].

Probably the most general question in this direction is whether every
rationally connected variety is birational to the underlying variety of a
numerical Calabi–Yau pair $(Y,\Delta)$. Here we allow the most general
definition of Calabi–Yau pairs, that is, $Y$ is normal, proper, $\Delta$ is a
pseudo-effective ${\mathbb{R}}$-divisor, $K_{Y}+\Delta$ is
${\mathbb{R}}$-Cartier and $K_{Y}+\Delta\equiv 0$, but we impose no
restrictions on the singularities of $Y$. (For most purposes the more
restrictive definition of Calabi–Yau pairs adopted in [KX16] is the best; but
the above numerical version also seems natural.) It turns out that the
singularities of $Y$ are not very important and the difference between
effective and pseudo-effective divisors also may not be significant, but
allowing divisors with coefficient $>1$ in $\Delta$ leads to many more cases;
see Example 4 and Definition 6. If $X$ is birational to a numerical Calabi–Yau
pair $(Y,\Delta)$ then the birational transform of $\Delta$ on $X$ is
frequently a quite interesting divisor. Understanding such divisors was a key
to proving unirationality of degree 1 conic bundle surfaces [KM17].

We discuss several methods to show that certain conic bundles (see Definition
5) are not birational to any numerical Calabi–Yau pair. Typical results are
the following.

###### Theorem 1

There are conic bundles $S\to{\mathbb{P}}^{1}$ defined over ${\mathbb{Q}}$
that are not birational to any numerical Calabi–Yau surface.

This property seems to depend very subtly on the coefficients involved in the
definition of $S$ and we give only sufficient conditions in Example 23. The
following are some concrete special cases.

###### Example 2

Let $p\geq 11$ be a prime such that $p\equiv-1\mod 4$ and choose
$m\in{\mathbb{Z}}$ not divisible by $p$. Then the conic bundle

$\Bigl{(}z^{2}=\Bigl{(}\tfrac{s^{p-1}-1}{s^{2}-1}\Bigr{)}x^{2}+\bigl{(}s^{p-1}-1+mp\bigr{)}y^{2}\Bigr{)}\subset{\mathbb{P}}^{2}_{xyz}\times{\mathbb{A}}^{1}_{s}$

is not birational—over ${\mathbb{Q}}$—to any numerical Calabi–Yau surface.

In dimension 3 we get the following.

###### Theorem 3

Let $X_{d,2}\subset{\mathbb{P}}^{2}\times{\mathbb{P}}^{2}$ be a general
hypersurface of bidegree $(d,2)$ over a field of characteristic $\neq 2,3,5$.
Then $X_{d,2}$ is not birational to the underlying variety of a numerical
Calabi–Yau pair for $d\geq 7$.

I could not write down explicit examples with smooth branch curve, but it is
easy to get many with reducible branch curve using Corollary 17.

It is quite likely that the theorem also holds over any infinite field. The
main arguments in our paper work whenever the characteristic is $\neq 2$, but
a key reference [Sar80] is stated in the literature only for characteristic
$\neq 2,3$ and we also use resolution of singularities.

The most important numerical invariant of a conic bundle
$\pi:X\to{\mathbb{P}}^{n}$ is the degree of its branch divisor
$B_{X}\subset{\mathbb{P}}^{n}$. Typical results say that for smooth branch
divisors the degree is the only important invariant and the higher the degree
of $B_{X}$, the more complicated $X$ is. This is, however, not the case for
our question. We see in Examples 20–21 that there are conic bundles
$\pi:X\to{\mathbb{P}}^{2}$ with smooth branch curve of arbitrary high degree
(resp. $\pi:S\to{\mathbb{P}}^{1}$ with many singular fibers) that are
birational to numerical Calabi–Yau varieties.

The following example shows that many varieties are birational to the
underlying variety of a smooth numerical Calabi–Yau pair.

###### Example 4

If $(Y,\Delta)$ is a numerical Calabi–Yau pair and $\Delta\neq 0$ then $Y$ is
uniruled by [MM86] but $Y$ need not be rationally connected. In fact, for any
smooth, projective variety $X$, the product $X\times{\mathbb{P}}^{1}$ is
birational to a numerical Calabi–Yau pair. To see this, let $H$ be an ample
divisor on $X$ such that $H-K_{X}$ is effective and note that the
anticanonical class of
${\mathbb{P}}_{X}\bigl{(}{\mathcal{O}}_{X}\oplus{\mathcal{O}}_{X}(H)\bigr{)}$
is effective. It is the sum of twice the negative section and of the pull-back
of $H-K_{X}$.

In particular, the conic bundles in Theorem 3 are not birationally ruled.
However, we use even stronger non-rationality results of [Sar80] during the
proof.

###### Definition 5

A conic bundle is a flat, proper morphism $\pi:X\to Z$ such that every fiber
of $\pi$ is isomorphic to a plane conic. We are mainly interested in the cases
when $Z$ is regular, but for the basic definition it is enough to assume that
$Z$ is normal. (For many purposes one should allow non-flat morphisms and more
singular fibers, but for us the restrictive version is more convenient.)

A conic bundle is called extremal if for every codimension 1 point $z\in Z$
the fiber $X_{z}$ is irreducible over $k(z)$. Equivalently, the relative
Picard number is 1.

A conic bundle is called minimal if it is extremal and has no rational
sections. (This is the “right” definition for conic bundles but note that a
product $Z\times{\mathbb{P}}^{1}\to Z$ is extremal but not minimal in our
sense.) If $X$ is regular then $\pi:X\to Z$ is minimal iff
$\operatorname{Pic}(X)=\pi^{*}\operatorname{Pic}(Z)\oplus{\mathbb{Z}}[\omega_{X/Z}^{-1}]$;
this is the key property that we are interested in. This is equivalent to
saying that if $L$ is any line bundle on $X$ and $C\subset X$ is a fiber then
$\deg(L|_{C})$ is even and if $C$ is reducible then $L|_{C}$ has the same
degree on both irreducible components.

The branch locus, denoted by $B_{X}\subset Z$, is the subscheme parametrizing
singular fibers of $\pi$. Set theoretically it consists of points $z\in Z$
such that $X_{z}:=\pi^{-1}(z)$ is singular. If $X$ is regular then $B_{X}$ is
reduced. In general, the scheme structure is given by the formula (25.3) which
also shows that $B_{X}$ has pure codimension 1 in $Z$, except possibly when
$\operatorname{char}k(Z)=2$. In order to avoid various complications, we
assume from now on that all residue characteristics are $\neq 2$. Our main
interest is in conic bundles over ${\mathbb{C}}$ or ${\mathbb{Q}}$, but we
will use some examples that are defined over ${\mathbb{Z}}_{p}$.

Assume that $Z$ is regular and let $z\in B_{X}$ be a regular point. Then
$X_{z}$ is a pair of lines and $X$ is regular along $X_{z}$; see Paragraph 25.
This defines a double cover $\tilde{B}_{X}\to B_{X}$ that is étale over the
regular locus of $B_{X}$. Then $X\to Z$ is minimal iff, for every irreducible
component $B_{i}\subset B_{X}$, the corresponding $\tilde{B}_{i}$ is also
irreducible.

I do not know a good general introduction to conic bundles, but everything can
be gleaned from the basic sources [Isk67, Bea77, Sar80, Isk87] or [Cor00,
Chap.4], [KSC04, 3.11-13]. The key results are discussed in Paragraphs 25–28.

###### Definition 6

Let $k$ be a field and $X$ a normal, proper $k$-variety of dimension $n$. A
divisor class $B$ is called effective (resp. ${\mathbb{Q}}$-effective) if $B$
is linearly (resp. ${\mathbb{Q}}$-linearly) equivalent to an effective
${\mathbb{Z}}$-divisor (resp. ${\mathbb{Q}}$-divisor). An
${\mathbb{R}}$-divisor $B$ is called pseudo-effective if its class in
$\operatorname{N}_{n-1}(X)_{{\mathbb{R}}}$ is a limit of
${\mathbb{Q}}$-effective divisors.

Birational transformation of divisors does not preserve linear equivalence, so
it is not very useful to ask whether a divisor is birationally effective or
not. However, the birational transform of a mobile linear system is well
defined and the canonical class makes sense on any birational model of $X$.

Let $|M|$ be a mobile linear system on $X$. We say that $-K_{X}+|M|$ is
birationally effective (resp. birationally ${\mathbb{Q}}$-effective or
birationally pseudo-effective) if there is a normal, proper $k$-variety
$\phi:X^{\prime}\stackrel{{\scriptstyle bir}}{{\sim}}X$ such that
$-K_{X^{\prime}}+\phi_{*}|M|$ is effective (resp. ${\mathbb{Q}}$-effective or
pseudo-effective).

For now our main interest is in the case $|M|=0$.

We stress that we do not impose a priori restrictions on the singularities of
$X^{\prime}$ but it is easy to improve them, at least in characteristic 0.

Assume that $-K_{X^{\prime}}\sim_{{\mathbb{Q}}}D^{\prime}$ is pseudo-effective
and let $p:X^{\prime\prime}\to X^{\prime}$ be a terminal modification [Kol13,
1.33]. Write
$-K_{X^{\prime\prime}}\sim_{{\mathbb{Q}}}E^{\prime\prime}+D^{\prime\prime}$
where $E^{\prime\prime}$ is $p$-exceptional and $D^{\prime\prime}$ is the
birational transform of $D^{\prime}$. By the Negativity lemma [KM98, 3.39] we
see that $E^{\prime\prime}$ is an effective ${\mathbb{R}}$-divisor, hence
$-K_{X^{\prime\prime}}$ is pseudo-effective.

If $K_{X^{\prime\prime}}$ is not numerically trivial, then it is also not
pseudo-effective, hence a suitable minimal model program terminates with a
Mori fiber space $X^{m}\to Z$. That is, $-K_{X^{m}}$ is relatively ample and
the relative Picard number of $X^{m}/Z$ is 1. Note that
$\phi:X^{\prime\prime}\dasharrow X^{m}$ is a birational contraction, that is,
$\phi^{-1}$ has no exceptional divisors. Thus if
$-K_{X^{\prime\prime}}\sim_{{\mathbb{Q}}}D^{\prime\prime}$ then
$-K_{X^{m}}=-\phi_{*}(K_{X^{\prime\prime}})\sim_{{\mathbb{Q}}}\phi_{*}(D^{\prime\prime})=:D^{m}$
shows that $-K_{X^{m}}$ is also pseudo-effective.

Thus the following holds.

Claim 6.1. Let $k$ be a field of characteristic 0 and $X$ a normal
$k$-variety. Then $-K_{X}$ is birationally pseudo-effective iff

  1. (a)

either $X$ has canonical singularities and $K_{X}$ is numerically trivial,

  2. (b)

or there is a projective variety $X^{m}$ with terminal singularities such that
$X^{m}$ is birational to $X$, $-K_{X^{m}}$ is pseudo-effective and there is a
Mori fiber space structure $\pi:X^{m}\to Z$.

This suggests that our question should be treated using the Noether–Fano
method. (See [KSC04, Chap.5] for an introduction and [Cor00] for a more
detailed treatment.) This is the approach taken in [Kry14] and the examples
given there are also not birational to a numerical Calabi–Yau pair. We work
with conic bundles. These have a rich birational geometry yet all of their
birational models are quite well understood.

The characteristic 0 assumption is only needed to guarantee that resolutions
and Mori fiber space models exist. Thus Claim 6 holds over any field $k$ if
$\dim X=2$ or if $\dim X=3$ and $\operatorname{char}k>5$.

Note also that in birational geometry one would usually like to control the
singularities of the pair $(X^{\prime},\Delta^{\prime})$, not just the
singularities of $X^{\prime}$. However, even if $\Delta^{\prime}$ is
effective, we do not assume that the coefficients in $\Delta^{\prime}$ are
$\leq 1$, thus a similar reduction to the log canonical case is not possible.

###### Remark 7

Our notion is close to the concept of Fano type varieties introduced in
[PS09], which asks for a normal, proper $k$-variety
$X^{\prime}\stackrel{{\scriptstyle bir}}{{\sim}}X$ such that
$-K_{X^{\prime}}\sim_{{\mathbb{Q}}}\Delta^{\prime}+H^{\prime}$ where
$(X^{\prime},\Delta^{\prime})$ is klt and $H^{\prime}$ is an ample
${\mathbb{Q}}$-divisor. (That is, $(X^{\prime},\Delta^{\prime})$ is a log-Fano
pair.) ¿From the technical point of view the difference between $-K_{X}$ being
birationally pseudo-effective and $X$ being of Fano type can be substantial,
but in our examples none of the difficulties appear.

######   8 (Outline of the proof of Theorems 1 and 3)

Start with $X\to{\mathbb{P}}^{2}$ over ${\mathbb{C}}$ and restrict to a
generic line ${\mathbb{P}}^{1}\subset{\mathbb{P}}^{2}$. We get a 2-dimensional
conic bundle $S\to{\mathbb{P}}^{1}$ over the function field
${\mathbb{C}}(s,t)$. A slight complication is that $-K_{X}|_{S}=-K_{S}+F$
where $F$ is a general fiber, but this is easy to deal with. Next we study
when $-K_{S}+F$ is birationally pseudo-effective for 2-dimensional conic
bundles $S\to{\mathbb{P}}^{1}$ over ${\mathbb{C}}(s,t)$, or, more generally,
over any field $k$. We show in Corollary 11 that this holds iff $S$ contains a
double section $C\subset S$ whose normalization has genus $\leq 2$. The double
section $C\subset S$ then extends to a generically finite double section
$D\subset X$ whose normalization is a birational to a K3 surface. (The
projection $\pi_{D}:D\to{\mathbb{P}}^{2}$ has degree 2 and the ramification
curve is a sextic, but $\pi_{D}$ need not be finite.) Then we study how the
branch curve of $X\to{\mathbb{P}}^{2}$ and the branch curve of
$D\to{\mathbb{P}}^{2}$ intersect.

In order to prove Theorem 1, we extend a 2-dimensional conic bundle over
${\mathbb{P}}^{1}_{{\mathbb{Q}}}$ to a 3-dimensional conic bundle over
${\mathbb{P}}^{1}_{{\mathbb{Z}}}$ and argue as above. The answer seems to
depend on subtle properties of $S\to{\mathbb{P}}^{1}_{{\mathbb{Q}}}$. In
particular, the branch divisor $B_{S}\subset{\mathbb{P}}^{1}_{{\mathbb{Q}}}$
alone is not enough to decide what happens; see Example 21.

## Surface conic bundles

Fix an arbitrary field $k$. Assume for simplicity that
$\operatorname{char}k\neq 2$. Let $S$ be a smooth surface over $k$ and
$S\to{\mathbb{P}}^{1}$ a minimal conic bundle. The number of singular
geometric fibers, which is the degree of the branch locus $B_{S}$, is denoted
by $\delta(S)$. A typical fiber is denoted by $F$. Thus
$(K_{S}^{2})=8-\delta(S)$ and $(K_{S}\cdot F)=-2$.

###### Lemma 9

Let $k$ be a field and $\pi:S\to{\mathbb{P}}^{1}$ a minimal conic bundle. Fix
a natural number $m$ such that $\delta(S)>12+6m$. Assume that $-K_{S}+mF$ is
pseudo-effective.

Then there is a unique irreducible curve $C\subset S$ such that $(C^{2})<0$.
Furthermore, $\pi:C\to{\mathbb{P}}^{1}$ has degree 2 and $|-K_{S}+mF|=C+|bF|$
for some $b\geq 0$. In particular, $-K_{S}+mF$ is effective.

  * Proof.

By assumption there is a sequence of effective ${\mathbb{Q}}$-divisors $D_{t}$
converging to $-K_{S}+mF$. Note that $(-K_{S}+mF)^{2}=8-\delta(S)+4m<0$, hence
$(D_{t}^{2})<0$ for some $t$. So there is a $k$-irreducible component
$C\subset\operatorname{Supp}D_{t}$ with $C^{2}<0$. Thus the cone of curves is
generated by $C$ and a fiber $F$.

Write $C\sim_{{\mathbb{Q}}}a(-K_{S}+m^{\prime}F)$. Here $a$ is an integer but
$m^{\prime}$ could be rational. The degree of the dualizing sheaf of $C$ is

$\deg\omega_{C}=2p_{a}(C)-2=C(C+K_{S})=a(a-1)(K_{S}^{2})+2a(2a-1)m^{\prime}.$

Since $C_{\bar{k}}$ has at most $2a$ irreducible components,
$\deg\omega_{C}\geq-4a$, hence

$a(a-1)(K_{S}^{2})+2a(2a-1)m^{\prime}\geq-4a.$

If $a\geq 2$ then this rearranges to

$\delta(S)-8=-(K_{S}^{2})\leq\bigl{(}4+\tfrac{2}{a-1}\bigr{)}m^{\prime}+\tfrac{4}{a-1}\leq
6m^{\prime}+4.$

Note that $-K_{S}+mF\sim_{{\mathbb{Q}}}\frac{1}{a}C+(m-m^{\prime})F$, hence
$m^{\prime}\leq m$ since $-K_{S}+mF$ is pseudo-effective. Thus, if
$\delta(S)>12+6m$ then $a=1$.

Note that we also could have used Bend-and-break (as stated in [Kol96,
II.5.5.3]) to show that $a\leq 2$.

###### Example 10

There are minimal conic bundles $S\to{\mathbb{P}}^{1}$ with $\delta(S)=12$ for
which $-2K_{S}$ is effective but $-K_{S}$ is not effective.

To construct such examples, let $Q\subset{\mathbb{P}}^{2}$ be an irreducible
degree 4 point. Then $B_{Q}{\mathbb{P}}^{2}\to{\mathbb{P}}^{1}$ is a minimal
conic bundle with with $\delta=3$. The exceptional curves give a conjugate set
of 4 sections, each with self-intersection $-1$.

Pull it back by a general degree $r$ map ${\mathbb{P}}^{1}\to{\mathbb{P}}^{1}$
to get $S\to{\mathbb{P}}^{1}$. We get a minimal conic bundle
$S\to{\mathbb{P}}^{1}$ with $\delta(S)=3r$ and a conjugate set of 4 sections,
each with self-intersection $-r$. The sum of these 4 sections is in
$|-2K_{S}+(r-4)F|$. Thus the bound $\delta(S)>12+6m$ in Lemma 9 is sharp.

(It is interesting to note that for $r=4$ contracting the 4 sections gives
$S\to T$ where $T$ is a singular Enriques surface. It has 4 singular points
and $2K_{T}\sim 0$.)

###### Corollary 11

Let $k$ be a field and $S\to{\mathbb{P}}^{1}$ a minimal conic bundle. Assume
that $\delta(S)>12+6m$. Then $-K_{S}+mF$ is birationally pseudo-effective iff
$S$ contains a double section $C\subset S$ whose normalization has genus $\leq
m+1$.

  * Proof.

Assume that $S\dasharrow S^{\prime}$ is birational and
$-K_{S^{\prime}}+mF^{\prime}$ is pseudo-effective. By [Isk67] (see also
Theorem 27), $S^{\prime}$ is another minimal conic bundle with the same
$\delta$, $S\to{\mathbb{P}}^{1}$ and $S^{\prime}\to{\mathbb{P}}^{1}$ have the
same generic fiber and $-K_{S^{\prime}}+m^{\prime}F^{\prime}$ is linearly
equivalent to an irreducible double section $C^{\prime}$ for some
$m^{\prime}\leq m$ by Lemma 9. Thus $C$ is obtained as the birational
transform of $C^{\prime}$. Its normalization has genus $\leq m+1$ by Lemma 12.

Conversely, let $C\subset S$ be a double section. We can resolve the
singularities of $C$ by performing elementary transformations at its singular
points; see Paragraph 26. (Clearly, these points are on smooth fibers.) At the
end we have $C^{\prime}\subset S^{\prime}$ and $C^{\prime}$ is smooth. If
$p_{a}(C^{\prime})\leq m+1$ then $C^{\prime}$ is a sub-curve of
$|-K_{S^{\prime}}+mF^{\prime}|$ by Lemma 12.

###### Lemma 12

Let $C\subset S$ be a double section. Then $C$ is a sub-curve of $|-K_{S}+mF|$
iff $p_{a}(C)\leq m+1$.

  * Proof.

$C\sim-K_{S}+bF$ for some $b\in{\mathbb{Z}}$. Thus
$2p_{a}(C)-2=C(C+K_{S})=b(C\cdot F)=2b$ hence $b\leq m$ iff $p_{a}(C)\leq
m+1$.

## Threefold conic bundles

We use Corollary 11 to prove a similar result for 3-dimensional conic bundles.

###### Theorem 13

Let $\pi:X\to{\mathbb{P}}^{2}$ be a smooth, minimal conic bundle over a field
of characteristic $\neq 2,3,5$ whose branch curve
$B_{X}\subset{\mathbb{P}}^{2}$ has degree $\geq 19$. The following are
equivalent.

  * 1.

$-K_{X}$ is birationally pseudo-effective.

  * 2.

There is a generically finite double section $D\subset X$ with normalization
$\tau:\bar{D}\to D$ such that the branch curve $B_{D}$ of
$\pi\circ\tau:\bar{D}\to{\mathbb{P}}^{2}$ has degree $\leq 6$.

  * 3.

$X$ is birational to a smooth, minimal conic bundle
$\pi^{\prime}:X^{\prime}\to{\mathbb{P}}^{2}$ (with the same branch curve) such
that $|-K_{X^{\prime}}|\neq\emptyset$.

  * Proof.

Assume that $\phi:X\dasharrow X^{\prime}$ shows that $-K_{X}$ is birationally
pseudo-effective. By Complement 28, we may choose $X^{\prime}$ such that
$\pi^{\prime}:X^{\prime}\to{\mathbb{P}}^{2}$ is also a conic bundle with the
same branch curve. Write $-K_{X^{\prime}}\sim_{{\mathbb{Q}}}\Delta^{\prime}$
where $\Delta^{\prime}$ is pseudo-effective.

Let $L\subset{\mathbb{P}}^{2}$ be the generic line. By restriction we get
$\pi^{\prime}_{S}:S^{\prime}\to L$. Since
$-K_{S^{\prime}}+F=-K_{X^{\prime}}|_{S^{\prime}}$, we see that
$-K_{S^{\prime}}+F\sim_{{\mathbb{Q}}}\Delta^{\prime}|_{S^{\prime}}$ is pseudo-
effective. Moreover, if $\Delta^{\prime}_{t}$ is a sequence of
${\mathbb{Q}}$-effective divisors converging to $\Delta^{\prime}$ then
$\Delta^{\prime}_{t}|_{S^{\prime}}$ is a sequence of ${\mathbb{Q}}$-effective
divisors converging to $\Delta^{\prime}|_{S^{\prime}}$.

Thus, by Lemma 9, $\Delta^{\prime}_{t}|_{S^{\prime}}$ has an irreducible
component $C^{\prime}\subset S^{\prime}$ with negative self-intersection and
its normalization has genus $\leq 2$ by Lemma 12. Since $L$ is the generic
line, $C^{\prime}$ is the restriction of an irreducible component $D^{\prime}$
of $\Delta^{\prime}_{t}$. Thus $D^{\prime}$ is a double section and its
birational transform on $X$ is a double section $D\subset X$. Furthermore, we
know that the preimage of $L$ in the normalization $\tau:\bar{D}\to D$ has
genus $\leq 2$. Thus $L$ intersects the branch curve of
$\pi\circ\tau:\bar{D}\to{\mathbb{P}}^{2}$ in $\leq 6$ points. This shows that
(1) $\Rightarrow$ (2).

Assume next that $\pi$ has a double section $D\subset X$ as in (2). By Lemma
14, there is a sequence of elementary transformations $X\dasharrow X^{\prime}$
such that the branch curve of $\pi^{\prime}:D^{\prime}\to{\mathbb{P}}^{2}$ has
degree $\leq 6$. By Lemma 15 this implies that $D^{\prime}$ is a sub-divisor
of $|-K_{X^{\prime}}|$. Thus (2) $\Rightarrow$ (3) and (3) $\Rightarrow$ (1)
holds by definition.

###### Lemma 14

Let $\pi:X\to Z$ be a conic bundle. Assume that $X$ and $Z$ are smooth and
$\dim Z=2$. Let $D\subset X$ be a generically finite double section. Then
there is a sequence of elementary transformations $X\dasharrow X^{\prime}$
such that the resulting $D^{\prime}\subset X^{\prime}$ is normal, except
possibly along finitely many fibers.

  * Proof.

Let $C\subset\operatorname{Sing}D$ be a curve not contained in a fiber of
$\pi$. Let $F^{\prime}\subset F\subset X$ be an irreducible component of a
fiber that is not contained in $D$ but meets $C$ at a point $p$. Since $D$ is
singular at $p$, the local intersection number $(D\cdot F^{\prime})_{p}$ is at
least 2. Since $D$ is a double section, $(D\cdot F)\leq 2$. Thus $F$ is
irreducible and $D\cap F=\\{p\\}$, hence $\pi_{C}:C\to\pi(C)$ is birational.
As we discuss in Paragraph 26, the elementary transformation centered at $C$
decreases the degree of the branch curve of $D\to{\mathbb{P}}^{2}$. After
finitely many such steps we get $\pi^{\prime}:X^{\prime}\to Z$ such that
$D^{\prime}\subset X^{\prime}$ is normal, except possibly along finitely many
fibers.

###### Lemma 15

Let $\pi:X\to{\mathbb{P}}^{2}$ be a minimal conic bundle and $D\subset X$ a
generically finite double section. Then $D$ is a sub-divisor of $|-K_{X}|$ iff
the branch curve $B_{D}\subset{\mathbb{P}}^{2}$ of
$\pi_{D}:D\to{\mathbb{P}}^{2}$ has degree $\leq 6$.

  * Proof.

Since $X$ is minimal, we know that $D\sim-K_{X}+b\pi^{*}H$ for some
$b\in{\mathbb{Z}}$ where $H$ is the class of a line in ${\mathbb{P}}^{2}$.
Thus $K_{D}\sim b\pi_{D}^{*}H$.

Let $D\to D^{\prime}\to{\mathbb{P}}^{2}$ denote the Stein factorization. Then
$K_{D}\sim b\pi_{D}^{*}H$ pushes forward to $K_{D^{\prime}}\sim
b\pi_{D^{\prime}}^{*}H$. By the Hurwitz formula
$K_{D^{\prime}}\sim\pi_{D^{\prime}}^{*}\bigl{(}K_{{\mathbb{P}}^{2}}+\tfrac{1}{2}B_{D}\bigr{)}$.
Thus $b\leq 0$ iff $\deg B_{D}\leq 6$.

## K3 surfaces on conic bundles

In order to prove Theorem 3, it remains to show that (2.) does not hold for
suitable branch curves $B\subset{\mathbb{P}}^{2}$. Example 20 shows that there
are conic bundles with high degree smooth branch curve for which $|-K_{X}|$
does contain a K3 surface. Thus we need to focus on more subtle properties of
$B$. We present 2 approaches. The first uses branch curves with many nodes; we
prove that all the nodes have to lie on a sextic curve. This leads to quite
explicit examples starting with $\deg B=12$; see Corollary 17.

Note also that, in a flat family of conic bundles, double sections are
parametrized by countably many components of the Chow variety. Thus if we find
one conic bundle without certain type of double sections then the very general
conic bundle also has no double sections of the same type.

The other approach proves that there has to be a sextic curve that is
everywhere tangent to $B_{X}$. This is easily seen to be a non-empty Zariski
open condition for $\deg B\geq 15$, though I could not write down explicit
examples. As pointed out by [Sar82, Sec.5], the results of [AM72, Sec.3] imply
that every smooth plane curve of degree $\geq 3$ is the branch curve of a
minimal conic bundle.

###### Lemma 16

Let $\pi:X\to Z$ be a minimal, smooth conic bundle with branch curve $B_{X}$.
Let $D\subset X$ be a double section with normalization $\tau:\bar{D}\to D$
and $B_{D}\subset Z$ the branch curve of $\pi|_{D}\circ\tau:\bar{D}\to Z$.
Then $\operatorname{Sing}(B_{X})\subset B_{D}$.

  * Proof.

If $p\notin B_{D}$ then both of the étale local branches of $\bar{D}\to Z$
give étale local sections of $\pi$. As we discuss in Claim 25, there are no
étale local sections over the nodes of $B_{X}$ for $X$ smooth.

###### Corollary 17

Let $X\subset{\mathbb{P}}^{2}_{\mathbf{x}}\times{\mathbb{P}}^{2}_{\mathbf{y}}$
be given by an equation of bidegree $(d,2)$

$\textstyle{\sum}_{i}g_{i}(x_{0},x_{1},x_{2})y_{i}^{2}=0.$

Assume that the curves
$B_{i}:=\bigl{(}g_{i}(x_{0},x_{1},x_{2})=0\bigr{)}\subset{\mathbb{P}}^{2}_{\mathbf{x}}$
are smooth and they intersect each other transversally in $3d^{2}$ distinct
points.

Let $D\subset X$ be a double section and
$B_{D}\subset{\mathbb{P}}^{2}_{\mathbf{x}}$ the branch curve of
$\pi|_{D}\circ\tau:\bar{D}\to Z$. Then $\deg B_{D}\geq 2d$.

  * Proof.

The assumptions imply that $X$ is smooth. The branch curve of $X$ is
$B_{X}=B_{0}+B_{1}+B_{2}$. By Lemma 16, $B_{D}$ passes through all the nodes
of $B$. In particular, $B_{D}$ intersects $B_{0}$ in the $2d^{2}$ points
$B_{0}\cap(B_{1}\cup B_{2})$. Thus $\deg B_{D}\geq 2d$ unless $B_{0}\subset
B_{D}$. We can repeat the argument for $B_{1}$ to get that $\deg B_{D}\geq 2d$
unless $B_{0}+B_{1}\subset B_{D}$. The letter case also implies that $\deg
B_{D}\geq 2d$.

For conic bundles with smooth branch curves we have a less precise condition.
(The 2 results are closely related since passing through a node guarantees
that the intersection multiplicity is $\geq 2$.)

###### Proposition 18

Let $\pi:X\to{\mathbb{P}}^{2}$ be a smooth, minimal conic bundle with branch
locus $B_{X}\subset{\mathbb{P}}^{2}$. Assume that there is no reduced curve
$C\subset{\mathbb{P}}^{2}$ of degree $\leq 6$ such that all points of $C\cap
B_{X}$ have intersection multiplicity $\geq 2$. Then $-K_{X}$ is not
effective.

  * Proof.

Assume to the contrary that $-K_{X}\sim D$ where $D$ is an effective
${\mathbb{Z}}$-divisor. Since $X$ is minimal, there are no rational sections.
Thus there is a unique irreducible component $D_{0}\subset D$ that is a double
section of $\pi$. Let $\tau:\bar{D}_{0}\to D_{0}$ denote the normalization and
let $B_{D}\subset{\mathbb{P}}^{2}$ be the branch curve of
$\tau\circ\pi|_{D_{0}}:\bar{D}_{0}\to{\mathbb{P}}^{2}$. Note that $D_{0}\sim-
K_{X}-mH$ for some $m\geq 0$ and $\deg B_{D}\leq 6$ by Lemma 15.

We claim that all points of $B_{D}\cap B_{X}$ have intersection multiplicity
$\geq 2$. To see this, assume to the contrary that there is a point $p\in
B_{D}\cap B_{X}$ where $B_{D},B_{X}$ are smooth and intersect transversally.
In a neighborhood of $p$ the surface $D_{0}$ is a smooth double cover ramified
along $B_{D}$. Then $(\pi|_{D_{0}})^{-1}(B_{X})$ is a smooth double cover of
$B_{X}$ ramified at $p$. On the other hand,
$(\pi|_{D_{0}})^{-1}(B_{X})=D_{0}\cap\pi^{-1}(B_{X})$ hence its normalization
factors through the étale cover $\tilde{B}_{X}\to B_{X}$. This is a
contradiction. (This argument in fact shows that these intersection
multiplicities are even.)

Next we check that the assumptions of Proposition 18 hold for a general branch
curve of degree $\geq 15$.

###### Lemma 19

Let $B\subset{\mathbb{P}}^{2}$ be a general curve of degree $d$. Then there is
no reduced curve $C\subset{\mathbb{P}}^{2}$ of degree $<\tfrac{1}{2}d-1$ such
that all points of $C\cap B$ have intersection multiplicity $\geq 2$.

  * Proof.

Fix a reduced curve $C$ and let $L$ be a line bundle of degree $m$. Let
$W(L)\subset H^{0}(C,L)$ denote the subvariety consisting of sections without
simple zeros. Note that any $s\in W(L)$ has at most $\frac{1}{2}m$ zeros. The
map that sends a section of $L$ to its set of zeros has 1-dimensional fibers.
Thus $\dim W(L)\leq\frac{1}{2}m+1$.

Fix a reduced plane curve $C\subset{\mathbb{P}}^{2}$ of degree $c$. Let
$W(C,d)$ denote the set of degree $d$ curves $B$ such that all points of
$C\cap B$ have intersection multiplicity $\geq 2$. Applying the above estimate
to $L={\mathcal{O}}_{{\mathbb{P}}^{2}}(d)|_{C}$ we get that $\dim
W(C,d)\leq\frac{1}{2}cd+\binom{d-c+2}{2}$. Next let $W(d)$ denote the set of
degree $d$ curves $B$ such that all points of $C\cap B$ have intersection
multiplicity $\geq 2$ for some curve $C$ of degree $c$. Then

$\dim W(d)\leq\tfrac{1}{2}cd+\tbinom{d-c+2}{2}+\tbinom{c+2}{2}-1.$

After expanding the binomials we see that

$\dim W(d)<\tbinom{d+2}{2}-1\quad\mbox{for $d>2c+2$.}\quad$

(Note that the bound $\tfrac{1}{2}d-1$ is sharp for $d=4$ but one should be
able to prove slightly better bounds for larger values of $d$.)

###### Example 20

Let $Z$ be a smooth variety and $E$ a rank 3 vector bundle one $Z$. Set
$P:={\mathbb{P}}_{Z}(E)$ with projection $\pi:P\to Z$ and note that
$\omega_{P/Z}\cong{\mathcal{O}}_{P}(-3)\otimes\pi^{*}\det E$. Let $L$ be a
line bundle on $Z$ and $X\subset P$ the zero set of a section of
${\mathcal{O}}_{P}(2)\otimes\pi^{*}L$. Then
$\omega_{X}\cong\bigl{(}{\mathcal{O}}_{P}(-1)\otimes\pi^{*}(L\otimes\det
E)\bigr{)}|_{X}$ and
$\pi_{*}\bigl{(}\omega_{X}^{-1}\bigr{)}\cong(\omega_{Z}\otimes L\otimes\det
E)^{-1}\otimes E$.

Note that
$H^{0}\bigl{(}P,{\mathcal{O}}_{P}(2)\otimes\pi^{*}L\bigr{)}=H^{0}\bigl{(}Z,L\otimes\operatorname{Sym}^{2}E\bigr{)}$
and
$H^{0}\bigl{(}X,{\mathcal{O}}_{X}(-K_{X})\bigr{)}=H^{0}\bigl{(}Z,\bigl{(}L(K_{Z})\otimes\det
E\bigr{)}^{-1}\otimes E\bigr{)}$. We would like $X$ to be smooth, this
suggests that $L\otimes\operatorname{Sym}^{2}E$ should be generated by global
sections, hence $L$ should be positive. By contrast, the condition
$|-K_{X}|\neq\emptyset$ suggests that $L$ should be negative. It seems that
both of these can be satisfied only if $E$ is rather unstable.

For example, take $Z={\mathbb{P}}^{2}$, set
$E={\mathcal{O}}_{{\mathbb{P}}^{2}}(c)\oplus{\mathcal{O}}_{{\mathbb{P}}^{2}}(3)\oplus{\mathcal{O}}_{{\mathbb{P}}^{2}}$
where $c>3$ and $L={\mathcal{O}}_{{\mathbb{P}}^{2}}$. Then
$L\otimes\operatorname{Sym}^{2}E$ is generated by global sections, so a
general section gives a smooth conic bundle $X$. Furthermore,

$\pi_{*}\bigl{(}\omega_{X}^{-1}\bigr{)}\cong{\mathcal{O}}_{{\mathbb{P}}^{2}}(3-3-c)\otimes
E\cong{\mathcal{O}}_{{\mathbb{P}}^{2}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{2}}(3-c)\oplus{\mathcal{O}}_{{\mathbb{P}}^{2}}(-c)$

has a unique section. Thus $X\to{\mathbb{P}}^{2}$ is a conic bundle such that
$|-K_{X}|$ contains a unique K3 surface. It is obtained by intersecting $X$
with the divisor in $P$ corresponding to the
${\mathcal{O}}_{{\mathbb{P}}^{2}}(3)\oplus{\mathcal{O}}_{{\mathbb{P}}^{2}}$
summand of $E$. The branch curve of $X$ is given by the equation

$\det\left(\begin{array}[]{ccc}g_{2c}&g_{c+3}&g_{c}\\\ g_{c+3}&g_{6}&g_{3}\\\
g_{c}&g_{3}&g_{0}\end{array}\right)\ =0,$

where $g_{i}:=g_{i}(x,y,z)$ denotes a homogeneous polynomial of degree $i$.
(We can thus assume that $g_{0}=1$.) Eliminating $g_{c},g_{3}$ in the last row
writes $g_{0}$ times the determinant, hence the equation of $B_{X}$, in the
form

$(g_{6}g_{0}-g_{3}^{2})(g_{2c}g_{0}-g_{c}^{2})-(g_{c+3}g_{0}-g_{c}g_{3})^{2}=0.$

Thus the degree 6 curve $g_{6}g_{0}-g_{3}^{2}=0$ intersects $B_{X}$ at the
points where $g_{c+3}g_{0}-g_{c}g_{3}=0$ and all intersection multiplicities
are even; as needed in Lemma 18.

## Examples of conic bundle surfaces

Working backwards from Theorem 3 we see that there are surface conic bundles
$S\to{\mathbb{P}}^{1}$ over ${\mathbb{C}}(s,t)$ such that $-K_{S}$ is not
birationally ${\mathbb{Q}}$-effective. We will exhibit similar examples over
${\mathbb{Q}}$ and ${\mathbb{Q}}_{p}$.

###### Example 21

Let $k$ be a field and $S\to{\mathbb{P}}^{1}$ a conic bundle over $k$ that
becomes trivial after a quadratic extension $k^{\prime}\supset k$. Then
$S_{k^{\prime}}$ has a section $C^{\prime}$, thus $S$ has a conjugate pair of
sections $C$. The normalization of $C$ has $p_{a}=-1$. Thus Corollary 11
implies that $S\to{\mathbb{P}}^{1}$ is birational to a conic bundle
$S^{\prime}\to{\mathbb{P}}^{1}$ such that $-K_{S^{\prime}}$ is effective.
(Typically the base locus of $|-K_{S}|$ consists of the disjoint conjugate
sections and the moving part of 2 fibers.)

There are such examples over ${\mathbb{Q}}$, even with arbitrary branch locus.
Let $g(s)\in{\mathbb{Q}}[s]$ be a polynomial of degree $2d$ with simple roots
only. Choose a prime $p$ such that $\sqrt{p}$ is not contained in the
splitting field of $g$. Let $S_{g}\to{\mathbb{P}}^{1}$ be the projective model
of the surface

$\bigl{(}g(s)z^{2}=x^{2}-py^{2}\bigr{)}\subset{\mathbb{P}}^{1}_{xyz}\times{\mathbb{A}}^{1}_{s}.$

Then $S_{g}\to{\mathbb{P}}^{1}$ is minimal and the singular fibers lie exactly
over the roots of $g(s)$. Thus there are surface conic bundles
$S\to{\mathbb{P}}^{1}$ with $\delta(S)$ arbitrarily large and branch locus
$B_{S}$ in general position for which $-K_{S}$ is birationally effective.

Note also that the above argument implies that $-K_{S}$ is birationally
effective for every conic bundle over ${\mathbb{R}}$ since every surface conic
bundle over ${\mathbb{C}}$ has a section. The situation over finite fields is
unclear to me.

The examples where $-K_{S}$ is not birationally ${\mathbb{Q}}$-effective rest
on the following observation.

###### Lemma 22

Let $Z$ be a 2-dimensional regular scheme and $\pi:X\to Z$ a conic bundle. Let
$C\subset Z$ be an irreducible 1-dimensional subscheme such that
$X_{C}:=\pi^{-1}(C)\to C$ has no rational sections. Let $W\subset C$ be the
set of points $z\in C$ such that $X_{z}$ is a double line and $X$ is regular
along $X_{z}$. Let $D\subset X$ be a double section with normalization
$\tau:\bar{D}\to D$ and $B_{D}\subset Z$ the branch locus of
$\pi|_{D}\circ\tau:\bar{D}\to Z$.

Then $C\not\subset B_{D}$ but $W\subset B_{D}$.

  * Proof.

If $C\subset B_{D}$ then $\operatorname{red}\bigl{(}D\cap X_{C}\bigr{)}$ is a
rational section of $X_{C}\to C$. This is contrary to our assumptions and
$W\subset B_{D}$ follows from Claim 25.3.

###### Example 23

We will apply Lemma 22 with $Z={\mathbb{P}}^{1}_{{\mathbb{Z}}_{p}}$ and
$C={\mathbb{P}}^{1}_{{\mathbb{F}}_{p}}$ for some prime $p\geq 3$. Fix a
natural number $d$ and set
$P:={\mathbb{P}}_{Z}\bigl{(}{\mathcal{O}}_{Z}\oplus{\mathcal{O}}_{Z}(d)\oplus{\mathcal{O}}_{Z}(d+1)\bigr{)}$.
The conic bundle $X\subset P$ will be given by a section of
${\mathcal{O}}_{P}(2)$. For simplicity we choose a section of
${\mathcal{O}}_{Z}+{\mathcal{O}}_{Z}(2d)\oplus{\mathcal{O}}_{Z}(2d+2)\subset\pi_{*}{\mathcal{O}}_{P}(2)$.
Choosing an affine coordinate $s$ on ${\mathbb{P}}^{1}$, one can give such an
$X$ by an equation

$X:=\bigl{(}z^{2}=a(s)x^{2}+b(s)y^{2}\bigr{)}\quad\mbox{where}\quad\deg
a(s)=2d,\deg b(s)=2d+2.$

We choose $a(s)$ and $b(s)$ as follows.

  * 1.

$\bar{a}(s)$ has only simple zeros where $\bar{\ }$ denotes reduction mod $p$,

  * 2.

$b(s)=(s^{2}-1)a(s)+pc(s)$ for some $c(s)$,

  * 3.

$s^{2}-1,\bar{a}(s),\bar{c}(s)$ are pairwise relatively prime and

  * 4.

$\bar{a}(1)$ is not a square in ${\mathbb{F}}_{p}$.

Claim 23.9. Let
$\pi:X_{{\mathbb{Q}}_{p}}\to{\mathbb{P}}^{1}_{{\mathbb{Q}}_{p}}$ be the
generic fiber of the above $X\to Z$ and $D_{{\mathbb{Q}}_{p}}\subset
X_{{\mathbb{Q}}_{p}}$ a double section with normalization
$\tau:\bar{D}_{{\mathbb{Q}}_{p}}\to D_{{\mathbb{Q}}_{p}}$. Then
$g(\bar{D}_{{\mathbb{Q}}_{p}})\geq d-1$.

  * Proof.

The closure of $D_{{\mathbb{Q}}_{p}}$ gives a double section $D\subset X$. Let
$B_{D}\subset Z$ be the branch locus of $\pi|_{D}\circ\tau:\bar{D}\to Z$. We
aim to apply Lemma 22.

Let $\alpha$ be a root of $a(s)$. Near $\alpha$ the equation of $X$ has the
form

$z^{2}=(s-\alpha)u_{1}x^{2}+\bigl{(}(s-\alpha)u_{2}+pu_{3}\bigr{)}y^{2},$

where the $u_{i}$ are units. Thus the fiber $X_{\alpha}$ is a double line and
$X$ is regular along $X_{\alpha}$ by (25). Over the point
$(s=1)\in{\mathbb{P}}^{1}_{{\mathbb{F}}_{p}}$ the fiber is
$(z^{2}=\bar{a}(1)x^{2})\subset{\mathbb{P}}^{2}_{{\mathbb{F}}_{p}}$. Since
$\bar{a}(1)$ is not a square, its only ${\mathbb{F}}_{p}$-point is
$(0{:}1{:}0)$ where $\bar{X}$ is smooth. Hence
$\bar{X}\to{\mathbb{P}}^{1}_{{\mathbb{F}}_{p}}$ has no sections. Thus Lemma 22
implies that

$2g(\bar{D}_{{\mathbb{Q}}_{p}})+2=\deg B_{D}\geq\deg a(s)=2d.$

The choices (1.–4.) can be satisfied for $a(s),b(s)\in{\mathbb{Q}}[s]$, hence
we proved the following more precise form of Theorem 1.

###### Corollary 24

For every $g$ there are conic bundles $S\to{\mathbb{P}}^{1}$ defined over
${\mathbb{Q}}$ with $\delta(S)=4g+6$ and such that every double section of $S$
has geometric genus $\geq g$.

For $g=2$ this gives conic bundles $S\to{\mathbb{P}}^{1}$ with $\delta(S)=14$
defined over ${\mathbb{Q}}$ such that $-K_{S}$ is not birationally pseudo-
effective.

## Birational maps of conic bundles

We summarize the results on birational maps of conic bundles that we used. As
before, all residue characteristics are assumed to be $\neq 2$.

######   25 (Extending conic bundles)

We will need to understand the following

Problem 25.1. Let $Z$ be a regular surface, $W\subset Z$ a finite subset and
$\pi^{0}:X^{0}\to Z^{0}:=Z\setminus W$ a conic bundle with branch locus
$B^{0}_{X}$. We would like to extend $\pi^{0}:X^{0}\to Z^{0}$ to a conic
bundle $\pi:X\to Z$ and control the singularities of $X$ in terms of $B_{X}$.
(This is also interesting if $\dim Z>2$ but the 2-dimensional case is
simpler.)

We may assume that $W=\\{p\\}$ is a single point and $Z$ is local. The push-
forward $E^{0}:=\pi^{0}_{*}\omega_{x^{0}/Z^{0}}^{-1}$ is a locally free sheaf
of rank 3. Set $E:=j_{*}E^{0}$ where $j:Z^{0}\lhook\joinrel\to Z$ is the
natural injection. Then $E$ is a reflexive sheaf but, since $Z$ is regular and
2-dimensional, $E$ is locally free, hence free. Set $P:={\mathbb{P}}_{Z}(E)$
and let $X\subset P$ be the closure of $X^{0}\subset P^{0}$. Choose an
isomorphism $P\cong{\mathbb{P}}^{2}_{Z}$, then $X$ is given by an equation

$\textstyle{\sum}_{ij}g_{ij}(z_{1},z_{2})x_{i}x_{j}=0,$ (25.2)

where $g_{ij}(z_{1},z_{2})\in{\mathcal{O}}_{Z}$. The scheme structure of the
branch locus is defined by

$\det\bigl{(}g_{ij}(z_{1},z_{2})\bigr{)}=0.$ (25.3)

The worst case is when the central fiber $X_{p}$ equals ${\mathbb{P}}^{2}$;
thus $\pi:X\to Z$ is not even equidimensional. If this happens then all the
$g_{ij}$ vanish at $p$ so $\det\bigl{(}g_{ij}(z_{1},z_{2})\bigr{)}\in
m_{p}^{3}$. Thus $B_{X}$ has a triple (or higher) point at $p$. Otherwise the
central fiber $X_{p}$ is conic.

$X_{p}$ is smooth iff $p\not\in B_{X}$.

Next assume that $X_{p}$ is a pair if lines. Possibly after a quadratic
residue field extension, in suitable formal coordinates we can diagonalize the
equation of $X$ as

$x_{0}^{2}=x_{1}^{2}+f(z_{1},z_{2})x_{2}^{2}\quad\mbox{and}\quad
B_{X}=\bigl{(}f(z_{1},z_{2})=0\bigr{)}$

where $f(0,0)=0$. We see that $X$ is regular along $X_{p}$ iff
$f\not\in(z_{1},z_{2})^{2}$; that is, iff $B_{X}$ is regular at $p$.
Furthermore, $(z_{1},z_{2})\mapsto(1{:}1{:}0;z_{1},z_{2})$ is a formal
section.

Finally consider the case when $X_{p}$ is a double line. Then
$\det\bigl{(}g_{ij}(z_{1},z_{2})\bigr{)}\in m_{p}^{2}$ and $B_{X}$ has a
double (or higher) point at $p$. If the branch curve has a node then, after a
quadratic residue field extension, in suitable formal coordinates we can write
$X$ as

$X=\bigl{(}x_{0}^{2}=z_{1}x_{1}^{2}+z_{2}x_{2}^{2}\bigr{)}.$

The fiber of the projection $\pi:X\to{\mathbb{A}}^{2}_{\mathbf{z}}$ is a
double line over the origin, a pair of lines over the coordinate axes
$(z_{1}z_{2}=0)$ and smooth otherwise. In contrast with the previous nodal
case, the equation has no solutions in the quotient field of the completion of
${\mathcal{O}}_{z,Z}$; see Claim 25.

We have thus proved the following.

Claim 25.4. Let $Z$ be a regular surface whose residue characteristics are
$\neq 2$, $B\subset Z$ a curve with only nodal singularities and $W\in Z$ a
finite subset containing the nodes of $B$. Let $\pi^{0}:X^{0}\to Z\setminus W$
be a conic bundle with (scheme theoretic) branch locus $B\setminus W$. Then
$X^{0}$ extends to a unique conic bundle $\pi:X\to Z$.

Furthermore, $X$ is regular iff the following holds: for every $p\in W$ that
is a node of $B$, the projection $\pi:X\to Z$ has no sections over the
quotient field of the strict henselization ${{\mathcal{O}}}^{\rm sh}_{p,Z}$ of
the local ring ${{\mathcal{O}}}_{p,Z}$.

We also used the following well known result.

Claim 25.5. Let $\bigl{(}R,m=(s,t)\bigr{)}$ be a regular local ring of
dimension 2. Then $x_{0}^{2}=sx_{1}^{2}+tx_{2}^{2}$ has no nonzero solutions
in the quotient field of $R$.

  * Proof.

After clearing denominators, we may assume that the $x_{i}$ are in $R$. Let
$c$ be the largest such that the $x_{i}$ are in $m^{c}$. We can thus write
$x_{i}=p_{i}(s,t)+r_{i}$ where $p_{i}$ is a homogeneous polynomial of degree
$c$ and $r_{i}\in m^{c+1}$. Then $x_{0}^{2}\in m^{2c}$ but
$sx_{1}^{2}+tx_{2}^{2}\in m^{2c+1}$. Thus in fact $p_{0}\equiv 0$ and so
$sp_{1}^{2}+tp_{2}^{2}\in m^{2c+2}$. Thus $sp_{1}^{2}+tp_{2}^{2}\equiv 0$
hence $p_{1}$ and $p_{2}$ are both identically 0. This is a contradiction.

######   26 (Elementary transformations of conic bundles)

Let $\pi:X\to Z$ be a conic bundle over a regular surface and $C\subset Z$ a
regular curve such that $\pi$ is smooth over $C$ and $s:C\to X_{C}$ a section
of $X_{C}\to C$. We first blow up $s(C)\subset X$ and then contract the
birational transform of $X_{C}$. We get another conic bundle
$\pi^{(s)}:X^{(s)}\to Z$. The rational map $X\dasharrow X^{(s)}$ is called the
elementary transformation with center $s(C)$.

Next consider the case when $C\subset Z$ is a geometrically reduced curve,
$\pi$ is smooth over the generic points of $C$ and $s:C\dasharrow X_{C}$ is a
rational section of $X_{C}\to C$. Then there is a finite subset $W\subset Z$
such that $Z^{0}:=Z\setminus W$, $C^{0}:=C\setminus W$ and
$X^{0}:=X\setminus\pi^{-1}(W)$ satisfy the previous assumptions. We can thus
construct the elementary transformation $\pi^{0,(s)}:X^{0,(s)}\to Z^{0}$ of
$\pi^{0}:X^{0}\to Z^{0}$ with center $s(C^{0})$. Finally, whenever possible,
we use the method of Paragraph 25 to extend $\pi^{0,(s)}:X^{0,(s)}\to Z^{0}$
to a conic bundle $\pi^{(s)}:X^{(s)}\to Z$ called the elementary
transformation with center $s(C)$.

Claim 26.1. Let $\pi:X\to Z$ be a conic bundle over a surface, $C\subset Z$ a
geometrically reduced curve and $s:C\dasharrow X_{C}$ a rational section of
$X_{C}\to C$. Assume that $X$ and $Z$ are regular and $\pi$ is smooth over the
generic points of $C$.

Then the elementary transformation with center $s(C)$ exists and it is a conic
bundle $\pi^{(s)}:X^{(s)}\to Z$ such that $X^{(s)}$ is also regular and has
the same branch curve as $X\to Z$.

The following result of [Sar80], whose idea goes back to [Isk67], describes
birational transformations of conic bundles over the same base.

Claim 26.2. Let $\pi_{i}:X_{i}\to Z$ be smooth conic bundles over a smooth
surface. Let $\phi:X_{1}\dasharrow X_{2}$ be a birational equivalence over
$Z$. That is, the following diagram commutes

$\begin{array}[]{ccc}X_{1}&\stackrel{{\scriptstyle\phi}}{{\dasharrow}}&X_{2}\\\
\pi_{1}\downarrow\hphantom{\pi_{1}}&&\hphantom{\pi_{2}}\downarrow\pi_{2}\\\
Z&=&Z.\\\ \end{array}$

Then $\phi$ is a composite of elementary transformations.

The key result about birational maps of conic bundles is the following. The
surface case is due to [Isk67]. (The minimal model program for surfaces over
any field is established in [Mor82], thus the arguments of [Isk67] extend to
any field.) The much harder 3-fold case is treated in [Sar80]. See [Cor95,
Cor00] for more conceptual proofs.

###### Theorem 27

Let $\pi:X\to Z$ be a minimal conic bundle over a field of characteristic
$\neq 2,3,5$ such that $X,Z$ are smooth and $B_{X}+4K_{Z}$ is effective. Let
$\pi^{\prime}:X^{\prime}\to Z^{\prime}$ be a Mori fiber space and
$\phi:X^{\prime}\dasharrow X$ a birational map. Then $\pi^{\prime}$ is also a
conic bundle, and there is a birational map $\phi_{Z}:Z^{\prime}\dasharrow Z$
such that the following diagram commutes

$\begin{array}[]{ccc}X^{\prime}&\stackrel{{\scriptstyle\phi}}{{\dasharrow}}&X\\\
\pi^{\prime}\downarrow\hphantom{\pi^{\prime}}&&\hphantom{\pi}\downarrow\pi\\\
Z^{\prime}&\stackrel{{\scriptstyle\phi_{Z}}}{{\dasharrow}}&Z.\\\ \end{array}$
(27.1)

For our applications we need the following more precise version for which I
could not find an explicit reference.

###### Complement 28

Assume in addition that $B_{X}+K_{Z}$ is ample. Then the diagram (27.1) can be
factored as

$\begin{array}[]{ccccc}X^{\prime}&\stackrel{{\scriptstyle\rho}}{{\dasharrow}}&X^{\prime\prime}&\stackrel{{\scriptstyle\tau}}{{\dasharrow}}&X\\\
\pi^{\prime}\downarrow\hphantom{\pi^{\prime}}&&\hphantom{\pi^{\prime\prime}}\downarrow\pi^{\prime\prime}&&\hphantom{\pi}\downarrow\pi\\\
Z^{\prime}&\stackrel{{\scriptstyle\rho_{Z}}}{{\to}}&Z&=&Z,\\\ \end{array}$
(28.1)

where $\tau:X^{\prime\prime}\dasharrow X$ is a composite of elementary
transformations, $\rho:X^{\prime}\dasharrow X^{\prime\prime}$ is a birational
contraction and
$-K_{X^{\prime\prime}}=\rho_{*}\bigl{(}-K_{X^{\prime}}\bigr{)}$.

  * Proof.

Let $\pi:X\to Z$ be a conic bundle over a field of characteristic 0 such that
$X,Z$ are smooth. Pick a point $z\in Z$ and let $p:Z_{1}:=B_{z}Z\to Z$ denote
the blow-up. The pull-back $X\times_{Z}Z_{1}$ is birational to a minimal conic
bundle $\pi_{1}:X_{1}\to Z_{1}$. We want to describe its branch locus
$B_{X_{1}}$. It is clear that $B_{X_{1}}\supset p^{-1}_{*}B_{X}$, the only
question is what happens with the exceptional curve $E_{z}$. The following
possibilities are listed in [Sar80, 2.4–2.5].

If $z\not\in B_{x}$ then $X_{1}=X\times_{Z}Z_{1}$ hence
$B_{X_{1}}=p^{-1}_{*}B_{X}$.

If $z\in B_{x}$ is a smooth point then $E_{z}\subset B_{X_{1}}$ iff the fiber
$X_{z}$ irreducible. (Thus again $B_{X_{1}}=p^{-1}_{*}B_{X}$ if the base field
is algebraically closed.)

If $z\in B_{x}$ is a singular point then $X_{1}=X\times_{Z}Z_{1}$ is singular.
Using the local equation $x_{0}^{2}=z_{1}x_{1}^{2}+z_{2}x_{2}^{2}$ we get
$x_{0}^{2}=z^{\prime}_{1}x_{1}^{2}+z^{\prime}_{1}z^{\prime}_{2}x_{2}^{2}$. We
can rewrite this as
$z^{\prime}_{1}(x^{\prime}_{0})^{2}=x_{1}^{2}+z^{\prime}_{2}x_{2}^{2}$ where
$x^{\prime}_{0}=x_{0}/z^{\prime}_{1}$. Thus
$B_{X^{\prime}}=p^{-1}_{*}B_{X}+E_{z}$. In all these cases we see that for any
sequence of blow ups $p_{r}:Z_{r}\to Z$ we have

$K_{Z_{r}}+B_{X_{r}}=p_{r}^{*}(K_{Z}+B_{X})+(\mbox{effective exceptional
divisor}).$ (28.2)

Applying this to a common resolution $Z^{\prime}\leftarrow Z_{r}\to Z$ we
conclude that if $K_{Z}+B_{X}$ is ample then $(Z,B_{X})$ is the unique log-
canonical model of $(Z_{r},B_{X_{r}})$. In particular, the rational map
$\phi_{Z}:Z^{\prime}\dasharrow Z$ in (27.1) is a morphism. This establishes
the bottom row of (28.1).

In order to get the top row, let $A^{\prime}$ be an ample divisor on
$Z^{\prime}$ and choose $m$ such that
$|H^{\prime}|:=|-K_{Z^{\prime}}+m(p^{\prime})^{*}A^{\prime}|$ is a very ample
linear system on $Z^{\prime}$. Its push-forward $\phi_{*}|H^{\prime}|$ is a
mobile linear system consisting of rational double sections of $\pi$. Its base
locus consists of some horizontal curves (that is curves $C_{i}\subset X$ such
that $C_{i}\to\pi(C_{i})$ is birational) and some vertical curves (that is
curves that are contracted by $\pi$). By a small variation of Claim 26, a
sequence of elementary transformations along the horizontal curves leads to a
factorization

$\phi:X^{\prime}\stackrel{{\scriptstyle\rho}}{{\dasharrow}}X^{\prime\prime}\stackrel{{\scriptstyle\tau}}{{\dasharrow}}X$
(28.3)

where $\tau$ is a composite of elementary transformations and the base locus
$\operatorname{Bs}\bigl{(}\rho_{*}|H^{\prime}|\bigr{)}$ is vertical. Thus
$\rho^{-1}$ gives an injection
$X^{\prime\prime}\setminus\operatorname{Bs}\bigl{(}\rho_{*}|H^{\prime}|\bigr{)}\lhook\joinrel\to
X^{\prime}$, hence it has no exceptional divisors. Thus $\rho$ is a birational
contraction and therefore
$-K_{X^{\prime\prime}}=\rho_{*}\bigl{(}-K_{X^{\prime}}\bigr{)}$.

## Acknowledgements

I thank A. Corti, Y. Gongyo, A. Skorobogatov, C. Xu and the referees for
helpful comments, corrections and references.

## References

  * [AM72] M. Artin and D. Mumford, _Some elementary examples of unirational varieties which are not rational_ , Proc. London Math. Soc. (3) 25 (1972), 75–95. MR-0321934
  * [Bea77] Arnaud Beauville, _Variétés de Prym et jacobiennes intermédiaires_ , Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309–391. MR-0472843
  * [CG13] Paolo Cascini and Yoshinori Gongyo, _On the anti-canonical ring and varieties of Fano type_ , Saitama Math. J. 30 (2013), 27–38 (2013). MR-3203718
  * [Che04] Ivan Cheltsov, _Conic bundles with a large discriminant_ , Izv. Math. 68 (2004), no. 2, 429–434. MR-2058006
  * [CK10] Ivan Cheltsov and Ilya Karzhemanov, _Halphen pencils on quartic threefolds_ , Adv. Math. 223 (2010), no. 2, 594–618. MR-2565543
  * [Cor95] Alessio Corti, _Factoring birational maps of threefolds after Sarkisov_ , J. Algebraic Geom. 4 (1995), no. 2, 223–254. MR-1311348
  * [Cor00] , _Singularities of linear systems and $3$-fold birational geometry_, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 259–312. MR-1798984
  * [Isk67] V. A. Iskovskikh, _Rational surfaces with a pencil of rational curves_ , Mat. Sb. (N.S.) 74 (116) (1967), 608–638. MR-0220734
  * [Isk87] , _On the rationality problem for conic bundles_ , Duke Math. J. 54 (1987), no. 2, 271–294. MR-899398
  * [KM98] János Kollár and Shigefumi Mori, _Birational geometry of algebraic varieties_ , Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. MR-1658959
  * [KM17] János Kollár and Massimiliano Mella, _Quadratic families of elliptic curves and unirationality of degree 1 conic bundles_ , Amer. J. Math. (2017), (to appear).
  * [Kol96] János Kollár, _Rational curves on algebraic varieties_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 32, Springer-Verlag, Berlin, 1996. MR-1440180
  * [Kol13] , _Singularities of the minimal model program_ , Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013, With the collaboration of Sándor Kovács. MR-3057950
  * [Kry14] I. Krylov, _Rationally connected non-Fano type varieties_ , ArXiv e-prints (2014).
  * [KSC04] János Kollár, Karen E. Smith, and Alessio Corti, _Rational and nearly rational varieties_ , Cambridge Studies in Advanced Mathematics, vol. 92, Cambridge University Press, Cambridge, 2004. MR-2062787
  * [KX16] János Kollár and Chenyang Xu, _The dual complex of Calabi-Yau pairs_ , Invent. Math. 205 (2016), no. 3, 527–557. MR-3539921
  * [MM86] Yoichi Miyaoka and Shigefumi Mori, _A numerical criterion for uniruledness_ , Ann. of Math. (2) 124 (1986), no. 1, 65–69. MR-847952
  * [Mor82] Shigefumi Mori, _Threefolds whose canonical bundles are not numerically effective_ , Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR-0662120
  * [Oka09] Takuzo Okada, _On the birational unboundedness of higher dimensional $\mathbb{Q}$-Fano varieties_, Math. Ann. 345 (2009), no. 1, 195–212. MR-2520058
  * [PS09] Yu. G. Prokhorov and V. V. Shokurov, _Towards the second main theorem on complements_ , J. Algebraic Geom. 18 (2009), no. 1, 151–199. MR-2448282
  * [Ryd06] Daniel Ryder, _Classification of elliptic and $K3$ fibrations birational to some $\mathbb{Q}$-Fano 3-folds_, J. Math. Sci. Univ. Tokyo 13 (2006), no. 1, 13–42. MR-2223680
  * [Sar80] V. G. Sarkisov, _Birational automorphisms of conic bundles_ , Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 918–945, 974. MR-587343
  * [Sar82] , _On conic bundle structures_ , Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 2, 371–408, 432. MR-651652