[title=Index of definitions and notations,intoc]

# Existence obstructions for separating morphisms and totally real pencils

Matilde Manzaroli

###### Abstract.

It goes back to Ahlfors that a real algebraic curve $C$ admits a separating
morphism $f$ to the complex projective line if and only if the real part of
the curve disconnects its complex part, i.e. the curve is separating. The
degree of such $f$ is bounded from below by the number $l$ of real connected
components of $\mathbb{R}C$. The sharpness of this bound is not a priori
clear. We prove that real algebraic separating curves, embedded in some
ambient surface and with $l$ bounded in a certain way, do not admit separating
morphisms of lowest possible degree. Moreover, this result of non-existence
can be applied to show that certain real separating plane curves of degree
$d$, do not admit totally real pencils of curves of degree $k$ such that
$kd\leq l$. Finally, we construct an explicit family of real separating plane
curves of increasing degree $2s+1$ such that each member of the family does
not admit totally real pencils of curves of degree lower or equal to
$\frac{s^{2}+1}{2s+1}$.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Theorem 1.13 and applications

##  1\. Introduction

A real algebraic variety is a compact complex algebraic variety $X$ equipped
with an anti-holomorphic involution $\sigma:X\rightarrow X$, called real
structure. The real part $\mathbb{R}X$ of $X$ is the set of points fixed by
the involution $\sigma$.

Let $C$ be any non-singular real algebraic curve. Denote by
$C_{1},\dots,C_{l}$ the connected components of $\mathbb{R}C$. Harnack-Klein’s
inequality [Har76, Kle73] bounds $l$ by the genus $g$ of $C$ plus one. If
$C\setminus\mathbb{R}C$ is connected, we say that $C$ is of type II or non-
separating, otherwise of type I or separating [Kle73]. Looking at the real
part of the curve and its position with respect to its complexification gives
information about $l$ and viceversa. For example, we know that if $l$ equals
$g+1$, then $C$ is of type I. Or, if $C$ is of type I then $l$ has the parity
of $g+1$. Rokhlin promoted and contributed to this new point of view; an
example of Rokhlin’s contribution is the introduction and the study of the
complex orientations of a separating real algebraic curve [Rok72]. Namely, if
$C$ is of type I, the two halves of $C\setminus\mathbb{R}C$ induce two
opposite orientations on $\mathbb{R}C$ called complex orientations of the
curve. Looking at complex orientations of separating real curves has allowed a
change of prospective and a remarkable progress in the study of real algebraic
curves, their topology and a refinement of their classifications.

######  Definition 1.1.

We say that a real morphism $f$ from a real algebraic curve $C$ to the complex
projective line $\mathbb{C}\mathbb{P}^{1}$ is separating if
$f^{-1}(\mathbb{R}\mathbb{P}^{1})=\mathbb{R}C$.

According to Ahlfors [Ahl50, §4.2], there exists a separating morphism
$f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ if and only if $C$ is separating. In
this paper, we focus on the relation of separating curves embedded in some
ambient surface and their separating morphisms; see Definition 1.1. Before
stating the main result of this paper, Theorem 1.13, in its generality, we
present it restricted to the case of real algebraic plane curves.

###  1.1. Real plane curves

Let us consider separating real algebraic plane curves $C$. Let $l$ denote the
number of the connected components of $\mathbb{R}C$. The following proposition
show that there is a relation between $l$ and the existence of separating
morphisms $f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ of degree $r(l)$.

######  Proposition 1.2 (Particular case of Theorem 1.13).

Let $C$ be a non-singular real algebraic plane curve of type I and with $l$
real connected components. Assume one of the following:

  1. (1)

The degree of $C$ is $2s+1$, for some $s\in\mathbb{Z}_{>1}$ and
$l+\varepsilon\leq s^{2}+s-2,$ where $\varepsilon\in\\{0,1\\}$ such that
$l+\varepsilon$ is even.

  2. (2)

The degree of $C$ is $2s$, for some $s\in\mathbb{Z}_{\geq 4}$ and
$l+\varepsilon\leq s^{2}-2s+2,$ where $\varepsilon\in\\{0,1\\}$ such that
$l-s+\varepsilon$ is even.

Then $C$ admits no separating morphisms of degree $l$.

Proposition 1.2 falls within the line of results relating topology, complex
orientations and properties of separating plane curves, such as Rokhlin’s
complex orientation formulas (2.1), (2.2), and such as [Ore21, Theorem 1.1],
where Orevkov shows that there are finer relations for the numbers which
intervene in (2.2).

As corollaries of Proposition 1.2, one can find obstructions for the
existences of totally real pencils of curves of a certain degree for given
separating plane curves; see Section 1.3.

###  1.2. Generalities of separating morphisms

A separating morphism $f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ is always
unramified once restricted to $\mathbb{R}C$; see [KS20a, Theorem 2.19].
Therefore, the restriction of $f$ to each connected component of $\mathbb{R}C$
is a covering map of $\mathbb{R}\mathbb{P}^{1}$. This implies that the degree
of a separating morphism is at least as big as the number of connected
components of $\mathbb{R}C$. Actually, the definition of separating morphism
is more general and includes real morphisms between any real algebraic
varieties of same dimension. In order to a have a general idea of the subject,
we refer the interested reader to [KS20a] and [KTM22]. In the context of this
paper, we only need Definition 1.1.

In [Hui01], [Gab06],[CH13], [Cop13], [KS20b], [Ore18] properties of separating
morphisms and their existence are treated. For example, [Gab06, Theorem 7.1]
states that a genus $g$ real separating curve with $l$ real connected
components admits a separating morphism of degree at most $\frac{g+l+1}{2}$.
Later, Coppens, in [Cop13], constructed, for every value $h$ between
$\tilde{l}$, the minimum for a separating morphism degree, and
$\frac{\tilde{g}+\tilde{l}+1}{2}$, a separating curve $\tilde{C}$ of genus
$\tilde{g}$ and with $\tilde{l}$ real connected components such that $h$ is
the smallest possible degree of a separating morphism of $\tilde{C}$. In
[KS20b], the authors fix a separating curve $C$ and study all separating
morphisms of $C$ as follows. Set $d_{i}(f)\in\mathbb{N}$ the degree of the
covering map $f:C_{i}\rightarrow\mathbb{R}\mathbb{P}^{1}$ and set
$d(f):=(d_{1}(f),\dots,d_{l}(f))$. Let us denote by
$t:=\sum_{i=1}^{l}d_{i}(f)$ the degree of $f$. The set $Sep(C)$ of all such
degree partitions forms a semigroup, called separating semigroup, and for all
element $d\in Sep(C)$, it is shown that $d+\mathbb{Z}^{l}_{\geq 0}$ is also
contained in $Sep(C)$. So, in order to understand $Sep(C)$ for a given
separating curve $C$, it is important to understand which minimal possible
element $Sep(C)$ contains, where with minimal we mean that $t$ is of minimal
possible value.

######  Remark 1.3.

Thanks to Harnack-Klein inequality, any real curve of genus $g$ cannot have
more than $g+1$ real connected components. By Riemann-Roch theorem, all non-
singular real curves with $l=g+1$ real connected components admits a
separating morphism of degree $l$. But, whenever $l<g+1$, it is not a priori
clear whether a separating morphism of degree $l$ exists.

###  1.3. Obstruction of existence of totally real pencils

Kummer and Shaw, in [KS20b], also prove that for all real separating curves
embedded in the complex projective plane, there exist infinitely many totally
real pencils of curves.

######  Theorem 1.4.

([KS20b, Theorem 1.6]) Let $C$ be a non-singular real algebraic plane curve of
type I. Then there exists an integer $k$ such that $\forall k^{\prime}\geq k$,
there are $f,g\in\mathbb{R}[x,y,z]_{k^{\prime}}$ such that $V(f,g)\cap
C=\emptyset$ and $V(\lambda f+\mu g)\cap C$ consists of real points only for
all $\lambda,\mu\in\mathbb{R}$ not both zero.

######  Definition 1.5.

Let $C$ be a non-singular real algebraic plane curve of type I. We say that
$C$ admits a totally real pencil of curves of degree $k$ if there exists an
integer $k$ such that there are $f,g\in\mathbb{R}[x,y,z]_{k}$ such that
$V(f,g)\cap C=\emptyset$ and $V(\lambda f+\mu g)\cap C$ consists of real
points only for all $\lambda,\mu\in\mathbb{R}$ not both zero.

In [KS20b, Question 1], the authors wonder which may be the minimal possible
value of $k$ in Theorem 1.4. An immediate corollary of Proposition 1.2 gives a
lower bound for $k$, as follows.

######  Corollary 1.6.

Let $C$ be a non-singular degree $d$ real algebraic plane curve of type I and
with $l$ real connected components. Assume that $C$ respects either $(1)$ or
$(2)$ of Proposition 1.2. Then $C$ admits no totally real pencil of curves of
degree $k$ such that $kd\leq l$.

######  Remark 1.7.

As remarked in [KS20b, Remark 3.5], according to [Ahl50, §4.2], real
separating plane curves admit separating morphisms (whose degree is a priori
unknown). Hence, there exists an integer $k$ such that there are
$f,g\in\mathbb{R}[x,y,z]_{k}$ such that $V(\lambda f+\mu g)\cap C$ consists of
real points only for all $\lambda,\mu\in\mathbb{R}$ not both zero. The
difference with Definition 1.5 is that $V(f,g)\cap C$ may be non-empty.
Analogously, [Gab06, Theorem 1.7] implies the existence of separating
morphisms of degree between $l$ and $\frac{g+l+1}{2}$ and, once again, this
does not imply that a given real separating plane curve of genus $g$ and with
$l$ real connected components, admits totally real pencils of curves of degree
$k(l,g)$ in the sense of Definition 1.5.

In order to have examples of separating real curves respecting the hypothesis
of Proposition 1.2, we construct an explicit family of real separating plane
curves of increasing degree $2s+1$ such that each member of the family admits
no totally real pencils of curves of degree lower than $\frac{s^{2}+1}{2s+1}$;
see Proposition 1.8.

In particular, Proposition 1.8 presents some explicit examples of real curves
in which the bound for the minimal degree of separating morphisms is higher
than the Brill-Noether bound. Namely, the Brill-Noether theorem states that a
generic curve of genus $g$ has gonality $\lfloor\frac{g+3}{2}\rfloor$. This
implies in particular that for a generic real curve of genus $g$, there cannot
be any separating morphism of degree less than $\frac{g+3}{2}$. For the real
separating plane curves $C$ of odd degree $2s+1$ constructed to prove
Proposition 1.8, this gonality $\displaystyle{\frac{2s^{2}-s+3}{2}}$ is
strictly smaller than $s^{2}+1$, which is the number of connected components
of $\mathbb{R}C$.

######  Proposition 1.8.

For all $s\in\mathbb{N}_{\geq 2}$, there exists a non-singular real plane
curve $B_{d}$ of degree $d=2s+1$, of type I and with $s^{2}+1$ real connected
components, which admit no totally real pencils of curves of degree $k$ such
that $k\leq\frac{s^{2}+1}{2s+1}$.

######  Remark 1.9.

We do not know yet whether the curves $B_{2s+1}$, constructed (see Lemma 3.2)
to prove Proposition 1.8, admit totally real pencils of degree
$k^{\prime}=\lceil\frac{s^{2}+1}{2s+1}\rceil$.

###  1.4. Statement of the main result and strategy

In this paper, we consider real separating curves embedded in some ambient
real surface and focus on their separating morphisms. The key tools of our
approach are the fact that all separating curves come equipped with two
possible complex orientations and the use of [Ore21, Theorem 3.2] as shown in
[Ore21, Example 3.3]. Before recalling [Ore21, Theorem 3.2] and state the main
result of this paper (Theorem 1.13), let us introduce some notation.

######  Notation 1.10.

Let $X$ be a smooth real algebraic surface and $C\subset X$ a non-singular
real algebraic curve of class $d$ in $H_{2}(X;\mathbb{Z})$. Let us consider a
real divisor $D$ belonging to the linear system $|C+K_{X}|$. Assume that $D$
has not $C$ as a component. This divisor can always be written as
$2D_{0}+D_{1}$ with the following properties. The divisor $D_{0}$ is an
effective divisor in $H_{2}(X;\mathbb{Z})$ and $D_{1}$ is a reduced curve
realising some class $\beta\in H_{2}(X;\mathbb{Z})$.

######  Example 1.11.

Let $C$ be a non-singular real algebraic curve of degree $2k$ in
$\mathbb{C}\mathbb{P}^{2}$. Then $D_{0}$ can be chosen as the divisor
realising the class of a plane curve of degree $k-2$ and $D_{1}$ as a line in
$\mathbb{C}\mathbb{P}^{2}$.

######  Theorem 1.12.

([Ore21, Theorem 3.2]) Let $X$ be a smooth real algebraic surface, let $C$ be
a smooth irreducible real separating curve on $X$ and $D$ be a real divisor on
$X$ as in Notation 1.10. Let us fix a complex orientation on $\mathbb{R}C$ and
an orientation $\mathcal{O}$ on
$\mathbb{R}X\setminus(\mathbb{R}C\cup\mathbb{R}D_{1})$ which changes each time
we cross $\mathbb{R}C\cup\mathbb{R}D_{1}$ at its smooth points. The latter
orientation induces a boundary orientation on
$\mathbb{R}C\setminus(\mathbb{R}C\cap D_{1})$. Let
$f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ be a separating morphism. Then it is
impossible that, for some $p\in\mathbb{R}\mathbb{P}^{1}$, the set
$f^{-1}(p)\setminus supp(D)$ is non-empty and the two orientations coincide at
each point of the set.

In [Ore21], Orevkov shows powerful applications of Theorem 1.12, such
as[Ore21, Theorem 1.1] and the construction of complex schemes (Definition
2.7) in the real projective plane which are realisable by real
pseudoholomorphic plane curves of odd degree and not realisable by real
algebraic plane curves of same degree. Moreover, other applications of [Ore21,
Theorem 3.2] are presented via examples. Therefore, the possible directions of
use of the content of [Ore21, Theorem 3.2] are still wide open. Now, let us
present the main result of this paper. We refer the reader to Notation 1.10.
Unless differently stated, we choose $\beta\in H_{2}(X;\mathbb{Z})$ in order
to maximise $\frac{D_{0}^{2}-D_{0}K_{X}}{2}\in\mathbb{Z}_{>0}$.

######  Theorem 1.13.

Let $X$ be a smooth real algebraic surface with holomorphic Euler
characteristic of the trivial bundle $\chi(0)\geq 1$ and $C\subset X$ a non-
singular real algebraic curve of class $d$ and of type I. Denote by $l$ the
number of connected components of $\mathbb{R}C$. Assume one of the following:

  1. (1)

$D_{1}$ can be chosen empty and
$2\lceil\frac{l}{2}\rceil\leq\frac{d^{2}-k_{X}^{2}}{4}$;

  2. (2)

$D_{1}$ cannot be chosen empty and $2\lceil\frac{l-m-\theta}{2}\rceil\leq
D_{0}^{2}-D_{0}K_{X}-2\theta$, where $\theta+m$ is the number of connected
components of $\mathbb{R}C$ which have non-empty intersection with
$\mathbb{R}D_{1}$, and $m=min(\frac{\beta^{2}-\beta K_{X}}{2},l)$;

Then there are no separating morphisms
$f:C\rightarrow\mathbb{C}\mathbb{P}^{1}$ of degree $l$.

######  Remark 1.14.

For example, Theorem 1.13 holds for separating real curves in any smooth
rational surface $X$, because the holomorphic Euler characteristic of the
trivial bundle equals one. In particular, Theorem 1.13 directly implies
Proposition 1.2. In fact, for plane curves of odd degree $2s+1$, the curve
$D_{1}$ can be chosen empty and
$2\lceil\frac{l}{2}\rceil\leq\frac{(2s+1)^{2}-9}{4}=s^{2}+s-2$. On the other
hand, for plane curves of even degree $2s$, the curve $D_{1}$ can be chosen to
be a real line so that $m+\theta\leq s$ with $m=2$, and $D_{0}=(s-2)[D_{1}]\in
H_{2}(\mathbb{C}\mathbb{P}^{2};\mathbb{Z})$. Hence

$2\Bigl{\lceil}\frac{l-s}{2}\Bigl{\rceil}\leq
l-2-\theta\leq(s-2)^{2}+3(s-2)-2\theta$

and one gets $l+\varepsilon\leq s^{2}-s-\theta$, where $\varepsilon=1$ if
$l-s$ is odd and $0$ otherwise. At worst, the number $\theta$ equals $s-2$ and
we obtain the bound given in the hypothesis of Proposition 1.2.

###  1.5. Acknowledgements

I would like to thank Mario Kummer and Kris Shaw for useful discussions. A
special thanks to Stepan Orevkov for addressing my attention to [Vir86,
(4.8)]. Moreover I would like to thank Erwan Brugallé and Hannah Markwig for
suggestions and remarks on a preliminary version of the paper. Thanks to
Athene Grant.

### Organisation of the paper

We start presenting some tools and terminology in Section 2. Then, in Section
3.1, we give a proof of Theorem 1.13. Finally, in Section 3.2, we present
examples and applications of Theorem 1.13, and we prove Proposition 1.8.

##  2\. Preliminaries

Let us consider non-singular real algebraic curves embedded in the complex
projective plane. The real part of real plane curves is homeomorphic to a
disjoint union of $S^{1}$ embedded in $\mathbb{R}\mathbb{P}^{2}$. Each $S^{1}$
can be embedded in $\mathbb{R}\mathbb{P}^{2}$ in two different ways: if it
realises the trivial-class in
$H_{1}(\mathbb{R}\mathbb{P}^{2};\mathbb{Z}/2\mathbb{Z})$, it is called oval,
otherwise it is called pseudo-line and denoted by $J$. If a non-singular real
plane curve has even degree then its real part consists of ovals only;
otherwise of exactly one pseudo-line and ovals.

An oval in $\mathbb{R}P^{2}$ separates two disjoint non-homeomorphic connected
components: the connected component homeomorphic to a disk is called interior
of the oval; the other one is called exterior of the oval. For each pair of
ovals, if one is in the interior of the other we speak about an injective
pair, otherwise a non-injective pair. For a separating real plane curve,
equipped with one of its two complex orientations, an injective pair of ovals
is said to be positive if the complex orientations of the ovals are induced by
an orientation of the annulus bounded by the ovals; otherwise negative. An
oval $\mathcal{O}$ of an odd degree separating curve is said positive if
$[\mathcal{O}]=-2[J]$ in $H_{1}(M)$, where $M$ is the closure of the non-
orientable component of $\mathbb{R}\mathbb{P}^{2}\setminus\mathcal{O}$; and
negative otherwise.

###  2.1. Rokhlin’s complex orientation formula

For any separating plane curve of even degree $2s$ and with $l$ real connected
components, Rokhlin in [Rok74] proved that

(2.1) $2(\Pi_{+}-\Pi_{-})=l-s^{2},$

where $\Pi_{+}$ and $\Pi_{-}$ denote the number of respectively positive and
negative injective pairs of ovals of the curve.

Later, for separating plane curves of odd degree $2s+1$ and with $l$ real
connected components, Mishachev [Mis75] proved that

(2.2) $\Lambda_{+}-\Lambda_{-}+2(\Pi_{+}-\Pi_{-})=l-1-s(s+1),$

where $\Lambda_{+}$ and $\Lambda_{-}$ denote respectively positive and
negative ovals of the curve.

In order to give examples of separating real curves respecting the hypothesis
of Proposition 1.2 and to prove Proposition 1.8, we need the following result.

######  Proposition 2.3.

[Vir86, Section (4.8)] Let $A_{1}$ and $A_{2}$ be separating real plane curves
of degree $d_{1}$ and $d_{2}$, transversal to each other and intersecting in
$r$ real points. Let $B$ be a real curve of degree $d=d_{1}+d_{2}$ obtained
from $A_{1}\cup A_{2}$ by a small perturbation such that some complex
orientations of $A_{1}$ and $A_{2}$ determine an orientation $\mathfrak{O}$ of
$\mathbb{R}B$. Then $B$ is of type I if and only if the pair
$(\mathbb{R}B,\mathfrak{O})$ respects Rokhlin’s complex orientation formula.

###  2.2. Encoding real schemes

In this section, it is introduced some notation and definitions used Section
3.2.

######  Definition 2.4.

Let $B_{1},\dots,B_{n}$ be any collection of disjoint ovals (and of a pseudo-
line) in $\mathbb{R}\mathbb{P}^{2}$. We call topological type the arrangement
$\mathcal{S}$ realised by the pair
$(\mathbb{R}\mathbb{P}^{2},\bigsqcup_{i=1}^{n}B_{i})$.

######  Notation 2.5 (Encoding topological types).

Let us consider collections of disjoint ovals in $\mathbb{R}\mathbb{P}^{2}$.
An empty union of ovals is denoted by $0$. We say that a union of $l$ ovals
realises $l$ if there are no injective pairs. The symbol
$\langle\mathcal{S}\rangle$ denotes the disjoint union of a non-empty
collection of ovals realising $\mathcal{S}$, and an oval forming an injective
pair with each oval of the collection. A collection of $h$ ovals in
$\mathbb{R}\mathbb{P}^{2}$ is called a nest of depth $h$ if any two ovals of
the collection form an injective pair. The disjoint union of any two
collections of ovals, realising respectively $\mathcal{S}^{\prime}$ and
$\mathcal{S}^{\prime\prime}$ in $\mathbb{R}\mathbb{P}^{2}$, is denoted by
$\mathcal{S}^{\prime}\sqcup\mathcal{S}^{\prime\prime}$ if none of the ovals of
one collection forms an injective pair with the ovals of the other one and
they are both non-empty collections. A disjoint union of the form
$\mathcal{S}^{\prime}\sqcup 0$ is still denoted $\mathcal{S}^{\prime}$.
Moreover, as previously mentioned, a pseudo-line in $\mathbb{R}\mathbb{P}^{2}$
is denoted by $J$.

######  Definition 2.6.

A topological type $\mathcal{S}$, up to homeomorphism of
$\mathbb{R}\mathbb{P}^{2}$, is called real scheme. Let
$A\subset\mathbb{C}\mathbb{P}^{2}$ be a non-singular real curve. We say that
$A$ has real scheme $\mathcal{S}$ if the pair
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}A)$ realises $\mathcal{S}$.

######  Definition 2.7.

A real scheme $\mathcal{S}$ in $\mathbb{R}\mathbb{P}^{2}$, where each
connected component is endowed with an orientation is called complex scheme.
Let $A\subset\mathbb{C}\mathbb{P}^{2}$ be a non-singular real separating
curve. We say that $A$ has complex scheme $\mathcal{S}$ if the pair
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}A)$ realises $\mathcal{S}$ as real scheme
and one of the two complex orientations of $\mathbb{R}A$ agrees with that of
$\mathcal{S}$.

##  3\. Theorem 1.13 and applications

Section 3.1 is uniquely devoted to the proof of Theorem 1.13. Then Section 3.2
presents the proof of Proposition 1.8 and some examples.

###  3.1. Proof of Theorem 1.13

###### Proof of Theorem 1.13.

Denote by $C_{i}$ the connected components of $\mathbb{R}C$ where $i=1,\dots
l$. Let us consider a real divisor $D$ belonging to the linear system
$|C+K_{X}|$. For the sake of contradiction, assume that there exists a
separating morphism $f:C\rightarrow\mathbb{P}^{1}$ of degree $l$. Then, for
all $p\in\mathbb{R}\mathbb{P}^{1}$, the set
$f^{-1}(p)=\\{p_{1},\dots,p_{l}\\}$ is a collection of $l$ real points such
that every $C_{i}$ contains exactly one $p_{i}$. Let us choose a curve $D_{1}$
and a curve of class $D_{0}$; see Notation 1.10. For our purposes, the choice
of $\beta\in H_{2}(X;\mathbb{Z})$, and consequently of $D_{1}$, is done in
order to maximise $\frac{D_{0}^{2}-D_{0}K_{X}}{2}\in\mathbb{Z}_{>0}$; where,
thanks to Riemann-Roch theorem and the hypothesis that the holomorphic Euler
characteristic of the trivial bundle of $X$ is bigger or equal to one, the
positive integer $\frac{D_{0}^{2}-D_{0}K_{X}}{2}$ is always less or equal than
the dimension of the linear system of curves in $X$ of class $D_{0}$.
Therefore, fixed a configuration of $\frac{D_{0}^{2}-D_{0}K_{X}}{2}$ points,
there always exists at least one curve of class $D_{0}$ passing through the
configuration.

Once $D_{1}$ is chosen, let us fix an orientation on
$\mathbb{R}X\setminus(\mathbb{R}C\cup\mathbb{R}D_{1})$ which changes each time
we cross $\mathbb{R}C\cup\mathbb{R}D_{1}$ at its smooth points; see [Ore21,
Proof of Theorem 3.2] for details. The latter orientation induces a boundary
orientation $\mathfrak{O}$ on $\mathbb{R}C$. Fix some $p$ in
$\mathbb{R}\mathbb{P}^{1}$.

Before treating cases $(1)$ and $(2)$ and prove the statement. Let us give an
idea of the strategy. In order to get a contradiction, we use Theorem 1.12.
Namely, define $Y_{p}$ (resp. $NY_{p}$) as the set of points in
$f^{-1}(p)\setminus supp(D)$ on which the complex orientation and the
orientation $\mathfrak{O}$ agree (resp. do not agree). Then, Theorem 1.12 tell
us that if the set $f^{-1}(p)\setminus supp(D)$ is non-empty, one has that
$Y_{p}$ and $NY_{p}$ are both non-empty.

First, let us treat case $(1)$ in which $D_{1}$ can be chosen empty. Fix one
of the two complex orientations on $\mathbb{R}C$. Then, remark that either the
number $V^{1}_{p}$ of $p_{i}$’s where the complex orientation and the
orientation $\mathfrak{O}$ agree, or the number $V^{2}_{p}$ of the $p_{i}$’s
where the two orientations do not agree, does not exceed
$\lceil\frac{l}{2}\rceil$. Given a collection $\mathcal{P}$ of
$\frac{d^{2}-K_{X}^{2}}{8}$ real points in $X$, there exists a real curve $A$
of class $D_{0}$ passing through $\mathcal{P}$. By hypothesis
$2\lceil\frac{l}{2}\rceil\leq\frac{d^{2}-K_{X}^{2}}{4}$, therefore it is
possible to choose such a collection $\mathcal{P}$ so that it contains at
least $\lceil\frac{l}{2}\rceil$ points of $f^{-1}(p)$ belonging to
$V^{j}_{p}$, for an opportune $j\in\\{1,2\\}$. It follows that the set
$f^{-1}(p)\setminus supp(D)$ is non-empty and either $NY_{p}$ or $Y_{p}$ is
empty, which contradicts Theorem 1.12.

Now, let us consider case $(2)$ in which $D_{1}$ is not empty. Let us pick
$D_{1}$ as the real curve passing through a given collection
$\tilde{\mathcal{P}}$ of $\frac{\beta^{2}-\beta K_{X}}{2}$ real points
contained in $m=min(\frac{\beta^{2}-\beta K_{X}}{2},l)$ connected components
$C_{j_{1}},\dots,C_{j_{m}}$. Moreover, choose the collection
$\tilde{\mathcal{P}}$ such that it contains the points
$p_{j_{1}},\dots,p_{j_{m}}$ of $f^{-1}(p)$. Now, we proceed as in the previous
case with the difference that one must take $D_{1}$ into account when looking
at the orientation $\mathfrak{O}$. Let $C_{j_{m+1}},\dots,C_{j_{m+\theta}}$ be
the connected components of
$\mathbb{R}C\setminus\bigsqcup_{i=j_{1}}^{j_{m}}C_{i}$ intersecting
$\mathbb{R}D_{1}$. Each connected components $C_{j_{m+i}}$, with
$i=1,\dots,\theta$, may contain, regardless of the chosen complex orientation
on $\mathbb{R}C$, a point of $f^{-1}(p)$ where the orientation $\mathfrak{O}$
and the complex orientation do not coincide. Fix one of the two complex
orientations on $\mathbb{R}C$. Then, remark that either the number $V^{1}_{p}$
of $p_{i}$’s contained in
$\mathbb{R}C\setminus\bigsqcup_{i=1}^{m+\theta}C_{j_{i}}$, where the complex
orientation and the orientation $\mathfrak{O}$ agree, or the number
$V^{2}_{p}$ of the $p_{i}$’s in
$\mathbb{R}C\setminus\bigsqcup_{i=1}^{m+\theta}C_{j_{i}}$, where the two
orientations do not agree, does not exceed $\lceil\frac{l-m-\theta}{2}\rceil$.
Therefore, we pick a collection $\mathcal{P}$ of
$\frac{D_{0}^{2}-D_{0}K_{X}}{2}$ real points such that $\theta$ of those are
the points of $f^{-1}(p)$ contained in the $C_{j_{m+i}}$’s, where
$i=1,\dots,\theta$; and the others include the points of $V^{j}_{p}$, for an
opportune $j\in\\{1,2\\}$. Once again, the set $f^{-1}(p)\setminus supp(D)$ is
non-empty and either $NY_{p}$ or $Y_{p}$ is empty, which contradicts Theorem
1.12. ∎

###  3.2. Examples and Applications

Let us consider real separating curves in the complex projective plane and
focus on [KS20b, Question 1]. A first known example [KS20b, Example 2.8]
concerns real separating plane curves of degree $2s$ or $2s+1$ having a a nest
of maximal depth $s$. Such curves admit a totally real pencil of lines. Some
more examples can be found in [Tou13], where for some pairs $(A,\mathcal{S})$,
where $A$ is a real separating plane sextic curve with $9$ real connected
components and $\mathcal{S}$ its real scheme in $\mathbb{R}\mathbb{P}^{2}$,
the minimal value for $k$ of Theorem 1.4, is shown to be equal to $3$.

$q$ Figure 1. $(\mathbb{R}\mathbb{P}^{2},\mathbb{R}A,\mathbb{R}L)$ of Example
3.1. Double arrows denote $\mathfrak{O}$, simple arrows the fixed complex
orientation of $\mathbb{R}A$ and $\bullet$ the points in $f^{-1}(p)$.

######  Example 3.1 (Application of Theorem 1.12).

Let $A$ be a non-singular plane quintic of type I realising the real scheme
$J\quad\sqcup\quad 4$. We use Theorem 1.12 in order to show that there exists
no separating morphism $f:A\rightarrow\mathbb{C}\mathbb{P}^{1}$ of degree $5$.
For the sake of contradiction assume that such a $f$ exists. By Rokhlin’s
complex orientation formula (2.2), among the ovals of $\mathbb{R}A$, three are
negative and one is positive. Fix $p\in\mathbb{R}\mathbb{P}^{1}$. Denote $q$
the point lying on the positive oval of $\mathbb{R}A$ and contained in
$f^{-1}(p)$. Let us consider Notation 1.10. Choose $D_{1}$ to be empty.
Therefore $D_{0}$ realises the class of a line. Fix an orientation
$\mathfrak{O}$ that changes every time time we cross $\mathbb{R}A$. Up to
choose one of the two complex orientations of $\mathbb{R}A$, one gets that the
fixed complex orientation and $\mathfrak{O}$ agree on the pseudo-line and the
three negative ovals of $\mathbb{R}A$. Hence, if we pick any line $L$ passing
through the point $q$, we get a contradiction thanks to Theorem 1.12; see Fig.
1,

A key information used to deal with the examples above is the fact that their
real schemes are given. In this paper, on the other hand, we focus on real
separating curves $C$ about which only the number of real connected components
is set. We manage to obstruct the existence of certain totally real pencils of
curves with respect to $C$, but we do not know yet how to compute the minimal
value for $k$ of Theorem 1.4. The rest of the section is dedicated to the
proof of Proposition 1.8. We start with a lemma, where a recursive
construction of curves produces odd degree plane curves respecting the
hypothesis of Proposition 1.8.

######  Lemma 3.2.

For all positive integer $s$ and $\varepsilon\in\\{0,1\\}$, there exist non-
singular real plane projective curves of degree $2s+\varepsilon$ of type I and
with $s^{2}+\varepsilon$ real connected components.

###### Proof.

For small positive integers $d\leq 4$, real maximal, therefore separating,
plane curves of degree $d$ have the required properties of the statement.
Therefore, we present a recursive construction of real plane curves $B_{d}$
with the wanted features via a variation of Harnack construction method
[Har76] combined with Proposition 2.3, which allows us to check, at every step
of the construction, that the realised curves are of type I. Set $l_{d}$ as
the number of connected components of $\mathbb{R}B_{d}$. Moreover if $d=2s+1$,
denote $\Lambda_{\pm}$ with $\Lambda^{s}_{\pm}$ for $B_{d}$. In the following,
whenever we consider a plane curve $A$, we denote its polynomial by
$a(x,y,z)$.

Let us fix a real line $L$.

  1. (i)

Step $0$ induction: In order to construct a curve with the wanted properties
for any given degree, we start from the following. One can construct a non-
singular real maximal cubic $B_{3}$ whose pseudo-line $J_{3}$ intersects
$\mathbb{R}L$ in $3$ distinct points; see Fig 2.1. Equip $\mathbb{R}B_{3}$
with one of its two complex orientations and $\mathbb{R}L$ with the complex
orientation opposite to that of $J_{3}$. Let $\mathcal{H}_{1},\mathcal{H}_{2}$
and $\mathcal{H}_{3}$ be the connected components of
$\mathbb{R}L\setminus\mathbb{R}B_{3}$. Up to fix $\mathcal{H}_{1}$, label the
$\mathcal{H}_{i}$’s following the order given by the fixed complex orientation
on $J_{3}$. Pick four real lines $L_{1},\dots,L_{4}$ intersecting pairwise
respectively $\mathcal{H}_{1},\mathcal{H}_{2}$. Now, we apply Brusotti’s
perturbation method [Bru21] to $B_{3}\cup L$ in a compatible way with the
fixed complex orientations of $B_{3}$ and $L$. Namely, we take a small
perturbation of $L\cup B_{3}$ replacing $b_{3}(x,y,z)l(x,y,z)$ with
$b_{4}(x,y,z):=b_{3}(x,y,z)l(x,y,z)+\varepsilon
l_{1}(x,y,z)\dots,l_{4}(x,y,z)$, where $\varepsilon\not=0$ is a sufficient
small real number. Up to a choice of the sign of $\varepsilon$, the real
curve, defined by $b_{4}(x,y,z)=0$, is a non-singular real curve $B_{4}$ of
degree $4$. In particular $B_{4}$ is maximal, therefore of type I, equipped
with a complex orientation $\mathfrak{O}_{4}$ coming form the perturbation;
see Fig. 2.2.

Take for $L$ the opposite complex orientation to the previously chosen one.
Set $\mathcal{H}_{1},\dots,\mathcal{H}_{4}$ for the connected components of
$\mathbb{R}L\setminus\mathbb{R}B_{4}$. Label the $\mathcal{H}_{i}$’s following
the order given by the fixed complex orientation on $\mathbb{R}L$. Without
loss of generality, we can say that $\mathcal{H}_{1}$ and $\mathcal{H}_{3}$
are in the interior of the ovals of $\mathbb{R}B_{4}$ intersecting
$\mathbb{R}L$. Pick five real lines $L_{1},\dots,L_{5}$ such that $L_{1}$
intersects $\mathcal{H}_{2}$ and the others $\mathcal{H}_{4}$. Now, we apply
Brusotti’s perturbation method to $L\cup B_{4}$ with respect to
$\bigcup_{i=1}^{5}L_{i}$, in a compatible way with the complex orientation
$\mathfrak{O}_{4}$ and the fixed complex orientation of $L$, and obtain a non-
singular real curve $B_{5}$ of degree $5$ whose real part is equipped with an
orientation $\mathfrak{O}_{5}$ coming from the perturbation. In order to
assure that $\mathfrak{O}_{5}$ is a complex orientation of $B_{5}$ and, hence
$B_{5}$ is of type I, it is enough to check that $B_{5}$ respects (2.2); see
Proposition 2.3. Because of the absence of nests in $\mathbb{R}B_{5}$ and the
number of positive and negative ovals of $\mathbb{R}B_{5}$ differ by $-2$, it
follows that $(B_{5},\mathfrak{O}_{5})$ respect Rokhlin’s complex orientation
formula.

Fig. 2.1: $(\mathbb{R}\mathbb{P}^{2},\mathbb{R}B_{3},\mathbb{R}L)$Fig. 2.3:
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}B_{5},\mathbb{R}L)$Fig. 2.2:
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}B_{4},\mathbb{R}L)$Fig. 2.4:
$(\mathbb{R}\mathbb{P}^{2},\mathbb{R}B_{6},\mathbb{R}L)$ Figure 2.
$\mathbb{R}L$ in red and $\mathbb{R}B_{d}$ in black, where, from top to bottom
and from left to right, $d=3,4,5,6$.

  2. (ii)

Inductive Step: Assume that there exists a non-singular real plane curve
$B_{2s-1}$ of degree $2s-1$ and of type I. Equip $\mathbb{R}B_{2s-1}$ with one
of its two complex orientations and $\mathbb{R}L$ with the complex orientation
opposite to that of $J_{2s-1}$. Moreover, assume that

     * •

the pseudo-line $J_{2s-1}$ intersects $\mathbb{R}L$ in $2s-1$ distinct points
such that, if one labels the intersections points $2s-1,\dots,1$ with respect
to the complex orientation on $J_{2s-1}$, the order of the labeled points
agrees with the complex orientation of $\mathbb{R}L$;

     * •

the number $l_{2s-1}$ equals $(s-1)^{2}+1$;

     * •

the curve $B_{2s-1}$ has no nests and

$\Lambda^{(s-1)}_{+}=\frac{(s-1)(s-2)}{2}\quad\text{and}\quad\Lambda^{(s-1)}_{-}=\frac{s(s-1)}{2}.$

Let $\mathcal{H}_{1},\dots,\mathcal{H}_{2s-1}$ be the connected components of
$\mathbb{R}L\setminus\mathbb{R}B_{2s-1}$. Up to fix $\mathcal{H}_{1}$, label
the $\mathcal{H}_{i}$’s following the order given by the fixed complex
orientation on $J_{2s-1}$. Pick $2s$ real lines $L_{1},\dots,L_{2s}$
intersecting pairwise
$\mathcal{H}_{j},\mathcal{H}_{j+1},\dots,\mathcal{H}_{j+s}$, for some
$j\in\\{1,\dots,2s+1\\}$. Now, we apply Brusotti’s perturbation method to
$L\cup B_{2s-1}$ with respect to $\bigcup_{i=1}^{2s}L_{i}$, in a compatible
way with the fixed complex orientations of $B_{2s-1}$ and $L$, and obtain a
degree $2s$ non-singular real algebraic curve $B_{2s}$, equipped with an
orientation $\mathfrak{O}_{2s}$. Let us check that (2.1) applies to the pair
$(B_{2s},\mathfrak{O}_{2s})$. The curve $B_{2s}$ has no nests and
$l_{2s}=l_{2s-1}-1+2s-1=s^{2}$. Hence, Proposition 2.3 implies that $B_{2s}$
is of type I. In addition, remark that $s$ ovals of $\mathbb{R}B_{2s}$
intersect $\mathbb{R}L$.

Take for $L$ the opposite complex orientation to the previously chosen one.
Set $\mathcal{H}_{1},\dots,\mathcal{H}_{2s}$ for the connected components of
$\mathbb{R}L\setminus\mathbb{R}B_{2s}$. Label the $\mathcal{H}_{i}$’s
following the order given by the fixed complex orientation on $\mathbb{R}L$.
Without loss of generality, we can say that $\mathcal{H}_{2h+1}$, for $0\leq
h\leq s$ are in the interior of the ovals of $\mathbb{R}B_{2s}$ intersecting
$\mathbb{R}L$. Pick $2s+1$ real lines $L_{1},\dots,L_{2s+1}$ such that $L_{h}$
intersects $\mathcal{H}_{2h}$, for all $1\leq h\leq s-1$ and all others lines
intersect $\mathcal{H}_{2s}$. Once again, apply Brusotti’s perturbation method
to $L\cup B_{2s}$ with respect to $\bigcup_{i=1}^{2s+1}L_{i}$, in a compatible
way with the fixed complex orientations of $B_{2s}$ and $L$. We get a degree
$2s+1$ non-singular real curve $B_{2s+1}$ whose real part is equipped with an
orientation $\mathfrak{O}_{2s+1}$ coming from the perturbation, and
$l_{2s+1}=l_{2s}+1=s^{2}+1$. Let us check that the pair
$(B_{2s+1},\mathfrak{O}_{2s+1})$ respects (2.2). First, observe that

$\Lambda^{s}_{+}=\Lambda^{(s-1)}_{+}+s-1\quad\text{and}\quad\Lambda^{s}_{-}=\Lambda^{(s-1)}_{-}+s.$

It follows that, because of the absence of nests in $\mathbb{R}B_{2s+1}$ and
the number of positive and negative ovals of $\mathbb{R}B_{2s+1}$ differ by
$-s$, the pair $(B_{2s+1},\mathfrak{O}_{2s+1})$ satisfies Rokhlin’s complex
orientation formula. In conclusion, one obtains a real curve of degree $2s+1$
which is separating and with $s^{2}+1$ real connected components.

∎

###### Proof of Proposition 1.8.

The non-singular real algebraic plane curves $B_{2s+1}$ of odd degree
$2s+1\geq 5$ and of type I, constructed in Lemma 3.2, satisfy the hypothesis
of Corollary 1.6. Therefore $B_{2s+1}$ admit no totally real pencils of curves
of degree lower or equal to $\frac{s^{2}+1}{2s+1}$. ∎

######  Remark 3.3.

Notice that the non-singular real algebraic plane curves $B_{2s}$ of even
degree $2s\geq 6$ and of type I constructed in Lemma 3.2, does not satisfy
hypothesis of Corollary 1.6. We do not know yet whether there is a way to
apply [Ore21, Theorem 3.2] to this particular case in order to show that
$B_{2s}$ admit no totally real pencils of curves of degree lower or equal to
$\frac{s}{2}$.

Mimic the constructions adopted in the proof of Lemma 3.2, one may also
construct an explicit family of real separating plane curves of increasing
even degree, which respect the hypothesis of Proposition 1.2 or even those in
Remark 1.14. Instead of presenting to the reader such a construction, that
would be a modified version of proof of Lemma 3.2, we conclude with an example
of separating real curves in a different ambient surface, respecting
hypothesis of Theorem 1.13 and, therefore, non admitting separating morphisms
of degree equal to the number of their real connected components.

######  Example 3.4.

Let $Q$ be $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$, equipped with the anti-
holomorphic involution $\sigma:Q\rightarrow Q$ sending $(x,y)$ to
$(\overline{y},\overline{x})$, where $x=[x_{0}:x_{1}]$ and $y=[y_{0}:y_{1}]$
are in $\mathbb{C}P^{1}$ and
$\overline{x}=[\overline{x_{0}}:\overline{x_{1}}]$ and
$\overline{y}=[\overline{y_{0}}:\overline{y_{1}}]$ are respectively the images
of $x$ and $y$ via the standard complex conjugation on $\mathbb{C}P^{1}$. The
real part of $Q$ is homeomorphic to a $2$-sphere $S^{2}$ and $Q$ is called
quadric ellipsoid. Moreover, the Euler characteristic of the trivial bundle of
$Q$ equals one; see Remark 1.14. A non-singular real algebraic curve A on $Q$
is defined by a bi-homogeneous polynomial of bidegree $(d,d)$

$P(x,y)=\sum\limits_{0\leq i,j\leq
d}a_{i,j}x_{0}^{i}x_{1}^{d-i}y_{0}^{j}y_{1}^{d-j}=0,$

where $d$ is a positive integer and the coefficients satisfy
$a_{i,j}=\overline{a_{j,i}}$. Real schemes realised by non-singular real
algebraic curves of bidegree $(d,d)$ on $Q$ are completely classified for
$d\leq 5$; see [GS80], [Mik94] and [Man21]. Moreover such classifications
distinguish the cases in which a given topological type may or may not be
realised by a real algebraic curve of type I.

For $d\leq 5$, we report in Table 1 a list of real schemes realised by type I
real algebraic curves of bidegree $(d,d)$, with $l$ real connected components,
where $l$ undergoes the hypothesis of Theorem 1.13. Hence, the curves do not
admit separating morphisms of degree $l$. In order to understand Table 1, let
us recall some notation. Let $A$ be a non-singular bidegree $(d,d)$ real curve
on $Q$. The connected components of $A$ are called ovals. An oval in $S^{2}$
bounds two disks; therefore, on $S^{2}$ interior and exterior of an oval are
not well defined. It follows that the encoding of real schemes on $S^{2}$ is
not well defined either and it depends on the choice of a point $p$ on
$S^{2}\setminus\mathbb{R}A$. Let us take $S^{2}$ deprived of $p$, which is
homeomorphic to $\mathbb{R}^{2}$ and let us call oval any circle embedded in
$\mathbb{R}^{2}$. Analogously to the case of $\mathbb{R}P^{2}$, in
$\mathbb{R}^{2}$ one define interior and exterior of an oval and
(non-)injective pairs for each pair of ovals. Up to choose a point, we can now
adopt the same notation introduced in Section 2.2.

d | Real schemes realised by type I curves | $l\leq$ | $r(D_{0},D_{1})$ | $D_{0}$ | $D_{1}$ | $m$ | $\theta$
---|---|---|---|---|---|---|---
$3$ | $\langle\langle 1\rangle\rangle$ | $3\leq$ | 3 | $-$ | $(1,1)$ | 3 | 0
$4$ | $\langle\langle\langle 1\rangle\rangle\rangle$ | $4\leq$ | 6 | $(1,1)$ | $-$ | $-$ | $-$
$4$ | $4\quad\sqcup\quad\langle 1\rangle$ | $6\leq$ | 6 | $(1,1)$ | $-$ | $-$ | $-$
$4$ | $3\quad\sqcup\quad\langle 2\rangle$ | $6\leq$ | 6 | $(1,1)$ | $-$ | $-$ | $-$
$5$ | $\langle\langle\langle\langle 1\rangle\rangle\rangle\rangle$ | $5\leq$ | 7 | $(1,1)$ | $(1,1)$ | 3 | 2
$5$ |  $\alpha\quad\sqcup\quad\langle\beta\rangle\quad\sqcup\quad\langle\gamma\rangle,$ for all $\alpha,\beta,\gamma$ such | $7\leq$ | 7 | $(1,1)$ | $(1,1)$ | 3 | 2
| that $\alpha=0\pmod{2}$ and $\alpha+\beta+\gamma=5$ |  |  |  |  |  |
Table 1.  For all $d\leq 5$, this is a list of real schemes realised by type I
real algebraic curves of bidegree $(d,d)$, with $l$ real connected components,
where $l$ undergoes the hypothesis of Theorem 1.13. Hence, the curves do not
admit separating morphisms of degree $l$. The symbol $r(D_{0},D_{1})$
represents the bound for the number $l$, depending on choices of
$D_{0},D_{1}$, for which the hypothesis of Theorem 1.13 are satisfied and such
that one maximises the number
$\frac{D_{0}^{2}-D_{0}K_{S}}{2}\in\mathbb{Z}_{>0}$.

######  Remark 3.5.

In the context of Example 3.4, one can construct recursively, in a way similar
to the proof of Lemma 3.2, for all $d\geq 6$, bidegree $(d,d)$ separating real
algebraic curves on the quadric ellipsoid not admitting separating morphism of
degree equal to the number of their real connected components. In fact,
variations of Harnack’s construction method, Brusotti’s perturbation method
and Rokhlin’s complex orientation formula [Zvo83, Ore07] are also available on
the quadric ellipsoid.

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Matilde Manzaroli, Universität Tübingen, Germany

E-mail address:  matilde.manzaroli’at’uni-tuebingen.de