A Viro-Zvonilov-type inequality for Q-flexible curves of odd degree

We define an analogue of the Arnold surface for odd degree flexible curves, and we use it to double branch cover $Q$-flexible embeddings, where $Q$-flexible is a condition to be added to the classical notion of a flexible curve. This allows us to obtain a Viro--Zvonilov-type inequality: an upper bound on the number of non-empty ovals of a curve of odd degree. We investigate our method for flexible curves in a quadric to derive a similar bound in two cases. We also digress around a possible definition of non-orientable flexible curves, for which our method still works and a similar inequality holds.

Let F ⊂ CP 2 be a flexible curve of odd degree m.We denote as ℓ ± and ℓ 0 the number of ovals of the curve RF ⊂ RP 2 that bound from the outside a component of RP 2 ∖ RF which has positive, negative or zero Euler characteristic, respectively.In particular, ℓ + is the number of empty ovals, and ℓ 0 + ℓ − is that of nonempty ones.O. Viro and V. Zvonilov [1992] proved the following upper bound for the number of non-empty ovals: 2 , with h(m) denoting the biggest prime power that divides m.Their proof relied on taking a branched cover of CP 2 , ramified over the surface F. Usually, it is a good choice to take doubly sheeted branched covers, but this is not possible in this setting where m is odd.Odd degree curves are a different story compared to even degree ones, one reason being the nonexistence of the Arnold surface in S 4 (RF is not null-homologous in H 1 (RP 2 ; Z/2), and neither is F in H 2 (CP 2 ; Z/2)).In the present paper, we give a definition of an analogue of the Arnold surface in CP 2 for odd degree curves.This means that, under a certain condition of being Q-flexible (up to taking another conic Q with empty real part and pseudo-holomorphic, this is always satisfied by pseudo-holomorphic curves), we are allowed to take the double branched cover of CP 2 ramified over a perturbation of this Arnold surface.This condition is also always satisfied by algebraic curves.We will show the following result, by methods analogous to Viro and Zvonilov.
Theorem 3.11.Let F be a Q-flexible curve of odd degree m.Then If equality holds, then the curve is type I.
It is worth mentioning that this is not quite Zvonilov's bound (m − 1)(m − 3)/4 [1979], which works for any flexible curve that intersects a real line generically (this condition being the degree one analogue of our Q-flexibility), and in particular for any pseudo-holomorphic curve.However, it appears that Q-flexibility and this condition by Zvonilov are independent for general flexible curves.
In Section 1, we discuss some constructions in CP 2 and CP 2 seen as 2-fold branched covers of the standard 4-sphere.In Section 2, we construct the Arnold surface for odd degree curves, and we describe the behavior of the real part of the curve under this construction.In Section 3, we prove the inequality.In Section 4, we review our method for curves in a quadric to produce a result which, to our knowledge, is new even for algebraic curves.In Section 5, we compare our inequality to Viro and Zvonilov's, and we investigate the possible notion of nonorientable flexible curves, for which our method still applies to derive a similar bound.

Preliminaries
Throughout this paper, all surfaces will be assumed to be connected, and all embeddings are smooth.The complex conjugation conj : CP 2 → CP 2 is defined in homogeneous coordinates by conj([z 0 : z 1 : z 2 ]) = [z 0 : z 1 : z 2 ], and RP 2 ⊂ CP 2 is the fix-point set Fix(conj).Here, Q will always denote a generic real conic with empty real part (for instance, the Fermat conic given by the equation z 2 0 + z 2 1 + z 2 2 = 0).In particular, it is a smoothly embedded 2-sphere Q ⊂ CP 2 which represents the homology class ] in H 2 (CP 2 ; Z) ∼ = Z, the choice of a generator being the homology class of any complex line.
Flexible and Q-flexible curves.A real plane algebraic curve is a real nonsingular homogenous polynomial X ∈ R[x 0 : x 1 : x 2 ].By the real part of the curve, we mean the set RX = [x 0 : and by the complexification of the curve, we mean Evidently, CX is invariant under complex conjugation.If m = deg(X) ⩾ 1, then we see that [CX] = m[CP 1 ] ∈ H 2 (CP 2 ; Z), and that CX is a surface of genus g = (m − 1)(m − 2)/2.Also, the tangent space of the complex curve is related to that of the real curve in the following sense: for all x ∈ RX, T x CX = T x RX ⊕ i • T x RX.
Curves with b 0 (RF) = g + 1 are called M-curves.On the other hand, curves with b 0 (RF) = 0 if m is even and b 0 (RF) = 1 if m ⩾ 3 is odd are called minimal curves (note that there are no minimal curves in degree one).
(ii) If m is even, then each component of RF is contractible in RP 2 , and if m is odd, all but one components of RF are.Contractible components of RF are called ovals.An odd degree curve can never have RF = ∅ (it always has the non-contractible component).
(iii) A flexible curve F is said to be a type I (resp.type II) curve if F ∖ RF has two connected components (resp. is connected).An M-curve is always type I, and a minimal curve is always type II.
What makes flexible curves so different from algebraic curves is the lack of rigidity, mainly seen with the Bézout theorem, which, in particular, implies that a degree m algebraic curve generically intersects Q transversely in exactly 2m points.
2m points, necessarily swapped pairwise by complex conjugation.

Two double branched covers.
There is a well-known diffeomorphism between CP 2 /conj and S 4 (see [Kuiper 1974]).We denote the associated (cyclic) 2-fold branched cover as p : (CP 2 , RP 2 ) → (S 4 , R), with R = p(RP 2 ) an embedded RP 2 in S 4 .We also let Q = p(Q), the image of the preferred conic under that branched cover.From the fact that Q does not intersect the branch locus RP 2 , we see that the restriction p : Q → Q is an unbranched 2-fold cover, with the conjugation being an orientation-preserving involution generating the group of deck transformations.As such, we see that Q is also an embedded RP 2 in S 4 .
Given a closed embedded surface F 2 in a (closed oriented) 4-manifold X 4 , we denote as e(X, F) the normal Euler number of the embedding F ⊂ X; that is, the Euler class of the normal bundle νF.It is also equal to the self-intersection number F • F, which is defined by counting signed intersection points between F and a small perturbation F ′ of F in the normal direction.If F is oriented, then it corresponds to the intersection form of X evaluated on [F] ∈ H 2 (X; Z).Proposition 1.3.We have the following normal Euler numbers: we have that the normal bundle ν RP 2 and the tangent bundle T RP 2 are anti-isomorphic, and thus, for the Euler class, e(ν RP 2 ) = −e(T RP 2 ) = −χ(RP 2 ) = −1.Finally, the computations of e(S 4 , R) and e(S 4 , Q) come from the next lemma.
), and given F an embedded closed surface in X, we denote as F the lift p −1 (F).
Proof.One has to inspect what happens in each case individually.In the first, note that the lift of a perturbation is a perturbation of the lift, and one can ensure that the self-intersection points occur away from the ramification locus B. As such, each of these points lifts to two intersections, and the orientations agree because f is orientation-preserving.
The second case can be deduced from the first.Let F′ be a small transverse perturbation of F. Letting τ : Y → Y denote the involution that spans Aut( f ), and letting ■ We now wish to consider the 2-fold branched cover of S 4 , ramified over Q this time.It is possible to make some computations to find an orientation-reversing involution of S 4 which swaps R and Q.Alternatively, taking any orientationreversing free involution of S 4 , this maps R to a projective plane with normal Euler number +2, and this is always isotopic to Q in S 4 .Tracking this isotopy produces the involution needed.As such, we see that the smooth 4-manifold obtained as the The two branched coverings of interest and their associated branch loci.The arrows marked −→ → denote an unbranched 2-fold cover from a 2-sphere to a real projective plane.
double branched cover of S 4 ramified along Q is diffeomorphic1 to CP2 .We let p : CP 2 → S 4 denote that double branched cover.
Define Q = p−1 (R).We see that RP 2 and Q are respectively embeddings of RP 2 and S 2 in CP 2 .Using Lemma 1.4 again, we can compute the normal Euler numbers.
In Figure 1 we depict a summary of the different maps in play.

The Arnold surface of an odd degree flexible curve
For a flexible curve F ⊂ CP 2 , let A + (F) = F/conj = p(F).It is an embedded surface in S 4 with boundary ∂ A + (F) ⊂ R identified with RF, and it is orientable if and only if F is type I.
If the curve has even degree, then RF is null-homologous, and thus exactly one component of RP 2 ∖ RF is non-orientable (it is a punctured Möbius band).Let RP 2 ± be the closure of the two possible subsets of − to be the one containing the punctured Möbius band (i.e.RP 2 + is orientable, and RP 2 − has exactly one non-orientable component).In the case where F is an algebraic curve of even degree, the polynomial P defining it can be chosen in such a way that In Figure 2, we depict such an example for an algebraic curve.
Definition 2.1.Given a flexible curve F of even degree, we let , and we call it the Arnold surface of F. In odd degrees, it is not possible to define a surface in this way; we need to go to CP 2 first.Take F to be a flexible curve of odd degree.We denote as J ⊂ RP 2 the non-contractible component of RF and as o an oval of RF.Let J + = p(J), o + = p(o), and set Observe that p restricts to an unbranched 2-fold covering p : J → J + and p : o → o + .Proposition 2.2.We have J ∼ = S 1 and o ∼ = S 1 ⊔ S 1 , and the unbranched coverings p : J → J + and p : o → o + are respectively the non-trivial and the trivial 2-fold coverings of the circle.
Proof.Isotope J in RP 2 to be J = RX with X ∈ R[x 0 : x 1 : x 2 ] a degree-one nonsingular homogeneous polynomial.Note that we do not need to look at what happens outside of RP 2 for the claim.In particular, CX ⋔ Q is two points.Letting G + = p(CX) and G = p−1 (G + ), we obtain Moreover, G + ⋔ Q is one point, for the two points in CX ⋔ Q are swapped pairwise by conjugation.In particular, the covering p : G → G + is a 2-fold branched cover of the disc G + (for in degree one, CX is a sphere and RX is type I), with one branch point in its interior.This is unique, and it is known to induce the non-trivial cover on the boundary, so the first claim follows (see Figure 3).
For the other claim, an oval o bounds a disc D embedded in RP 2 , and is thus disjoint from Q. Therefore, the disc D/conj ⊂ R bounded by o + lifts in CP 2 to two disjoint discs in Q.This means that p : ō → o + is the trivial covering, and ō is two circles.■ We let RF be the set The previous statement implies that every oval of RF gets doubled in RF, whereas the non-contractible component J does not.Proof.This comes from the observation that the covering p : Q → Q is the quotient of the 2-sphere Q by a fixed-point free involution (that is, the antipodal map), as well as the fact that p : (RP 2 , RF) → (R, p(RF)) is a diffeomorphism of the pair.■ This means that the real scheme RF can be seen doubled in RF, as Figure 4 depicts.Now, define A + (F) = p−1 (A + (F)) = p−1 (F/conj).For the analogue of RP 2 + , there are two subsets Q ± of Q ∖ RF that have ∂ Q ± = RF, and those are diffeomorphic, exchanged by "symmetry" of Q along J.To be more precise, we . The set Q + , shaded, for an algebraic curve of degree 7 with real scheme ⟨J ⊔ 2 ⊔ 1⟨1⟩⟩ (in Viro notation), obtained as a perturbation of three ellipses and a line.
denote as Q ± the closure of these two sets, with a choice involved in labeling one Q + and the other Q − .
Definition 2.4.The Arnold surface of a flexible curve F of odd degree is the surface

Proving the inequality
The idea is that we would like to take the 2-fold branched cover of CP 2 ramified along the Arnold surface.This is not yet possible in this odd degree setting, for the surface A(F) is not null-homologous in H 2 (CP 2 ; Z/2) (we will see that it has an odd self-intersection number).In fact, this limitation is what led Viro and Zvonilov to consider h(m)-sheeted branched covers, where h(m) denotes the highest prime power that divides m.However, in our favorable setting, we can perturb the Arnold surface, with the important feature that it preserves the structure of the curve RF inside Q.One last remark is that we could not apply the same construction to a Q-flexible curve F ⊂ CP 2 directly, because the conic Q has an even homology class.
Branching over the Arnold surface.We are first interested in computing the normal Euler number of A(F) ⊂ CP 2 .Recall that if F ⊂ X is a closed surface in a closed oriented 4-manifold, then the Euler class e(νF) ∈ H 2 (F; Z w ) corresponds to the self-intersection of F (here, Z w means coefficients twisted by w 1 (νF), and w 1 (X) = 0 implies w 1 (νF) = w 1 (F), from which twisted Poincaré duality readily gives H 2 (F; Z w ) ∼ = Z).
In the case where ∂ F ̸ = ∅ however, one needs to choose a fixed nonvanishing section θ of νF| ∂ F , and consider a relative Euler class (see [Šarafutdinov 1973]): This Euler class corresponds to the integer obstruction to extend this section θ to the whole νF.However, if one needs to glue two surfaces F 1 and F 2 along their common boundary ∂ F 1 = ∂ F 2 and compute the Euler number of F 1 ∪ ∂ F 2 in terms of relative Euler numbers of F 1 and F 2 , there are two things to be careful about: (1) The bundle Λ = (νF 1 ∩ νF 2 )| ∂ F i over ∂ F i needs to be rank one.
(2) This bundle Λ needs to have a non-vanishing section θ .
If both conditions are satisfied, the section θ gives rise to the same section of This can be used to define relative Euler numbers e θ (X, F i ).Since, in the closed case, the number e(X, F) does not depend on the choice of the (possibly vanishing) global section of νF, we obtain the relation For instance, if F ⊂ CP 2 is a flexible curve of even degree m = 2k with nonempty real part RF, one sees that Λ = (ν RP 2 + ∩ νF)| RF is the trivial line bundle over RF.If θ denotes a section of the normal bundle RF in RP 2 , then iθ is a section of Λ, and letting R + = RP 2 + /conj and A + (F) = F/conj, it also induces a section θ of because F is closed, and now a flexible curve of odd degree, the normal bundle of RF in RP 2 is a non-trivial line bundle over RF (to be more precise, exactly one connected component of this bundle is the non-orientable line bundle over the circle: the component associated to the pseudo-line J ⊂ RF).As such, there is no nonvanishing section θ of Λ, and it does not give rise to a section iθ of νF| RF .However, the subbundle iΛ ⊂ νF| RF can be seen as a field of lines of νF| RF (instead of a section being a vector field).
In general, let Λ ⊂ νF| ∂ F be a line sub-bundle.As done in [Guillou and Marin 1980, §3], one can still consider the integer obstruction ẽΛ (X, to extend this field of line to the whole νF.In the case where Λ does have a section θ , we have ẽΛ (X, F) = 2e θ (X, F).
Back to where F is a flexible curve of odd degree, and letting because F is closed.This means that, in the above sense, we have e(S 4 , A + (F)) = m 2 /2, although this is a non-integer value.
To ease out the exposition, we will allow ourselves to write half-integer Euler numbers and to use Lemma 1.4 with half-integers.It will be understood that we use the obstruction ẽ when needed.We will also omit the choice of the field of lines in the subscript, as all surfaces will ultimately become closed at the end of computations.
Proof.Recall that we defined A + (F) = p−1 (A + (F)), and Using Lemma 1.4 twice, we compute that e(CP 2 , A + (F)) = m 2 .Now, we simply make use of the fact that e(CP and because Q + and Q − are swapped by the (orientation-preserving) involution of CP 2 spanning Aut( p), we obtain from which we derive e(CP 2 , Q ± ) = −2.Alternatively, this can be obtained from the following lemma.
Proof.The submanifold RP 2 ⊂ CP 2 being Lagrangian, and the covering p : CP 2 → S 4 being branched exactly on p(RP 2 ), we observe that νR ∼ = −T R in S 4 .However, the covering p : Because A(F) has an odd self-intersection, we see that it cannot be nullhomologous in H 2 (CP 2 ; Z/2).In fact, because this group has rank one, being Z/2-null-homologous is equivalent to having an even self-intersection.There is another surface, however, which is not null-homologous and transverse to A(F): the surface RP 2 .If F is a Q-flexible curve of odd degree m, the transverse intersection The surface A(F) ∪ RP 2 is therefore immersed with m transverse crossings only.
We will describe how to resolve those double points to obtain an embedded surface.Firstly, in a closed oriented 4-manifold X, let Σ ⊂ X be the image of a closed surface through an immersion, with only one transverse self-intersection point x ∈ Σ.Take B ⊂ X to be a small 4-ball around x, which meets Σ in two disks intersecting transversely at their common center x.The boundary of those discs is a Hopf link ∂ B ∩ Σ ⊂ ∂ B ∼ = S 3 , which bounds a Hopf band H ⊂ B. We call the surface Σ ′ defined by a choice of a gluing of a Hopf band H to Σ ∖ B a smoothing of the singularity of the immersed surface Σ ⊂ X.
Lemma 3.3.The resulting surface Σ ′ is an embedded surface in X with χ(Σ ′ ) = χ(Σ) − 1, e(X, Σ ′ ) = e(X, Σ) ± 2, and we have freedom in the choice.Proof.Regarding the claim about the normal Euler numbers, we use similar arguments as in [Yamada 1995, §5].Note that if B is a small 4-ball around the double point x ∈ Σ, then the Hopf link ∂ B ∩ Σ comes with two possible choices of . The two possible smoothings of a singularity of an immersion, given by both choices of orientation of the Hopf link.
orientations.Each determines a unique (up to isotopy fixing the boundary) oriented Hopf band H inducing that orientation.A transverse push-off s(Σ) of Σ can be assumed to be parallel to Σ near x, and the intersection s(Σ) ∩ Σ ∩ H is two points with the same sign.Finally, we see that those signs are opposite to one another in both choices of orientations of ∂ B ∩ Σ (see Figure 5).The fact that χ(Σ ′ ) = χ(Σ)−1 is simply a matter of using the formula χ(A∪B) = χ(A) + χ(B) − χ(A ∩ B) twice (here, all the sets involved are cellular subspaces).Indeed, if H denotes the Hopf band that is glued to Σ ∖ B, then by noting that B ∩ Σ is topologically a wedge of two discs.■ Consider F ⊂ CP 2 a Q-flexible curve of odd degree m.The Arnold surface A(F) needs not be orientable, and as said before, there is no 2-fold branched cover of (CP 2 , A(F)).Recall that A(F) ⋔ RP 2 is m points, and as such, A(F) ∪ RP 2 is an immersed surface with m double points.Applying the previous smoothing of the singularities at each of those m points, this yields a surface X (F) ⊂ CP 2 , with Here, r is not free to take all the possible values in {−m, . . ., m}.However, the extremal values ±m are always realizable.Define X (F) to be the one where we pick up a +2 every time (that is, r = +m).Two applications of the topological Riemann-Hurwitz formula give χ(A(F)) = χ(F) − m + 1.Therefore, we have Take Y 4 to be the 2-fold cover of CP 2 branched over X (F).This has been made possible because the surface X (F) has zero homology mod 2: [X (F)] = 0 ∈ H 2 (CP 2 ; Z/2) (see [Gompf and Stipsicz 1999, §6.3] or [Nagami 2000, Corollary 2.10]).Indeed, and A(F) intersects RP 2 in an odd number m of points.Therefore, we deduce We use a generalisation of the Gysin sequence, as stated in [Lee and Weintraub 1995, Theorem 1]: Here, H 1 (Y, * ; Z/2) ∼ = H1 (Y ; Z/2) the reduced homology group, and we have H 1 (CP 2 , X (F); Z/2) = 0, by looking at the homology long exact sequence of the pair (CP 2 , X (F)).This provides H 1 (Y ; Z/2) = 0, as claimed.For b 3 (Y ) = 0, this is a consequence of b 1 (Y ) = 0 and Poincaré duality.■ An educated guess is that Y may be simply-connected, just like the usual branched cover of CP 2 branched over an algebraic curve {P(x 0 , x 1 , x 2 ) = 0}, given as the algebraic surface {P(x 0 , [Wilson 1978]).However, we have enough information to compute all the homological invariants of Y that will be useful.We recall the Hirzebruch formula for the signature of 2-fold branched covers.
Theorem 3.5 [Hirzebruch 1969, Section 3].Let f : (Y, B) → (X, A) be a cyclic 2fold branched cover, with X and Y both closed oriented 4-manifolds, A a closed surface and f orientation-preserving.Then, we have Proposition 3.6.We have where b + 2 (Y ) and b − 2 (Y ) respectively denote the maximal ranks of the subspaces of H 2 (Y ; Z) on which the intersection form Q Y is positive and negative definite.
Proof.First, the topological Riemann-Hurwitz formula again yields Next, we use the Theorem 3.5 with the branched cover Θ to obtain Proving the inequality.We will now mostly mimic the proof of Viro and Zvonilov [1992].Note that the construction of X (F) from A(F) ∪ RP 2 happens away from a neighborhood Q.In particular, we still see RF embedded inside X (F).Given an oval o ⊂ RF, recall that RP 2 ∖ o has two connected components, one of which is a punctured disc (the other being a punctured Möbius band).Letting C(o) ⊂ R be the image of that component under p : CP 2 → S 4 , we see that p−1 (C(o)) ⊂ Q is diffeomorphic to two disjoint copies of C(o).We denote as C ± (o) each of these copies, with the property that C ± (o) ⊂ Q ± (see Figure 6).The same construction works for J: there are two path-connected subsets D ± (J) of Q ± that have J as a part of their boundary.Letting D(J) = Θ −1 (D − (J)), we have that D(J) is a surface of genus e the number of exterior ovals in RF (those not included in any other), and the restriction Θ : D(J) → D − (J) is again the quotient of Σ e by reflection along a plane in the middle.
Given an oval o ⊂ RF, we denote as the Euler characteristic of the connected subset of RP 2 ∖ RF bounded by o from outside.Similarly, we let χ(J) = χ(D − (J)).One remarks that χ(o) ⩽ 1, with equality if and only if o is empty, and that χ(J) = 1 − e with e the number of exterior ovals.
Proposition 3.8.Let o, o ′ ⊂ RF be ovals, and denote again by J the non-contractible component of RF. (1 4(e − 1). ( We can now prove an analogue to their Corollary 1.5.C. } has rank at least ℓ (where ℓ is the number of ovals o 1 , . . ., o ℓ of the curve).If the family has rank ℓ + 1, then the curve is type I.
Proof.We can apply Lemma 3.9 in our setting, where ν = Θ : Y → (CP 2 , X (F)) and h = 2.We then see that α2 : is injective, because CP 2 is connected and H 3 (Y ; Z/2) = 0 (Proposition 3.4).Noting that α2 (C − (o)) = C(o) and α2 (D − (J)) = D(J), the claim follows from the very same arguments as in [Viro and Zvonilov 1992, §2.4].■ Recall that ℓ ± and ℓ 0 denote the number of ovals of the curve that bound from the outside a component of RP 2 ∖RF with positive/negative or zero Euler characteristic, respectively.The previous results finally wraps up to yield our main theorem.
Theorem 3.11.Let F be a Q-flexible curve of odd degree m.Then If equality holds, then the curve is type I.
Proof.Take the maximal subset , and let r = rank(P).Then, we obtain and similarly for D(J), observe that P has exactly ℓ 0 + ℓ − + 1 elements (assuming that there is at least one oval to have D(J) ∈ P; if there are none, the theorem is vacuous).Therefore, because of Corollary 3.10, we deduce r ⩾ #P − 1 = ℓ 0 + ℓ − .This produces , which is the claimed inequality.The extremal case also follows from an almost word-for-word proof as in [Viro and Zvonilov 1992].■

Curves on a quadric
We investigate our method for flexible curves in CP 1 × CP 1 , with either of its antiholomorphic involutions c 1 (x, y) = ( x, ȳ) or c 2 (x, y) = ( ȳ, x).This is motivated by recent work from Zvonilov [2022], which generalizes [Viro and Zvonilov 1992] to flexible curves on almost-complex 4-manifolds.For a survey of results regarding curves in CP 1 × CP 1 , we refer the reader to [Matsuoka 1991] or [Gilmer 1991].We will also need the following result.
Theorem 4.1.[Letizia 1984, §3] There are diffeomorphisms More precisely, the differential structure on CP 1 × CP 1 ∖ Fix(c i )/c i extends to the standard one on S 4 or CP 2 , respectively.
Note that in the present work, we do not make any assumption regarding gcd(a, b) with [F] = (a, b) ∈ H 2 (CP 1 × CP 1 ), contrary to [Zvonilov 2022] where there is no result if gcd(a, b) = 1.
Theorem 4.5.Let F be a Q-flexible curve in the hyperboloid with bidegree (a, b) where both a and b are odd.Let ℓ ± and ℓ 0 denote the number of ovals of the curve that bound from the outside a subset with positive, negative or zero Euler characteristic, respectively.Then Note that H 2 (X; Z) is a Z ⊕ Z spanned by the homology classes of algebraic curves of bidegree (1, 0) and (0, 1).We have a notion of a (Q-)flexible curve in this setting too.Definition 4.2.Let F ⊂ X be a closed, connected and oriented surface.We call F a bidegree (a, b) flexible curve if the following conditions hold: (1) conj(F) = F. (2 Note that if both a and b are odd, then RF is some number of ovals (nullhomologous curves in R), and some non-zero number of parallel copies of a curve with homology class (α, β ) in H 1 (R; Z) ∼ = Z ⊕ Z, where 0 ⩽ α ⩽ a and 0 ⩽ β ⩽ b are both odd and coprime, and π 1 (R) = H 1 (R; Z) ∼ = Z ⊕ Z is spanned by the real parts of bidegree (1, 0) and (0, 1) algebraic curves.In the case of an oval o, the complement R ∖ o has two connected components, one of which is a disk and is called the interior of that oval, and we say that o bounds it from the outside.
We observe that R is a null-homologous torus, and Q is a torus with homology class (2, 2), both in H 2 (X; Z).Therefore e(X, R) = 0 and e(X, Q) = 8.Denoting as p : X → X/c 1 ∼ = S 4 the 2-fold branched cover, we set R = p(R) and Q = p(Q).Observe that R is a torus and Q is a Klein bottle.Finally, letting p : X → S 4 be the 2-fold branched cover of S 4 ramified along Q (which exists because . The branched covers in the case of CP 1 × CP 1 with its hyperboloid structure, with the same notation conventions as in Figure 1. Consecutive applications of Lemma 1.4 yield e(S 4 , R) = 0, e(S 4 , Q) = 4, e(X, R) = 2 and e(X, Q) = 0.
The situation is depicted in Figure 8.The topological Riemann-Hurwitz formula gives χ(X) = 4, and Theorem 3.5 provides σ (X) = −2.A similar reasoning as in Proposition 3.4 ensures that H 1 (X; Z/2) = 0, and thus that H 1 (X; Z) is torsion.In particular, This suggests that X may be diffeomorphic to CP 2 #CP 2 , but this will not be needed.Proof.Assume it corresponds to the subgroup 2Z ⊕ Z (the argument is the same with the other).Let γ be a curve with homology class (0, 1) in R. Its pre-image is therefore two parallel copies of it.The situation is depicted in Figure 9. Now, let C be a generic bidegree (0, 1) algebraic curve, so that . The unbranched 2-fold covering of the torus corresponding to the subgroup 2Z ⊕ Z, and its effect on the curve with homology class (0, 1).
covering that restricts to an unbranched covering of the boundary.An application of the Riemann-Hurwitz formula gives χ(A + (C)) = 1, and A + (C) has at most two boundary components.Therefore, there is no other choice but the same situation as depicted in Figure 3.That is, A + (C) is a disk, and p−1 This is excluded, by assumption.The same argument with a bidegree (1, 0) algebraic curve works for the subgroup Z ⊕ 2Z.■ The covering corresponding to the subgroup G is depicted in Figure 10.If a curve γ ⊂ R has homology class (α, β ) with both α and β odd (and coprime), then its pre-image is two parallel copies p−1 (γ) ⊂ Q.
Recalling that RF is some ovals and some number of parallel copies of an (α, β ) curve with α and β coprime and odd, we have the following immediate facts: (1) Each copy of the (α, β ) curve is doubled (indeed, the homotopy class of that curve belongs to the subgroup G).

−→ p
Figure 10.The unbranched covering p : On the left, the pre-image of the (3, 1) curve is two parallel copies of a (1, −2) curve.It is understood that the two tori are represented by the two squares, whose opposite sides are identified.
(3) The preimage respects mutual position of ovals, as in Proposition 2.3 (that is, an oval inside another lifts to two copies inside the other two copies).
Hence, we see that Q ∖ RF has two diffeomorphic subsets Q ± with the property ∂ Q ± = RF (we provide an example in Figure 11).We therefore have Therefore, we can define the Arnold surface of the curve as A(F) = A + (F)∪Q + .
Note that A(F) ⋔ R is a + b points (coming from the 2(a + b) points in F ⋔ Q).
We consider the immersed surface A(F) ∪ R, and we let X (F) be the smoothing of its singularities, as provided by Lemma 3.3, where we choose the smoothing that satisfies e(X, X (F)) = e(X, A(F) ∪ R) + 2(a + b).
Proposition 4.4.Let F ⊂ X be a Q-flexible curve of bidegree (a, b) with both a and b odd.The surface X (F) has zero homology in H 2 (X; Z/2) and satisfies Proof.Computing χ(X (F)) and e(X, X (F)) is straight-forward.To prove that [X (F)] = 0 ∈ H 2 (X; Z/2), it suffices to show that A(F) and R are homologous mod 2. Note that by the previous computations, b 2 (X) = −σ (X) = 2, so that X is a negative definite smooth 4-manifold.By virtue of Donaldson's theorem, this means that the intersection form of X is, up to a change of basis, that of CP 2 #CP 2 .
We consider a basis of H 2 (X, Z/2) ∼ = Z/2 ⊕ Z/2 that diagonalizes this intersection form, and we will show that A(F) and R both realize the homology class (1, 1) in H 2 (X; Z/2).
Because e(X, A(F)) and e(X, R) are both even, this rules out the two classes (1, 0) and (0, 1).As such, it suffices to show that A(F) and R are both not nullhomologous in H 2 (X; Z/2) (if A(F) was null-homologous, we could directly take the 2-fold covering of X ramified along A(F), without adding R).
Since a and b are both odd, this means that 2ab ̸ ≡ 0 mod 4. As such, we cannot have q([A(F)]) = 0 and q( This implies the following result. Theorem 4.5.Let F be a Q-flexible curve in the hyperboloid with bidegree (a, b) where both a and b are odd.Let ℓ ± and ℓ 0 denote the number of ovals of the curve that bound from the outside a subset with positive, negative or zero Euler characteristic, respectively.Then: Proof.We denote as Θ : Y → X the double branched cover of (X, X (F)).For any oval o ⊂ RF, R ∖ RF has exactly two path-connected components that have o as a part of their boundary.One is a punctured disc, and the other is a punctured torus.We denote as C(o) ⊂ R the image under p : X → S 4 of the punctured disc component, and as For an analogue of D(J), there is a subtlety.Indeed, in RF, there may be several parallel copies of an (α, β )-curve in H 1 (R; Z), where α and β are both odd and coprime.Each of these curves will lift in Q to two copies.
If RF contains ovals, then it is possible to choose one connected component D − of Q ∖ RF that has one of those curves as a boundary component, and at least one oval as another boundary component, and which is included in Q − .Define D = Θ −1 (D − ).By computations analogous to the CP 2 case, we have the following: ( (2) Q Y ( D, D) ⩽ 0. ( We can now apply Lemma 3.9 to the family composed of the collection of the C(o) and of D. ■ Curves on an ellipsoid.We now consider the other anti-holomorphic involution c 2 : ([x 0 : x 1 ], |y 0 : This time, we have and X/c 2 ∼ = CP 2 .Algebraic curves in (X, c 2 ) necessarily have a bidegree of the form (m, m) for some m ⩾ 1.Consider a purely imaginary bidegree (2, 2) algebraic curve Q, and define flexible curves and Q-flexible curves as before.Note that we still keep the same basis for H 2 (X; Z) as in the case of the hyperboloid.
Theorem 4.6.Let F be a bidegree (m, m) Q-flexible curve on the ellipsoid, with m odd.Let ℓ ± and ℓ 0 denote the number of connected components of R ∖ RF with positive, negative or zero Euler characteristic.Then We have e(X, R) = −2 and e(X, Q) = 8, because [R] = (±1, ∓1) ∈ H 2 (X; Z) (depending on a choice of orientation) and [Q] = (2, 2).Denoting the branched cover as p : X → CP 2 , we see that, letting In particular, Q is a null-homologous Klein bottle in H 2 (CP 2 ; Z/2), because it has even normal Euler number.This means that there is a well-defined 2-fold branched cover p : A direct computation provides χ(X) = 6 and σ (X) = −4, with H 1 (X) torsion.This is evidence to think that X ∼ = 4CP 2 .What will be useful is knowing that X is negative definite, and so has intersection form −I 4 by Donaldson's theorem.
. The two possible subsets Q ± , shaded.It is understood that the two spheres in the first row are Q 1 , and the two in the second are Q 2 .
This time, the restriction p : Q → R is a two-fold covering of the 2-sphere, and is necessarily trivial.We set Q = Q 1 ⊔ Q 2 .Let τ : X → X be the involution of X spanning Aut( p).Denote as R 1 and R 2 the two subsets of R ∖ p(RF) with ∂ R i = p(RF), and define We refer to Figure 12 for a representation.Of course, this definition depends on the choices of the labeling Q i of the two components of Q, as well as the choice of the labeling of the R i .But ultimately, the inequality we obtain will not depend on these choices.
This allows for a definition of A(F) such that e(Q + ) = 1 2 e(Q) and χ(Q Another key difference from the cases of CP 2 and of the hyperboloid is that the second homology H 2 (X; Z/2) now has rank four (the intersection form of X is −I 4 ).To show that A(F) and R are homologous mod 2 and not null-homologous, we need to eliminate more cases.We consider a basis of H 2 (X; Z) that diagonalizes the intersection form of X.It also descends to a basis of H 2 (X; Z/2).If (a, b, c, d) ∈ H 2 (X; Z/2) denotes the homology class of A(F) or R, with a, b, c, d ∈ {0, 1}, then the fact that e(X, A(F)) and e(X, R) are even implies that a + b + c + d ≡ 0 mod 2. There are 8 remaining cases: (0, 0, 0, 0), (1, 1, 1, 1), and the six cases of the type Figure 13.The core of a Möbius strip can be seen as a real line in the associated real projective plane.
That is, either X (F) is null-homologous, in which case Assuming that X (F) is characteristic, the Guillou-Marin congruence (Theorem 5.10) applies and gives β (X, X (F)) ≡ −(m + 1) 2 ≡ 0 or 4 mod 8, by inspection of the squares of odd integers mod 8.Because the surface X (F) has high genus, this method will a priori not yield any contradiction.
Proposition 4.7.The surface A(F) is homologous to R mod 2.
Proof.We start by describing the generators of the homology H 2 (X; Z/2).Consider a complex line CP 1 ⊂ CP 2 such that CP 1 ∩ Q = ∅.This means that CP 1 lifts to two spheres S 1 and S 2 in X, each with e(X, S i ) = −1.Moreover, because they are disjoint, we have Q X,Z/2 (S 1 , S 2 ) ≡ 0 mod 2, i.e. they are linearly independent.
There are two more generators that come from the following construction.
Q is a Klein bottle, which can be seen as the desingularization of two real projective planes R 1 and R 2 , with R 1 ⋔ R 2 = { * } and e(CP 2 , R i ) = 1.By Lemma 3.3, we see that this is possible from the computation e(CP 2 , Q) = 4. Let x ∈ R 1 ⋔ R 2 be the transverse intersection, and let D i ⊂ R i be a small disc centered at x. R i ∖ D i is a Möbius strip, whose core ℓ i can be seen as a real projective line in R i (see Figure 13).
This real line separates a complex line L i into two components is an even number of points, and thus Q Z/2 (Σ i , S j ) ≡ 0 mod 2. As such, (S 1 , S 2 , Σ 1 , Σ 2 ) is a basis for the homology H 2 (X; Z/2).From [Nagami 2000, Lemma 3.4], the surface p−1 (CP 1 ) ∪ R is mod 2 characteristic in X, and as such, we have In order to show that A(F) and R are homologous mod 2, it suffices to prove that Q X,Z/2 (A(F), Σ i ) ≡ 0 mod 2 for i = 1, 2. Equivalently, we need to show that A(F) ⋔ Σ i is an even number of points for i = 1, 2. Intersection points in A(F) ⋔ Σ i come in two types: (1) intersections between p−1 (F + ) and Σ i ; this number equals that of intersection points between F + and CP 1 , which is itself even because e(CP 2 , F + ) = 2m 2 ; (2) intersections between Q + and Σ i , itself also equal to #R ⋔ CP 1 , which is even because e(CP 2 , R) = −4.■ To prove Theorem 4.6, we apply the same method as before.Given a connected component U ⊂ R ∖ RF (or equivalently, U ⊂ R ∖ p(RF)), the lift p−1 (U) is two disjoint copies of U. We let C ± (U) denote those copies, with the condition that , with Θ : Y → X the double branched cover of X ramified over X (F).We have e(Y, C(U)) = −4χ(U).
If Q Y is the intersection form of Y , and if U and V are two distinct components of R ∖ RF, then there are two possibilities: (1) with {i, j} = {1, 2}.In particular, we still have Finally, one applies the same arguments as before to the family { C(U)} χ(U)⩽0 to obtain the claimed bound.

Further comments
Other ways to resolve the singularities.In order to take the 2-fold branched cover, we added RP 2 to A(F).This led us to resolve the m singularities that arose.As suggested by Zvonilov in a personal communication, one could be tempted to use blow-ups and see what effect this has.But in order to ensure that the new surface X (F) is still connected, we cannot blow-up all m singularities.Doing this procedure to m − 1 of those, and gluing a Hopf band for the last as we did previously, leads to the very same bound.That is, the 4-manifold Y which is the double branched cover of (mCP  m) .
That is, h(m) is the largest prime power that divides m.Viro and Zvonilov's inequality, which holds for flexible curves, is We denote as V Z(m) and S(m) the bounds obtained by Viro and Zvonilov and ours, respectively.That is, For infinitely many degrees m, one has S(m) < V Z(m).But in infinitely many others (e.g. when m is a prime power), the converse holds.However, both are far from sharp estimates that can be obtained from considerations for algebraic curves that come from Bézout theorem computations.That is, there are degrees m for which V Z(m) and S(m) are both not realized as upper bounds for ℓ 0 + ℓ − .For instance, Zvonilov [1979] has the sharper estimate, valid for pseudoholomorphic curves, If one starts with Viro and Zvonilov's inequality in the case where m + 2 is a prime power and the curve is Q-flexible of degree m, then one can perturb its union with the conic Q into a non-singular degree m + 2 flexible curve (which will have the same real set, for RQ = ∅), and obtain That is, one can derive our Theorem 3.11 from Viro and Zvonilov's when m + 2 is a prime power.On a side note, if the famous twin prime conjecture happens to be true, this means that there are infinitely many degrees m for which V Z(m) < S(m) and the bound S(m) is a corollary of their bound.By some easy number-theoretic considerations, one can show that there are infinitely many odd degrees m such that neither of m and m + 2 are prime powers, and for which S(m) < V Z(m).Indeed, the difference of the upper bounds in both inequalities is With m p = 1287×429 12p+1 , one has 5|m p +2 and 7|m p +2, and h(m p ) ∈ o(m 19/40 p ).
In particular, the difference diverges to +∞ on the degrees m p .
The same conclusion can be derived when comparing the inequalities of Theorems 4.5 and 4.6 with Zvonilov's work [2022].It turns out that, for a curve of bidegree (a, b) with a and b coprime, Zvonilov has no possibility to take a cyclic covering, and there is no inequality in those cases.

Non-orientable flexible curves.
There is a new object that could be interesting to study: nonorientable flexible curve.The motivation comes from the observation that in the operation of taking double branched covers, orientability of the ramification locus is disregarded.This is not the case for other cyclic branched covers (and the methods from [Viro and Zvonilov 1992] cannot apply to non-orientable surfaces).We propose the following non-orientable analogue of Definition 1.1.Definition 5.1.Let F ⊂ CP 2 be a closed, connected and non-orientable surface.We call F a non-orientable degree m and genus h flexible curve if the following conditions hold: What plays the role of asking that the integral homology class of F is m times a generator [CP 1 ] in H 2 (CP 2 ; Z) is the condition e(CP 2 , F) = m 2 .In the traditional orientable case, we also had the condition that χ(F) = −m 2 + 3m.This was a requirement of extremality in the genus bound proved by Kronheimer and Mrowka.Theorem 5.2 Thom conjecture, [Kronheimer and Mrowka 1994].Let F ⊂ CP 2 be a smoothly embedded oriented and connected surface with One could ask whether the implication holds for closed, connected nonorientable surfaces F smoothly embedded in CP 2 .In fact, self-intersection numbers of non-orientable surfaces need not be squares.Given any m ∈ Z, set Σ(m) to be the collection of all smoothly embedded, connected and non-orientable surfaces F ⊂ CP 2 with e(CP 2 , F) = m.We can define the following non-orientable genus function of CP 2 : χ(F).
For the other claims, the classical proofs for flexible curves, found, for instance in Viro's lecture notes, work word for word.■ We call a nonorientable flexible curve of degree m Q-flexible if, as before, the intersection F ⋔ Q is 2m points.Then, we have the following result.
Theorem 5.6.Let F ⊂ CP 2 be a non-orientable Q-flexible of odd degree m.Then Proof.The only difference with traditional flexible curves is that one needs to do all the computations in terms of χ(F).Indeed, starting at the level of S 4 , the surface F/conj needs not be orientable anymore.One checks that, for the Arnold surface, we have and for the smoothing X (F), we obtain χ(X (F)) = χ(F) − 3m + 2 and e(CP 2 , X (F)) = m 2 + 2m − 1.This gives, denoting as Y the double branched cover of (CP Note that χ(F) ⩽ 0 is necessarily even, as seen in Proposition 5.4.■ In regards to Theorem 5.3, we conjecture the following.
Conjecture 5.7.The lower bounds for g over non-negative odd integers are equalities.
If this holds, then one may add to the definition of a nonorientable flexible curve F of degree m that they must satisfy the extremal bound χ(F) = g(m).In this case, Theorem 5.6 becomes (2) construct a surface realizing that upper bound.
To this end, we will use the following.
This allows us to obtain the upper bounds g(−k) ⩽ 2 − k + ℓ 2 for k ∈ N ⋆ and ℓ ∈ {0, 1} having the same parity as k.Indeed, if F ⊂ CP 2 has e(CP 2 , F) = −k, then ℓ(F) = ℓ, because [F] ̸ = 0 ∈ H 2 (CP 2 ; Z/2) if and only if −k is odd, in which case a complex line is an integral lift of F with minimal self-intersection.
Another method (which worked for the orientable Thom conjecture in degree 4, for instance) is to make use of homological information of the double branched cover of CP 2 ramified along F.More precisely, we have the following.Proposition 5.9.Let F ⊂ CP 2 be a closed, connected surface such that [F] = 0 ∈ H 2 (CP 2 ; Z/2) (or equivalently, such that e(CP 2 , F) is even).Then  (x) .
In particular, from χ(F) ⩽ 1 always, we see that the only non-trivial bounds are for g(7) and g(9).
A simple calculation as before for the possible values of β (CP 2 , K) gives that, in the case of the projective plane, β (CP 2 , RP 2 ) = 1, a contradiction.
The previous calculations gave β (CP 2 , K) ∈ {0, 2, 6}, which is a contradiction.■ Now, the only things that remain to do are to construct surfaces that realize the upper bounds obtained thus far, and compute upper bounds for g in the special cases not covered yet.For constructions of surfaces, we will make use of local surfaces.Recall the so-called Whitney-Massey theorem.
All tuples (e, χ) satisfying this condition are realizable by a closed, connected, nonorientable surface.
A surface F embedded in the 4-ball B 4 that realizes an admissible tuple (e, χ) will be called a local surface.Note that the previous theorem ensures that those always exist for any admissible tuple.

Figure 3 .
Figure 3.The restrictions p : CX → G + and p : G → G + , the second one being a branched covering.On the right, the rotation by 180 • generates the group of deck transformations.

Figure 6 .Figure 7 .
Figure 6.Using the same scheme ⟨J ⊔ 2 ⊔ 1⟨1⟩⟩ as in the example of Figure 4, we take o to be the only non-empty oval.In the shaded regions, we depict C ± (o), where part of the boundary ∂C ± (o) is o.

Proof.
For the first claim, observe e(CP 2 ,C − (o)) = −2χ(o) and e(CP 2 , D − (J)) = −2χ(J), by using Lemma 3.2.Next, from Lemma 1.4, we can see that e(Y, C(o)) = 2e(CP 2 ,C − (o)) and e(Y, D(J)) = 2e(CP 2 , D − (J)).To derive Q Y ( C(o), C(o)) and Q Y ( D(J), D(J)), we remark that C(o) and D(J) are orientable surfaces, so the self-intersection and the evaluation of the intersection form agree.For the second claim, distinct ovals o and o ′ cannot satisfy C − (o) ∩C − (o ′ ) ̸ = ∅, even if one is included inside the other (but, it is possible that C − (o) ∩C + (o ′ ) ̸ = ∅).The same goes for C − (o) ∩ D − (J) = ∅.As such, the surfaces C(o), C(o ′ ) and D(J) are non-intersecting in Y .■ The homology classes of the surfaces C(o i ), i ∈ [[1, ℓ]], and D(J) were respectively denoted as β i and β 0 by Viro and Zvonilov (where ℓ denotes the number of ovals in RF).They showed the following result.Lemma 3.9 [Viro and Zvonilov 1992, Lemma 1.3].Let h = p r be a prime power.Let ν : Y → X be an h-sheeted cyclic covering between two n-manifolds, branched over a codimension-two subset A ⊂ X.Let B ⊂ X be a membrane, let b be the class in H k (X, A) determined by B, and let β be the class in H k (Y ) determined by ν −1 (B), oriented coherently with B. Let τ : Y → Y be a generator of Aut(ν), and let ρ = 1 − τ ∈ (Z/p)[Aut(ν)].Recall the Smith long exact sequence in homology (with coefficients in Z/p): Consider a Q-flexible curve F ⊂ X of bidegree(a, b), where a and b are both odd.In particular:χ(F) = −2ab + 2a + 2b and e(X, F) = 2ab.Letting A + (F) = p(F) and A + (F) = p−1 (A + (F)), one checks that χ(A + (F)) = −2ab + a + b and e(A + (F)) = 2ab.In order to understand RF = p−1 (∂ A + (F)) ⊂ Q, it is necessary to describe the unbranched 2-fold covering p : Q → R, which is a non-trivial 2-fold cover of the torus (non-triviality can be deduced by the same argument as in the proof of the next proposition).There are only three such coverings, each given by the subgroups2Z ⊕ Z, Z ⊕ 2Z and G = {(x, y) ∈ Z 2 | x + y ≡ 0 mod 2}.Proposition 4.3.The covering p : Q → R corresponds to the subgroup G.
2 , X (F)) still has b + 2 (Y ) = (m − 1) 2 /4.Comparisons of our inequality.Given a prime number p and an integer m ∈ N ⋆ , we denote as ν p (m) = max{n ∈ N | p n divides m} the p-adic valuation of m.Define the function h : N ⋆ → N by h(m) = max p prime p ν p ( Q-flexible curves.Proof of Theorem 5.3.The two steps of the proof are to (1) obtain upper bounds for χ(F) given e(CP 2 , F) = m, and
• generates the group of deck transformations.If o 1 ⊂ o 2 (where inclusion means that o 1 is contained in the orientable component of RP 2 ∖ o 2 ), then o 1 ⊂ o 2 , in the following sense: Q ∖ o 2 has three components, one being a cylinder containing J, and the other two being discs each containing a component of o 1 .