Real algebraic curves with large finite number of real points
Erwan Brugallé, Alex Degtyarev, Ilia Itenberg, Frédéric Mangolte
RREAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBEROF REAL POINTS
ERWAN BRUGALL´E, ALEX DEGTYAREV, ILIA ITENBERG, AND FR´ED´ERIC MANGOLTE
Abstract.
We address the problem of the maximal finite number of real points of a real algebraiccurve (of a given degree and, sometimes, genus) in the projective plane. We improve the knownupper and lower bounds and construct close to optimal curves of small degree. Our upper bound issharp if the genus is small as compared to the degree. Some of the results are extended to other realalgebraic surfaces, most notably ruled. Introduction A real algebraic variety ( X, c ) is a complex algebraic variety equipped with an anti-holomorphicinvolution c : X → X , called a real structure . We denote by R X the real part of X , i.e. , the fixedpoint set of c . With a certain abuse of language, a real algebraic variety is called finite if so is itsreal part. Note that each real point of a finite real algebraic variety of positive dimension is in thesingular locus of the variety.1.1. Statement of the problem.
In this paper we mainly deal with the first non-trivial case,namely, finite real algebraic curves in C P . (Some of the results are extended to more generalsurfaces.) The degree of such a curve C ⊂ C P is necessarily even, deg C = 2 k . Our primaryconcern is the number | R C | of real points of C . Problem 1.1.
For a given integer k ≥
1, what is the maximal number δ ( k ) = max {| R C | : C ⊂ C P a finite real algebraic curve, deg C = 2 k } ?For given integers k ≥ g ≥
0, what is the maximal number δ g ( k ) = max {| R C | : C ⊂ C P a finite real algebraic curve of genus g , deg C = 2 k } ?(See Section 2 for our convention for the genus of reducible curves.)The Petrovsky inequalities (see [Pet38] and Remark 2.3) result in the following upper bound: | R C | ≤ k ( k −
1) + 1 . Currently, this bound is the best known. Furthermore, being of topological nature, it is sharp in therealm of pseudo-holomorphic curves. Indeed, consider a rational simple Harnack curve of degree 2 k in C P (see [Mik00, KO06, Bru15]); this curve has ( k − k −
1) solitary real nodes (as usual, bya node we mean a non-degenerate double point, i.e. , an A -singularity) and an oval (see Remark2.3 for the definition) surrounding ( k − k −
2) of them. One can erase all inner nodes, leavingthe oval empty. Then, in the pseudo-holomorphic category , the oval can be contracted to an extrasolitary node, giving rise to a finite real pseudo-holomorphic curve C ⊂ C P of degree 2 k with | R C | = k ( k −
1) + 1. a r X i v : . [ m a t h . AG ] J u l ERWAN BRUGALL´E, ALEX DEGTYAREV, ILIA ITENBERG, AND FR´ED´ERIC MANGOLTE
Principal results.
For the moment, the exact value of δ ( k ) is known only for k ≤
4. Theupper (Petrovsky inequality) and lower bounds for a few small values of k are as follows: k δ ( k ) ≤ δ ( k ) ≥ k = 1 , k = 6 is given by Proposition 4.7, and all other cases are covered by Theorem 4.5.Asymptotically, we have 43 k (cid:46) δ ( k ) (cid:46) k , where the lower bound follows from Theorem 4.5.A finite real sextic C with | R C | = δ (3) = 10 was constructed by D. Hilbert [Hil88]. A finitereal octic C with | R C | = δ (4) = 19 could easily be obtained by perturbing a quartuple conic,although we could not find such an octic in the literature. The best previously known asymptoticlower bound δ ( k ) (cid:38) k is found in M. D. Choi, T. Y. Lam, B. Reznick [CLR80].With the genus g = g ( C ) fixed, the upper bound δ g ( k ) ≤ k + g + 1is also given by a strengthening of the Petrovsky inequalities (see Theorem 2.5). In Theorem 4.8,we show that this bound is sharp for g ≤ k − g fixed) and Corollary 2.6; an asymptotic lower bound is given by Theorem 4.2 (which also coversarbitrary projective toric surfaces), and a few sporadic constructions are discussed in Sections 5, 6.1.3. Contents of the paper.
In Section 2, we obtain the upper bounds, derived essentially fromthe Comessatti inequalities. In Section 3, we discuss the auxiliary tools used in the constructions,namely, the patchworking techniques, bigonal curves and dessins d’enfants , and deformation to thenormal cone. Section 4 is dedicated to curves in C P : we recast the upper bounds, describe ageneral construction for toric surfaces (Theorem 4.2) and a slight improvement for the projectiveplane (Theorem 4.5), and prove the sharpness of the bound δ g ( k ) ≤ k + g + 1 for curves of smallgenus. In Section 5, we consider surfaces ruled over R , proving the sharpness of the upper boundsfor small bi-degrees and for small genera. Finally, Section 6 deals with finite real curves in theellipsoid.1.4. Acknowledgments.
Part of the work on this project was accomplished during the second andthird authors’ stay at the
Max-Planck-Institut f¨ur Mathematik , Bonn. We are grateful to the MPIMand its friendly staff for their hospitality and excellent working conditions. We extend our gratitudeto Boris Shapiro, who brought the finite real curve problem to our attention and supported ourwork by numerous fruitful discussions. We would also like to thank Ilya Tyomkin for his help inspecializing general statements from [ST06] to a few specific situations.2.
Strengthened Comessatti inequalities
Let (
X, c ) be a smooth real projective surface. We denote by σ ± inv ( X, c ) (respectively, σ ± skew ( X, c ))the inertia indices of the invariant (respectively, skew-invariant) sublattice of the involution c ∗ : H ( X ; Z ) → H ( X ; Z ) induced by c . The following statement is standard. Proposition 2.1 (see, for example, [Wil78]) . One has σ − inv ( X, c ) = 12 ( h , ( X ) + χ ( R X )) − , σ − skew ( X, c ) = 12 ( h , ( X ) − χ ( R X )) , EAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBER OF REAL POINTS 3 where h • , • are the Hodge numbers and χ is the topological Euler characteristic. Corollary 2.2 (Comessatti inequalities) . One has − h , ( X ) ≤ χ ( R X ) ≤ h , ( X ) . Remark 2.3.
Let C ⊂ C P be a smooth real curve of degree 2 k . Recall that an oval of C is aconnected component o ⊂ R C bounding a disk in R P ; the latter disk is called the interior of o .An oval o of C is called even (respectively, odd ) if o is contained inside an even (respectively, odd)number of other ovals of C ; the number of even (respectively, odd) ovals of a given curve C isdenoted by p (respectively, n ). The classical Petrovsky inequalities [Pet38] state that p − n ≤ k ( k −
1) + 1 , n − p ≤ k ( k − . These inequalities can be obtained by applying Corollary 2.2 to the double covering of C P branchedalong C ⊂ C P (see e.g. [Wil78], [Man17, Th. 3.3.14]).The Comessatti and Petrovsky inequalities, strengthened in several ways (see, e.g. , [Vir86]), havea variety of applications. For example, for nodal finite real rational curves in C P we immediatelyobtain the following statement. Proposition 2.4.
Let C ⊂ C P be a nodal finite rational curve of degree k . Then, | R C | ≤ k + 1 .Proof. Denote by r the number of real nodes of C , and denote by s the number of pairs of complexconjugate nodes of C . We have r + 2 s = ( k − k − Y be the double coveringof C P branched along the smooth real curve C t ⊂ C P obtained from C by a small perturbationcreating an oval from each real node of C . The union of r small discs bounded by R C t is denotedby R P ; let ¯ c : Y → Y be the lift of the real structure such that the real part projects onto R P .Each pair of complex conjugate nodes of C gives rise to a pair of ¯ c ∗ -conjugate vanishing cycles in H ( Y ; Z ); their difference is a skew-invariant class of square −
4, and the s square − h , ( Y ) = 3 k − k + 2 (see, e.g. [Wil78]), Corollary 2.2 implies that χ ( R Y ) ≤ h , ( Y ) − s = 3 k ( k −
1) + 2 − s = k + 1 + r. Thus, r ≤ k + 1. (cid:3) The above statement can be generalized to the case of not necessarily nodal curves of arbitrarygenus in any smooth real projective surface.Recall that the geometric genus g ( C ) of an irreducible and reduced algebraic curve C is the genusof its normalization. If C is reduced with irreducible components C , . . . , C n , the geometric genusof C is defined by g ( C ) = g ( C ) + . . . + g ( C n ) + 1 − n. In other words, 2 − g ( C ) = χ ( ˜ C ), where ˜ C is the normalization.Define also the weight (cid:107) p (cid:107) of a solitary point p of a real curve C as the minimal number of blow-ups at real points necessary to resolve p . More precisely, (cid:107) p (cid:107) = 1 + (cid:80) (cid:107) p i (cid:107) , the summation runningover all real points p i over p of the strict transform of C blown up at p . For example, the weightof a simple node equals 1, whereas the weight of an A n − -type point equals n . If | R C | < ∞ , wedefine the weighted point count (cid:107) R C (cid:107) as the sum of the weights of all real points of C .The topology of the ambient complex surface X is present in the next statement in the form ofthe coefficient T , ( X ) = 16 (cid:0) c ( X ) − c ( X ) (cid:1) = 12 (cid:0) σ ( X ) − χ ( X ) (cid:1) = b ( X ) − h , ( X )of the Todd genus (see [Hir86]). ERWAN BRUGALL´E, ALEX DEGTYAREV, ILIA ITENBERG, AND FR´ED´ERIC MANGOLTE
Theorem 2.5.
Let ( X, c ) be a simply connected smooth real projective surface with non-emptyconnected real part. Let C ⊂ X be an ample reduced finite real algebraic curve such that [ C ] = 2 e in H ( X ; Z ) . Then, we have (1) | R C | ≤ (cid:107) R C (cid:107) ≤ e + g ( C ) − T , ( X ) + χ ( R X ) − . Furthermore, the inequality is strict unless all singular points of C are double.Proof. Since [ C ] ∈ H ( X ; Z ) is divisible by 2, there exists a real double covering ρ : ( Y, ¯ c ) → ( X, c )ramified at C and such that ρ ( R Y ) = R X . By the embedded resolution of singularities, we can finda sequence of real blow-ups π i : X i → X i − , i = 1 , . . . , n , real curves C i = π ∗ C i − mod 2 ⊂ X i , andreal double coverings ρ i = π ∗ i ρ i − : Y i → X i ramified at C i such that the curve C n and surface Y n are nonsingular. (Here, a real blow-up is either a blow-up at a real point or a pair of blow-ups attwo conjugate points. By π ∗ i C i − mod 2 we mean the reduced divisor obtained by retaining the oddmultiplicity components of the divisorial pull-back π ∗ i C i − .)Using Proposition 2.1, we can rewrite (1) in the form e + g ( C ) + h , ( X ) − T , ( X ) + 2 χ ( R X ) − σ − inv ( X, c ) − (cid:107) R C (cid:107) ≥ . We proceed by induction and prove a modified version of the latter inequality, namely,(2) e i + g ( C i ) + h , ( X i ) − T , ( X i ) + b − ( Y i ) + 2 χ ( R X i ) − σ − inv ( X i , c ) − (cid:107) R C i (cid:107) ≥ , where [ C i ] = 2 e i ∈ H ( X i ; Z ) and b − ( · ) is the dimension of the ( − ρ ∗ on H ( · ; C ).For the “complex” ingredients of (2), it suffices to consider a blow-up π : ˜ X → X at a singularpoint p of C , not necessarily real, of multiplicity O ≥
2. Denoting by C (cid:48) the strict transform of C ,we have ˜ C = π ∗ C mod 2 = C (cid:48) + εE , where E = π − ( p ) is the exceptional divisor and O = 2 m + ε , m ∈ Z , ε = 0 ,
1. Then, in obvious notation, e = ˜ e + m , g ( C ) = g ( ˜ C ) + ε, h , ( X ) = h , ( ˜ X ) − , T , ( X ) = T , ( ˜ X ) + 1 . Furthermore, from the isomorphisms H ( ˜ Y , ˜ ρ ∗ E ) = H ( Y, p ) = H ( Y ) we easily conclude that b − ( Y ) ≥ b − ( ˜ Y ) − b − ( ˜ ρ ∗ E ) ≥ b − ( ˜ Y ) − m − . It follows that, when passing from ˜ X to X , the increment in the first five terms of (2) is at least( m − + ε − ≥ −
1; this increment equals ( −
1) if and only if p is a double point of C .For the last three terms, assume first that the singular point p above is real. Then χ ( R X ) = χ ( R ˜ X ) + 1 , σ − inv ( X i , c ) = σ − inv ( ˜ X i , ˜ c ) , (cid:107) R C (cid:107) = (cid:107) R ˜ C (cid:107) + 1 , and the total increment in (2) is positive; it equals 0 if and only if p is a double point.Now, let π : ˜ X → X be a pair of blow-ups at two complex conjugate singular points of C . Then χ ( R X ) = χ ( R ˜ X ) , σ − inv ( X i , c ) = σ − inv ( ˜ X i , ˜ c ) − , (cid:107) R C (cid:107) = (cid:107) R ˜ C (cid:107) , and, again, the total increment is positive, equal to 0 if and only if both points are double.To establish (2) for the last, nonsingular, curve C n , we use the following observations: • χ ( Y n ) = 2 χ ( X n ) − χ ( C n ) (the Riemann-Hurwitz formula); • σ ( Y n ) = 2 σ ( X n ) − e n (Hirzebruch’s theorem); • b ( Y n ) − b ( X n ) = b − ( Y n ), as b +11 ( Y n ) = b ( X n ) via the transfer map; • χ ( R Y n ) = 2 χ ( R X n ), since R C n = ∅ and R Y n → R X n is an unramified double covering.Then, (2) takes the form σ − inv ( Y n , ¯ c n ) ≥ σ − inv ( X n , c n ) , which is obvious in view of the transfer map H ( X n ; R ) → H ( Y n ; R ): this map is equivariant andisometric up to a factor of 2. EAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBER OF REAL POINTS 5
Thus, there remains to notice that b − ( Y ) = 0. Indeed, since C = C is assumed ample, X (cid:114) C has homotopy type of a CW-complex of dimension 2 (as a Stein manifold). Hence, so does Y (cid:114) C ,and the homomorphism H ( C ; R ) → H ( Y ; R ) is surjective. Clearly, b − ( C ) = 0. (cid:3) Corollary 2.6.
Let ( X, c ) and C ⊂ X be as in Theorem 2.5. Then, we have | R C | ≤ e − e · c ( X ) − T , ( X ) + χ ( R X ) , the inequality being strict unless each singular point of C is a solitary real node of R C .Proof. By the adjunction formula we have g ( C ) ≤ e − e · c ( X ) + 1 − | R C | , and the result follows from Theorem 2.5. (cid:3) Remark 2.7.
The assumptions π ( X ) = 0 and b ( R X ) = 1 in Theorem 2.5 are mainly used toassure the existence of a real double covering ρ : Y → X ramified over a given real divisor C . Ingeneral, one should speak about the divisibility by 2 of the real divisor class | C | R , i.e. , class of realdivisors modulo real linear equivalence. (If R X (cid:54) = ∅ , one can alternatively speak about the setof real divisors in the linear system | C | or a real point of Pic( X ).) A necessary condition is thevanishing [ C ] = 0 ∈ H n − ( X ; Z / Z ) , [ R C ] = 0 ∈ H n − ( R X ; Z / Z ) , where n = dim C ( X ) and [ R C ] is the homology class of the real part of (any representative of) | C | (the sufficiency of this condition in some special cases is discussed in Lemma 3.4 below). If notempty, the set of double coverings ramified over C and admitting real structure is a torsor over thespace of c ∗ -invariant elements of H ( X ; Z / Z ).The proof of the following theorem repeats literally that of Theorem 2.5. Theorem 2.8.
Let ( X, c ) be a smooth real projective surface and C ⊂ X an ample finite reducedreal algebraic curve such that the class | C | R is divisible by . A choice of a real double covering ρ : Y → X ramified over C defines a decomposition of R X into two disjoint subsets R X + = ρ ( R Y ) and R X − consisting of whole components. Then, we have (cid:107) R C ∩ R X + (cid:107) − (cid:107) R C ∩ R X − (cid:107) ≤ e + g ( C ) − T , ( X ) + χ ( R X + ) − χ ( R X − ) − , the inequality being strict unless all singular points of C are double. (cid:3) Construction tools
Patchworking.
If ∆ is a convex lattice polygon contained in the non-negative quadrant( R ≥ ) ⊂ R , we denote by Tor(∆) the toric variety associated with ∆; this variety is a surfaceif ∆ is non-degenerate. In the latter case, the complex torus ( C ∗ ) is naturally embedded in Tor(∆).Let V ⊂ ( R ≥ ) ∩ Z be a finite set, and let P ( x, y ) = (cid:80) ( i,j ) ∈ V a ij x i y j be a real polynomial in twovariables. The Newton polygon ∆ P of P is the convex hull in R of those points in V that correspondto the non-zero monomials of P . The polynomial P defines an algebraic curve in the 2-dimensionalcomplex torus ( C ∗ ) ; the closure of this curve in Tor(∆ P ) is an algebraic curve C ⊂ Tor(∆ P ). If Q is a quadrant of ( R ∗ ) ⊂ ( C ∗ ) and ( a, b ) is a vector in Z , we denote by Q ( a, b ) the quadrant { ( x, y ) ∈ ( R ∗ ) | (( − a x, ( − b y ) ∈ Q } . If e is an integral segment whose direction is generated by a primitive integral vector ( a, b ), weabbreviate Q ( e ⊥ ) := Q ( b, − a ). A real algebraic curve C ⊂ Tor(∆) is said to be -finite (respectively, -finite ) if the intersection of the real part R C with the positive quadrant ( R > ) (respectively, theunion ( R > ) ∪ ( R > ) (1 ,
0) is finite.
ERWAN BRUGALL´E, ALEX DEGTYAREV, ILIA ITENBERG, AND FR´ED´ERIC MANGOLTE
Fix a subdivision S = { ∆ , . . . , ∆ N } of a convex polygon ∆ ⊂ ( R ≥ ) such that there existsa piecewise-linear convex function ν : ∆ → R whose maximal linearity domains are precisely thenon-degenerate lattice polygons ∆ , . . . , ∆ N . Let a ij , ( i, j ) ∈ ∆ ∩ Z , be a collection of real numberssuch that a ij (cid:54) = 0 whenever ( i, j ) is a vertex of S . This gives rise to N real algebraic curves C k , k = 1 , . . . , N : each curve C k ⊂ Tor(∆ k ) is defined by the polynomial P ( x, y ) = (cid:88) ( i,j ) ∈ ∆ k ∩ Z a ij x i y j with the Newton polygon ∆ k .Commonly, we denote by Sing( C ) the set of singular points of a curve C . If C ⊂ Tor(∆) and e ⊂ ∆ is an edge, we put T e ( C ) := C ∩ D ( e ), where D ( e ) is the toric divisor corresponding to e .Assume that each curve C k is nodal and Sing( C k ) is disjoint from the toric divisors of Tor(∆ k )(but C k can be tangent with arbitrary order of tangency to some toric divisors). For each inneredge e = ∆ i ∩ ∆ j of S , the toric divisors corresponding to e in Tor(∆ i ) and Tor(∆ j ) are naturallyidentified, as they both are Tor( e ). The intersection points of C i and C j with these toric divisorsare also identified, and, at each such point p ∈ Tor( e ), the orders of intersection of C i and C j withTor( e ) automatically coincide; this common order is denoted by mult p and, if mult p >
1, the point p is called fat . Assume that mult p is even for each fat point p and that the local branches of C i and C j at each real fat point p are in the same quadrant Q p ⊂ ( R ∗ ) .Each edge E of ∆ is a union of exterior edges e of S ; denote the set of these edges by { E } and,given e ∈ { E } , let k ( e ) be the index such that e ⊂ ∆ k ( e ) . The toric divisor D ( E ) ⊂ Tor(∆) isa smooth real rational curve whose real part R D ( E ) is divided into two halves R D ± ( E ) by theintersections with other toric divisors of Tor(∆); we denote by R D + ( E ) the half adjacent to thepositive quadrant of ( R ∗ ) . Similarly, the toric divisor D ( e ) ⊂ Tor(∆ k ( e ) ) is divided into R D ± ( e ). Theorem 3.1 (Patchworking construction; essentially, Theorem 2.4 in [Shu06]) . Under the assump-tions above, there exists a family of real polynomials P ( t ) ( x, y ) , t ∈ R > , with the Newton polygon ∆ , such that, for sufficiently small t , the curve C ( t ) ⊂ Tor(∆) defined by P ( t ) has the followingproperties: • the curve C ( t ) is nodal and Sing( C ( t ) ) is disjoint from the toric divisors; • if all curves C , . . . , C N are -finite ( respectively, -finite ) , then so is C ( t ) ; • there is an injective map Φ : N (cid:97) k =1 Sing( C k ) → Sing( C ( t ) ) , such that the image of each real point is a real point of the same type ( solitary/non-solitary ) and in the same quadrant of ( R ∗ ) , and the image of each imaginary point is imaginary; • there is a partition Sing( C ( t ) ) (cid:114) image of Φ = (cid:97) p Π p ,p running over all fat points, so that | Π p | = 2 m − if mult p = 2 m . The points in Π p areimaginary if p is imaginary and real and solitary if p is real; in the latter case, ( m − ofthese points lie in Q p and the others m points lie in Q p ( e ⊥ p ) , where p ∈ Tor( e p ) ; • for each edge E of ∆ , there is a bijective map Ψ E : (cid:97) e ∈{ E } T e ( C k ( e ) ) → T E ( C ( t ) ) preserving the intersection multiplicity and the position of points in R D ± ( · ) or D ( · ) (cid:114) R D ( · ) . EAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBER OF REAL POINTS 7
Proof.
To deduce the statement from [Shu06, Theorem 2.4], one can use Lemma 5.4(ii) in [Shu05]and the deformation patterns described in [IKS15], Lemmas 3.10 and 3.11 ( cf . also the curves C ∗ , , in Lemma 3.2 below). (cid:3) Bigonal curves via dessins d’enfants . We denote by Σ n , n ≥
0, the Hirzebruch surfaceof degree n , i.e. , Σ n = P ( O C P ( n ) ⊕ O C P ). Recall that Σ = C P × C P and Σ is the blow-upof C P at a point. The bundle projection induces a map π : Σ n → C P , and we denote by F afiber of π ; it is isomorphic to C P . The images of O C P and O C P ( n ) are denoted by B and B ∞ ,respectively; these curves are sections of π . The group H (Σ n ; C ) = H , ( X ; C ) is generated by theclasses of B and F , and we have[ B ] = n, [ B ∞ ] = − n, [ F ] = 0 , B ∞ ∼ B − nF, c (Σ n ) = 2[ B ] + (2 − n )[ F ] . (If n >
0, the exceptional section B ∞ is the only irreducible curve of negative self-intersection.) Inother words, we have D ∼ aB + bF for each divisor D ⊂ Σ n , and the pair ( a, b ) ∈ Z is calledthe bidegree of D . The cone of effective divisors is generated by B ∞ and F , and the cone of ampledivisors is { aB + bF | a, b > } .In this section, we equipp C P with the standard complex conjugation, and the surface Σ n withthe real structure c induced by the standard complex conjugation on O C P ( n ). Unless n = 0, this isthe only real structure on Σ n with nonempty real part. In particular c acts on H (Σ n ; C ) as − Id,and so σ − inv ( X, c ) = 0. The real part of Σ n is a torus if n is even, and a Klein bottle if n is odd.In the former case, the complement R Σ n (cid:114) ( R B ∪ R B ∞ ) has two connected components, which wedenote by R Σ n, ± . Lemma 3.2.
Given integers n > , b ≥ , and ≤ q ≤ n + b − , there exists a real algebraicrational curve C = C n,b,q in Σ n of bidegree (2 , b ) such that (see Figure 1):(1) all singular points of C are n + 2 b − solitary nodes; n + b + q of them lie in R Σ n, + , andthe other n + b − q − lie in R Σ n, − ;(2) the real part R C has a single extra oval o , which is contained in R Σ n, − ∪ R B ∪ R B ∞ anddoes not contain any of the nodes in its interior;(3) each intersection p ∞ := o ∩ B ∞ and p := o ∩ B consists of a single point, the multiplicitybeing b and n + 2 b − q , respectively; the points p and p ∞ are on the same fiber F . n + b + q (cid:122) (cid:125)(cid:124) (cid:123) R B R B ∞ (cid:124) (cid:123)(cid:122) (cid:125) n + b − q − p ∞ p R C n,b,q Figure 1.
This curve can be perturbed to a curve (cid:101) C n,b,q ⊂ Σ n satisfying conditions (1) and (2) and thefollowing modified version of condition (3):(˜3) the oval o intersects B ∞ and B at, respectively, b and n + b − q simple tangency points. Note that C n,b,q intersects B in q additional pairs of complex conjugate points. ERWAN BRUGALL´E, ALEX DEGTYAREV, ILIA ITENBERG, AND FR´ED´ERIC MANGOLTE
Proof.
Up to elementary transformations of Σ n (blowing up the point of intersection C ∩ B ∞ andblowing down the strict transforms of the corresponding fibers) we may assume that b = 0 and,hence, C is disjoint from B ∞ . Then, C is given by P ( x, y ) = 0, where(3) P ( x, y ) = y + a ( x ) y + a ( x ) , deg a i ( x ) = 2 in. (Strictly speaking, a i are sections of appropriate line bundles, but we pass to affine coordinates andregard a i as polynomials.) We will construct the curves using the techniques of dessins d’enfants ,cf. [Ore03, DIK08, Deg12]. Consider the rational function f : C P → C P given by f ( x ) = a ( x ) − a ( x ) a ( x ) . (This function differs from the j -invariant of the trigonal curve C + B by a few irrelevant factors.)The dessin of C is the graph D := f − ( R P ) decorated as shown in Figure 2. In addition to × -, ◦ -,0 1 ∞ Figure 2.
Decoration of a dessinand • -vertices, it may also have monochrome vertices, which are the pull-backs of the real criticalvalues of f other that 0, 1, or ∞ . This graph is real, and we depict only its projection to the disk D := C P /x ∼ ¯ x , showing the boundary ∂D by a wide grey curve: this boundary corresponds tothe real parts R C ⊂ R Σ n → R P . Assuming that a , a have no common roots, the real specialvertices and edges of D have the following geometric interpretation: • a × -vertex x corresponds to a double root of the polynomial P ( x , y ); the curve is tangentto a fiber if val x = 2 and has a double point of type A p − , p = val x , otherwise; • a ◦ -vertex x corresponds to an intersection R C ∩ R B of multiplicity val x ; • the real part R C is empty over each point of a solid edge and consists of two points overeach point of any other edge; • the points of R C over two × -vertices x , x are in the same half R Σ n, ± if and only if onehas (cid:80) val z i = 0 mod 8, the summation running over all • -vertices z i in any of the two arcsof ∂D bounded by x , x .(For the last item, observe that the valency of each • -vertex is 0 mod 4 and the sum of all valenciesequals 2 deg f = 8 n ; hence, the sum in the statement is independent of the choice of the arc.)Now, to construct the curves in the statements, we start with the dessin (cid:101) D n, , shown in Figure 3,left: it has 2 n • -vertices, 2 n ◦ -vertices, and (2 n +1) × -vertices, two bivalent and (2 n −
1) four-valent,numbered consecutively along ∂D . To obtain (cid:101) D n, ,q , we replace q disjoint embraced fragments withcopies of the fragment shown in Figure 3, right; by choosing the fragments replaced around even-numbered × -vertices, we ensure that the solitary nodes would migrate from R Σ n, − to R Σ n, + .Finally, D n, ,q is obtained from (cid:101) D n, ,q by contracting the dotted real segments connecting the real ◦ -vertices, so that the said vertices collide to a single (8 n − q )-valent one. Each of these dessins D gives rise to a (not unique) equivariant topological branched covering f : S → C P (cf. [Ore03,DIK08, Deg12]), and the Riemann existence theorem gives us an analytic structure on the sphere S making f a real rational function C P → C P . There remains to take for a a real polynomial witha simple zero at each (double) pole of f and let a := a (1 − f ). (cid:3) Generalizing, one can consider a geometrically ruled surface π : Σ n ( O ) := P ( O ⊕ O B ) → B , where B is a smooth compact real curve of genus g ≥ O is a line bundle, deg O = n ≥
0. If O isalso real, the surface Σ n ( O ) acquires a real structure; the sections B and B ∞ are also real and wecan speak about R B , R B ∞ . The real line bundle O is said to be even if the GL (1 , R )-bundle R O EAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBER OF REAL POINTS 91 (cid:27)(cid:26) (cid:27)(cid:26) ... ... ... (cid:26) (cid:27) ⇐ = Figure 3.
The dessin (cid:101) D n, , and its modificationsover R B is trivial (cf. Remark 2.7). In this case, the real part R Σ n ( O ) is a disjoint union of tori,one torus T i over each real component R i B of B , and each complement T ◦ i := T i (cid:114) ( R B ∪ R B ∞ ) ismade of two connected components (open annuli).A smooth compact real curve B of genus g is called maximal if it has the maximal possible numberof real connected components: b ( R B ) = g + 1. Lemma 3.3.
Let n , g be two integers, n ≥ g − ≥ . Then there exists an even real line bundle O of degree deg O = 2 n over a maximal real algebraic curve B of genus g , and a nodal real algebraiccurve C n ( g ) ⊂ Σ n ( O ) realizing the class B ] ∈ H (Σ n ( O ); Z ) such that(1) R C n ( g ) ∩ T consists of n solitary nodes, all in the same connected component of T ◦ ;(2) R C n ( g ) ∩ T is a smooth connected curve, contained in a single connected component of T ◦ except for n real points of simple tangency of C n and B ;(3) R C n ( g ) ∩ T i , i ≥ , is a smooth connected curve, contained in a single connected componentof T ◦ except for one real point of simple tangency of C n and B . Note that we can only assert the existence of a ruled surface Σ n ( O ): the analytic structure on B and line bundle O are given by the construction and cannot be fixed in advance. Proof.
We proceed as in the proof of Lemma 3.2, with the “polynomials” a i sections of O ⊗ i in (3)and half-dessin D n ( g ) /c B in the surface D := B/c B , which, in the case of maximal B , is a diskwith g holes; as above, we have ∂D = R B . The following technical requirements are necessary andsufficient for the existence of a topological ramified covering f : B → C P (see [DIK08, Deg12]) with B the orientable double of D : • each region (connected component of D (cid:114) D ) should admit an orientation inducing on theboundary the orientation inherited from R P (the order on R ), and • each triangular region ( i.e. , one with a single vertex of each of the three special types × , ◦ ,and • in the boundary) should be a topological disk.(For example, in the dessins (cid:101) D n, ,q in Figure 3 the orientations are given by a chessboard coloringand all regions are triangles.)The curve C n ( g ) as in the statement is obtained from the dessin D n ( g ) constructed as follows. If g = 1, then D n (1) is the dessin in the annulus shown in Figure 4, left (which is a slight modificationof (cid:101) D n, ,n − in Figure 3): it has 2 n real four-valent × -vertices, n inner four-valent • -vertices, and 2 n ◦ -vertices, n real four-valent and n inner bivalent. (Recall that each inner vertex in D doubles in B ,so that the total valency of the vertices of each kind sums up to 8 n = 2 deg f , as expected.) This ... ...... ...... ... n ⇐ = Figure 4.
The dessin D n (1) and its modificationsdessin is maximal in the sense that all its regions are triangles. To pass from D n (1) to D n (1 + q ), q ≤ n , we replace small neighbourhoods of q inner ◦ -vertices with the fragments shown in Figure 4,right, creating q extra boundary components.Each dessin D n ( g ) satisfies the two conditions above and, thus, gives rise to a ramified covering f : B → C P . The analytic structure on B is given by the Riemann existence theorem, and O isthe line bundle O B ( P ( f )), where P ( f ) is the divisor of poles of f . (All poles are even.) Then, thecurve in question is given by “equation” (3), with the sections a i ∈ H ( B ; O ⊗ i ) almost determinedby their zeroes: Z ( a ) = P ( f ) and Z ( a ) = Z (1 − f ). Further details of this construction (in themore elaborate trigonal case) can be found in [DIK08, Deg12]. (cid:3) Next few lemmas deal with the real lifts of the curves constructed in Lemma 3.3 under a ramifieddouble covering of Σ n ( O ). First, we discuss the existence of such coverings, cf . Remark 2.7. Lemma 3.4.
Let Σ n ( O ) be a real ruled surface over a real algebraic curve B such that R B (cid:54) = ∅ ,and let D be a real divisor on X . Then there exists a real divisor E on X such that | D | R = 2 | E | R if and only if [ R D ] = 0 ∈ H ( R X ; Z / Z ) .Proof. By [Har77, Proposition 2.3], we havePic(Σ n ( O )) (cid:39) Z B ⊕ Pic( B ) , and this isomorphism respects the action induced by the real structures. Let | D | = m | B | + | D | . Then m = [ R D ] ◦ [ R F ] mod 2, where F is the fiber of the ruling over a real point p ∈ R B , and D = D ◦ B ∞ , so that [ R D ] = [ R D ] ◦ [ R B ∞ ]. There remains to observe that | D | R is divisible by 2in R Pic( B ) if and only if [ R D ] = 0 ∈ H ( B ; Z / Z ). The “only if” part is clear, and the “if” partfollows from the fact that D can be deformed, through real divisors, to (deg D ) p . (cid:3) Lemma 3.5.
Let X := Σ n ( O ) be a real ruled surface over a real algebraic curve B such that R B (cid:54) = ∅ , and let C be a reduced real divisor on X such that [ R C ] = 0 ∈ H ( R X ; Z / Z ) . Then, forany surface S ⊂ R X such that ∂S = R C , there exists a real double covering Y → X ramified over C such that R Y projects onto S .Proof. Pick one covering Y → X , which exists by Lemma 3.4, and let S be the projection of R Y .We can assume that S ∩ T = S ∩ T for one of the components T of R X . Given another component T i , consider a path γ i connecting a point in T i to one on T , and let ˜ γ i = γ i + c ∗ γ i ; in view of theobvious equivariant isomorphism H ( Y ; Z / Z ) (cid:39) H ( B ; Z / Z ), these loops form a partial basis for EAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBER OF REAL POINTS 11 the space of c ∗ -invariant classes in H ( X ; Z / Z ). Now, it suffices to twist Y ( cf . Remark 2.7) bya cohomology class sending ˜ γ i to 0 or 1 if S ∩ T i coincides with S ∩ T i or with the closure of itscomplement, respectively. (cid:3) Lemma 3.6.
Let n , g be two integers, n ≥ g − ≥ , and let B , O , and C n ( g ) ⊂ Σ n ( O ) be as inLemma 3.3. Then there exists a real double covering Σ n ( O (cid:48) ) → Σ n ( O ) ramified along B ∪ B ∞ andsuch that the pullback of C n ( g ) is a finite real algebraic curve C (cid:48) n ( g ) ⊂ Σ n ( O (cid:48) ) with | R C (cid:48) n ( g ) | = 5 n − g . Proof.
By Lemma 3.5, there exists a real double covering Σ n ( O (cid:48) ) → Σ n ( O ) ramified along thecurve B ∪ B ∞ , such that the pull back in Σ n ( O (cid:48) ) of the curve C n ( g ) from Lemma 3.3 is a finitereal algebraic curve C (cid:48) n ( g ). Each node of C n ( g ) gives rise to two solitary real nodes of C (cid:48) n ( g ), andeach tangency point of C n ( g ) and R B gives rise to an extra solitary node of C (cid:48) n ( g ). (cid:3) Deformation to the normal cone.
We briefly recall the deformation to normal cone con-struction in the setting we need here, and refer for example to [Ful84] for more details. Given X a non-singular algebraic surface, and B ⊂ X a non-singular algebraic curve, we denote by N B/X the normal bundle of B in X , its projective completion by E B = P ( N B/X ⊕ O B ), and we define B ∞ = E B (cid:114) N B/X . Note that if both X and B are real, then so are E B and B ∞ .Let X be the blow up of X × C along B ×{ } . The projection X × C → C induces a flat projection σ : X → C , and one has σ − ( t ) = X if t (cid:54) = 0, and σ − (0) = X ∪ E B . Furthermore, in this latter case X ∩ E B is the curve B in X , and the curve B ∞ in E B . Note that if both X and B are real, and ifwe equip C with the standard complex conjugation, then the map σ is a real map.Let C = C X ∪ C B be an algebraic curve in X ∪ E B such that:(1) C X ⊂ X is nodal and intersects B transversely;(2) C B ⊂ E B is nodal and intersects B ∞ transversely; let a = [ C B ] ◦ [ F ] in H ( E B ; Z );(3) C X ∩ B = C B ∩ B ∞ = C X ∩ C B .In the following two propositions, we use [ST06, Theorem 2.8] to ensure the existence of a deforma-tion C t in σ − ( t ) within the linear system | C X + aB | of the curve C in some particular instances.We denote by P the set of nodes of C (cid:114) ( X ∩ E B ), and by I X (resp. I B ) the sheaf of ideals of P ∩ X (resp. P ∩ E B ). Proposition 3.7.
In the notation above, suppose that X ⊂ C P is a quadric ellipsoid, and that B is a real hyperplane section. If C is a finite real algebraic curve, then there exists a finite realalgebraic curve C in X in the linear system | C X + aB | such that | R C | = | R C | . Proof.
One has the following short exact sequence of sheaves0 −→ O ( C X ) ⊗ I X −→ O ( C X ) −→ O P∩ X −→ . (To shorten the notation, we abbreviate O ( D ) = O X ( D ) for a divisor D ⊂ X when the ambientvariety X is understood.) Since H ( X, O ( C X )) = 0, one obtains the following exact sequence0 −→ H ( X, O ( C X ) ⊗I X ) −→ H ( X, O ( C X )) −→ H ( P ∩ X, O P∩ X ) −→ H ( X, O ( C X ) ⊗I X ) −→ . The surface C P × C P is toric and it is a classical application of Riemann-Roch Theorem that H ( X, O ( C X ) ⊗ I X ) has codimension |P ∩ X | in H ( X, O ( C X )) (see for example [Shu99, Lemma 8and Corollary 2]). Since h ( P ∩ X, O P∩ X ) = |P ∩ X | , we deduce that H ( X, O ( C X ) ⊗ I X ) = 0 . The curve B is rational, and the surface E B is the surface Σ . In particular, E B is a toric surfaceand B ∞ is an irreducible component of its toric boundary. Hence we analogously obtain H ( E B , O ( C B − B ∞ ) ⊗ I B ) = 0 . Hence by [ST06, Theorem 3.1], the proposition is now a consequence of [ST06, Theorem 2.8]. (cid:3)
Recall that H ( E B , O ( C B ) ⊗ I B ) is the set of elements of H ( E B , O ( C B )) vanishing on P ∩ E B . Proposition 3.8.
Suppose that X = C P , that B is a non-singular real cubic curve, and that C X = ∅ . If C B is a finite real algebraic curve and if H ( E B , O ( C B ) ⊗ I B ) is of codimension |P| in H ( E B , O ( C B )) , then there exists a finite real algebraic curve C in C P of degree a such that | R C | = | R C B | . Proof.
Recall that E B is a ruled surface over B , i.e. , is equipped with a C P -bundle π : E B → B .By [Har77, Lemma 2.4], we have H i ( E B , O ( C B )) (cid:39) H i ( B ∞ , π ∗ O ( C B )) , i ∈ { , , } . In particular the short exact sequence of sheaves0 −→ O ( C B − B ∞ ) −→ O ( C B ) −→ O B ∞ −→ −→ H ( E B , O ( C B − B ∞ )) −→ H ( E B , O ( C B )) −→ H ( B ∞ , O B ∞ ) −→−→ H ( E B , O ( C B − B ∞ )) −→ H ( E B , O ( C B )) ι −−→ H ( B ∞ , O B ∞ ) −→ . Furthermore, by [GP96, Proposition 3.1] we have H ( E B , O ( C B − B ∞ )) = 0, hence the map ι isan isomorphism.On the other hand, the short exact sequence of sheaves0 −→ O ( C B ) ⊗ I B −→ O ( C B ) −→ O P −→ −→ H ( E B , O ( C B ) ⊗ I B ) −→ H ( E B , O ( C B )) r −−→ H ( P , O P ) −→−→ H ( E B , O ( C B ) ⊗ I B ) ι −−→ H ( E B , O ( C B )) −→ . By assumption, the map r is surjective, so we deduce that the map ι is an isomorphism.We denote by (cid:101) L the invertible sheaf on the disjoint union of E B and C P and restricting to O ( C B ) and O C P on E B and C P respectively. Finally, we denote by L the invertible sheaf on σ − (0) for which C is the zero set of a section. The natural short exact sequence0 −→ L ⊗ I B −→ (cid:101) L ⊗ I B −→ O B −→ −→ H ( σ − (0) , L ⊗ I B ) −→ H ( E B , O ( C B ) ⊗ I B ) ⊕ H ( C P , O C P ) r −−→ H ( B, O B ) −→−→ H ( σ − (0) , L ⊗ I B ) −→ H ( E B , O ( C B ) ⊗ I B ) ι −→ H ( B, O B ) −→ H ( σ − (0) , L ⊗ I B ) −→ . The restriction of the map r to the second factor H ( C P , O C P ) is clearly an isomorphism, hencewe obtain the exact sequence0 −→ H ( σ − (0) , L ⊗I B ) −→ H ( E B , O ( C B ) ⊗I B ) ι −→ H ( B, O B ) −→ H ( σ − (0) , L ⊗I B ) −→ . Since ι = ι ◦ ι is an isomorphism, we deduce that H ( σ − (0) , L ⊗ I B ) = 0. Now the propositionfollows from [ST06, Theorem 2.8]. (cid:3) EAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBER OF REAL POINTS 13 Finite curves in C P In the case X = C P , Theorem 2.5 and Corollary 2.6 specialize as follows. Theorem 4.1.
Let C ⊂ C P be a finite real algebraic curve of degree k . Then, | R C | ≤ k + g ( C ) + 1 , (4) | R C | ≤ k ( k −
1) + 1 . (5)In the rest of this section, we discuss the sharpness of these bounds.4.1. Asymptotic constructions.
The following asymptotic lower bound holds for any projectivetoric surface with the standard real structure.
Theorem 4.2.
Let ∆ ⊂ R be a convex lattice polygon, and let X ∆ be the associated toric surface.Then, there exists a sequence of finite real algebraic curves C k ⊂ X ∆ with the Newton polygon ∆( C k ) = 2 k ∆ , such that lim k →∞ k | R C k | = 43 Area(∆) , where Area(∆) is the lattice area of ∆ . Remark 4.3.
In the settings of Theorem 4.2, assuming X ∆ smooth, the asymptotic upper boundfor finite real algebraic curves C ⊂ X ∆ with ∆( C ) = 2 k ∆ is given by Theorem 2.5: | R C | (cid:46)
32 Area(∆) . Proof of Theorem 4.2.
There exists a (unique) real rational cubic C ⊂ ( C ∗ ) such that • ∆( C ) is the triangle with the vertices (0 , , , • the coefficient of the defining polynomial f of C at each corner of ∆( C ) equals 1; • R C ∩ R > is a single solitary node. Figure 5.
Figure 5 shows a tilling of R by lattice congruent copies of ∆( C ). Intersecting this tilling with k ∆ and making an appropriate adjustment in the vicinity of the boundary, we obtain a convexsubdivision of k ∆ containing k Area(∆) + O ( k ) copies of ∆( C ). Now, for each of these copies,we consider an appropriate monomial multiple of either f ( x, y ) or f (1 /x, /y ). Applying Theorem3.1, we obtain a real polynomial f k whose zero locus in R > consists of k Area(∆) + O ( k ) solitarynodes. There remains to let C k = { f k ( x , y ) = 0 } . (cid:3) Corollary 4.4.
There exists a sequence of finite real algebraic curves C k ⊂ C P , deg C k = 2 k , suchthat lim k → + ∞ k | R C k | = 43 . In the next theorem, we tweak the “adjustment in the vicinity of the boundary” in the proof ofTheorem 4.2 in the case X ∆ = C P . Theorem 4.5.
For any integer k ≥ , there exists a finite real algebraic curve C ⊂ C P of degree k such that | R C | = l − l + 2 if k = 3 l, l + 4 l + 3 if k = 3 l + 1 , l + 12 l + 6 if k = 3 l + 2 . Proof.
Following the proof of Theorem 4.2, we use the subdivision of the triangle k ∆ (with thevertices (0 , k, , k )) shown in Figure 6. In the t -axis ( t = x or y ), each segment oflength 1, 2 or 3 bears an appropriate monomial multiple of 1, ( t − or ( t − ( t + 1), respectively.Thus, each segment (cid:96) of length 2 or 3 gives rise to a point of tangency of the t -axis and the curve { f k = 0 } , resulting in two extra solitary nodes of C k . Similarly, each vertex of k ∆ contained in asegment of length 1 gives rise to an extra solitary node of C k . (cid:3) a) k = 3 l b) k = 3 l + 1 c) k = 3 l + 2 Figure 6.
Remark 4.6.
The construction of Theorem 4.5 for k = 3 , A curve of degree . The construction given by Theorem 4.5 is the best known if k ≤
5. If k = 6, we can improve it by 2 more units. Proposition 4.7.
There exists a finite real algebraic curve C ⊂ C P of degree such that | R C | = 45 . Proof.
Let C (cid:48) = C (cid:48) (1) be a finite real algebraic curve in Σ ( O (cid:48) ) as in Lemma 3.6. Let us denoteby P the set of nodes of C (cid:48) , and by I the sheave of ideals on Σ ( O (cid:48) ) defining P . Since R B (cid:54) = ∅ ,there exists a real line bundle (cid:32)L of degree 3 over B such that O (cid:48) = (cid:32)L ⊗ . This bundle (cid:32)L embeds B into C P as a real cubic curve for which (cid:32)L ⊗ = O is the normal bundle. The proposition will thenfollow from Proposition 3.8 once we prove that H (Σ ( O (cid:48) ) , O (4 B ) ⊗ I ) is of codimension 45 in H (Σ ( O (cid:48) ) , O (4 B )). Let us show that this is indeed the case, i.e. , let us show that given any node p of C (cid:48) , there exists an algebraic curve in O (4 B ) on Σ ( O (cid:48) ) passing through all nodes of C (cid:48) but p . EAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBER OF REAL POINTS 15
Recall that there exists a real double covering ρ : Σ ( O (cid:48) ) → Σ ( O (cid:48)⊗ ) ramified along B ∪ B ∞ withrespect to which C (cid:48) is symmetric, and that C (cid:48) has 18 pairs of symmetric nodes and 9 nodes on B .By Riemann-Roch Theorem, for any line bundle O over B of degree n ≥
1, and given any set P of n − n ( O ) and any disjoint finite subset P of Σ n ( O ), there exists analgebraic curve in O ( B ) containing P and avoiding P . As a consequence, there exists a symmetriccurve in O (2 B ) on Σ ( O (cid:48) ) passing through any 16 pairs of symmetric nodes of C (cid:48) and avoiding allother nodes of C (cid:48) . Altogether, we see that given any node p of C ’, there exists a reducible curvein O (4 B ) on Σ ( O (cid:48) ), consisting in the union of a symmetric curve in O (2 B ) and two curves in O ( B ), and passing through all nodes of C (cid:48) but p . (cid:3) Curves of low genus.
Here we show that inequality (4) of Theorem 4.1 is sharp when thedegree is large compared to the genus.
Theorem 4.8.
Given integers k ≥ and ≤ g ≤ k − , there exists a finite real algebraic curve C ⊂ C P of degree k and genus g such that | R C | = k + g + 1 . Proof.
Consider a real rational curve C ⊂ C with the following properties: • the Newton polygon of C is the triangle with the vertices (0 , , k −
2) and (2 k − , • C intersects the axis y = 0 in a single point with multiplicity 2 k − • R C ∩ { y > } consists of ( k − k −
3) solitary nodes.Such a curve exists: for example, one can take a rational simple Harnack curve with the prescribedNewton polygon (see [Mik00, KO06, Bru15]). Shift the Newton polygon ∆( C ) by 2 units up andplace in the trapezoid with the vertices (0 , k, k − , ,
2) a defining polynomial ofthe curve (cid:101) C ,k − ,g +1 given by Lemma 3.2. Applying Theorem 3.1, we obtain a real rational curve C ⊂ C such that • R C ∩ { y > } consists of ( k − k −
3) + 2 k + g − • C intersects the line y = 0 in k − g − g + 1 additionalpairs of complex conjugated points.If C is given by an equation f ( x, y ) = 0 positive on y >
0, we define C as the curve f ( x, y ) = 0.Each node p ∈ { y > } of C gives rise to two solitary real nodes of C , and each tangency point of C and the axis y = 0 gives rise to an extra solitary node of C . The genus g ( C ) = g is given by theRiemann–Hurwitz formula applied to the double covering C → C : its normalization is branched atthe 2( g + 1) points of transverse intersection of C and the axis y = 0. (cid:3) Finite curves in real ruled surfaces
We use the notation B, O , B , F, Σ n ( O ) introduced in Section 3.2. A real algebraic curve C inΣ n ( O ) realizing the class u [ B ] + v [ F ] ∈ H (Σ n ( O ); Z ) may be finite only if both u = 2 a and v = 2 b are even. General results of the previous sections specialize as follows. Theorem 5.1.
Let C ⊂ Σ n ( O ) be a finite real algebraic curve, [ C ] = 2 a [ B ]+2 b [ F ] ∈ H (Σ n ( O ); Z ) , a > , b > . Then, | R C | ≤ na + 2 ab + g ( C ) + 1 − g ( B ) , (6) | R C | ≤ na (3 a −
1) + 3 ab − ( a + b ) + 1 + ( a − g ( B ) . (7) Proof.
The statement is an immediate consequence of Theorem 2.8 and Corollary 2.6: due to Lemma3.5, we can choose R X + = R X and R X − = ∅ . (cid:3) As in the case of C P , we do not know whether the upper bounds (6) and (7) are sharp ingeneral. In the rest of the section, we discuss the special cases of small a or small genus. The twonext propositions easily generalize to ruled surfaces over a base of any genus (in the same sense asexplained after Lemma 3.3). For simplicity, we confine ourselves to the case of a rational base. Proposition 5.2 ( a = 1) . Given integers b, n ≥ , there exists a finite real algebraic curve C ⊂ Σ n of bidegree (2 , b ) such that | R C | = n + 2 b .Proof. A collection of n + 2 b generic real points in Σ n determines a real pencil of curves of bidegree(1 , b ), and one can take for C the union of two complex conjugate members of this pencil. (cid:3) Proposition 5.3 ( a = 2) . Given integers b, n ≥ , and − ≤ g ≤ n + b − , there exists a finite realalgebraic curve C ⊂ Σ n of bidegree (4 , b ) and genus g such that | R C | = 4 n + 4 b + g + 1 . In particular, if b + n ≥ , then there exists a finite real algebraic curve C ⊂ Σ n of bidegree (4 , b ) such that | R C | = 5 n + 5 b − . Proof.
We argue as in the proof of Lemma 3.6, starting from the curve (cid:101) C n,b,g +1 given by Lemma 3.2.The genus g ( C ) is computed by the Riemann–Hurwitz formula. (cid:3) All rational ruled surfaces are toric, and Theorem 4.2 takes the following form.
Theorem 5.4.
Given integers a > and b ≥ , there exists a sequence of finite real algebraic curves C k ⊂ Σ n of bidegree ( ka, kb ) such that lim k → + ∞ k | R C k | = 43 ( na + 2 ab ) . (cid:3) Furthermore, the proof of Theorem 4.8 extends literally to curves in Σ n . Theorem 5.5 (low genus) . Given integers a > , b, n ≥ , and − ≤ g ≤ n ( a −
1) + b − , thereexists a finite real algebraic curve C ⊂ Σ n of bidegree (2 a, b ) and genus g such that | R C | = na + 2 ab + g + 1 . (cid:3) Finite curves in the ellipsoid
The algebraic surface Σ = C P × C P has two real structures with non-empty real part, namely c h ( z, w ) = (¯ z, ¯ w ) and c e ( z, w ) = ( ¯ w, ¯ z ). The first one was considered in Section 5. In this section,Σ is assumed equipped with the real structure c e , and we have R Σ = S .6.1. General bounds.
Let e and e be the classes in H (Σ ; Z ) represented by the two rulings.The action of c e on H (Σ ; Z ) is given by c e ( e i ) = − e − i , and so σ − inv (Σ , c e ) = 1.The classes in H (Σ ; Z ) realized by real algebraic curves are those of the form m ( e + e ). Forany m ≥
1, a real algebraic curve of bidegree ( m, m ) may have finite real part.
Theorem 6.1.
Let C be a reduced finite real algebraic curve in (Σ , c e ) of bidegree ( m, m ) , with m ≥ . Then (8) | R C | ≤ k + g ( C ) + 3 if m = 2 k k + 4 k + g ( C ) if m = 2 k + 1 . EAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBER OF REAL POINTS 17
In particular we have (9) | R C | ≤ k − k + 2 if m = 2 k k + 2 k if m = 2 k + 1 . Proof.
In order to apply Theorem 2.5, we note that T , (Σ ) = − h , (Σ ) = − , c e ) being a sphere, χ ( R Σ ) = 2.The case when m = 2 k is then provided by Theorem 2.5 and Corollary 2.6. Indeed, in this case,[ C ] = m ( e + e ) = 2 k ( e + e ) and letting e = k ( e + e ), we get e = 2 k and e · c (Σ ) =2 k ( e + e )( e + e ) = 4 k .So suppose that m = 2 k + 1 and let p ∈ R C . Let E and E be a pair of conjugate generatriceswhich meet C at p . Let (cid:101) C = C ∪ E ∪ E and let C be the strict transform of (cid:101) C in the blow-upΣ of Σ at p . The class of the auxiliary curve (cid:101) C in H (Σ ; Z ) is then [ (cid:101) C ] = 2( k + 1)( e + e ). Let e = ( k +1)( e + e ), we get e = 2( k +1) . Let e be half the class of C in H (Σ ; Z ), we get e ≤ e − p is of multiplicity at least 4 in (cid:101) C . Furthermore, we have g ( C ) = g ( (cid:101) C ) = g ( C ) − | R C | = | R (cid:101) C | = | R C | + 1. In order to apply Theorem 2.5 for the curve C on Σ , it remains to notethat T , (Σ ) = T , (Σ ) − χ ( R Σ ) = χ ( R Σ ) −
1. Hence we obtain (8) from Theorem 2.5applied to the curve C on Σ .To get (9), it suffices to remark that, C being a curve of bidegree (2 k + 1 , k + 1), we have g ( C ) ≤ k − | R C | . (cid:3) Remark 6.2.
Let us consider the following problem: given a smooth real projective surface (
X, c )and a homology class d ∈ H ( X ; Z ), what is the maximal possible number of intersection pointsbetween C and R X for a non-real algebraic curve C in X realizing the class d ?Since any two distinct irreducible algebraic curves in X intersect positively, any non-real irre-ducible algebraic curve C in X intersects c ( C ) in − [ C ] · c ∗ [ C ] points, and so intersects R X in atmost − [ C ] · c ∗ [ C ] points. It is easy to see that this upper bound is sharp in C P . Interestingly,Theorem 6.1 shows that this trivial upper bound is not sharp in the case of the quadric ellipsoid.Any irreducible algebraic curve C in Σ realizing the class ( m − ,
1) with m ≥ C and c e ( C ) is a real algebraic curve of geometric genus − m, m ), Theorem 6.1 implies that | C ∩ R Σ | ≤ k + 2 if m = 2 k k + 4 k − m = 2 k + 1 , whereas ( m − , · (1 , m −
1) = m − m + 2 is at least twice as large.Next theorem is an immediate consequence of Theorem 4.2 and Proposition 3.7 Theorem 6.3.
There exists a sequence of finite real algebraic curves C m of bidegree ( m, m ) in thequadric ellipsoid such that lim m →∞ m | R C m | = 43 . Curves of low bidegree.
Next statement shows in particular that Theorem 6.1 is not sharpfor m = 2 and m = 5. Proposition 6.4.
For m ≤ , the maximal possible value δ e ( m ) of | R C | for a finite real algebraiccurve of bidegree ( m, m ) in the quadric ellipsoid is m δ e ( m ) 1 2 5 10 15 Proof.
We start by constructing real algebraic curves with a number of real points as stated in theproposition. For m ≤
4, such a curve is constructed by taking the union of two complex conjugatedcurves of bidegree ( m − ,
1) and (1 , m −
1) intersecting R Σ in ( m − + 1 points. For m ≤ m − m − , m = 4, consider 8 points in R P such that there exists a non-real rational cubic C ⊂ C P passing through these 8 points (such configuration of 8 points exist). Since C has a unique nodalpoint, it has to be non-real. Furthermore, since C intersects R P in an odd number of points, ithas to intersect R P in a ninth point. Hence the union of C with its complex conjugate is a realalgebraic curve of degree 6 with 9 solitary points and two complex conjugate nodal points. Denoteby O the line passing through the two latter. Blowing up the two nodes and blowing down the stricttransform of O , we obtain a real algebraic curve of bidegree (4 ,
4) in the quadric ellipsoid whose realpart has exactly 10 points.The case m = 5 is treated by applying the deformation to the normal cone construction to anon-singular real hyperplane section B , with R B (cid:54) = ∅ , in the quadric ellipsoid X . Here we usenotations from Section 3.3. According to Proposition 5.3, there exists a real algebraic curve C B ofbidegree (4 ,
2) in E B = Σ whose real part consists of 14 solitary nodes. Let C X be a reduciblecurve of bidegree (1 ,
1) in X passing through X ∩ E B ∩ C B , and let us define C = C X ∪ C B . Thecurve C is a finite real algebraic curve with | R C | = 15, hence Proposition 3.7 ensures the existenceof a finite real algebraic curve C of bidegree (5 ,
5) in X with | R C | = 15.We now prove that there does not exist finite real algebraic curves of bidegree m ≤ m, m ) with m = 1 or m = 2 has at most 1 or 2 real pointsrespectively. According to Theorem 6.1, a finite real algebraic curve of bidegree (3 , ,
4) or (5 , ,
10, or 16 real points respectively. Suppose thatthere exists a real algebraic curve of bidegree (5 ,
5) in the quadric ellipsoid with 16 real points. Bythe genus formula, this curve is rational and its 16 real points are all ordinary nodes. By a smallperturbation creating an oval for each node, we obtain a non-singular real algebraic curve of bidegree(5 ,
5) in the quadric ellipsoid whose real part consists of exactly 16 connected components, each ofthem bounding a disc in the sphere. This contradicts the congruence [Mik91, Theorem 1b)]. (cid:3)
References [Bru15] E. Brugall´e. Pseudoholomorphic simple Harnack curves.
Enseign. Math. , 61(3-4):483–498, 2015.[CLR80] Man Duen Choi, Tsit Yuen Lam, and Bruce Reznick. Real zeros of positive semidefinite forms. I.
Math. Z. ,171(1):1–26, 1980.[Deg12] Alex Degtyarev.
Topology of algebraic curves: An approach via dessins d’enfants , volume 44 of
De GruyterStudies in Mathematics . Walter de Gruyter & Co., Berlin, 2012.[DIK08] Alex Degtyarev, Ilia Itenberg, and Viatcheslav Kharlamov. On deformation types of real elliptic surfaces.
Amer. J. Math. , 130(6):1561–1627, 2008.[Ful84] W. Fulton.
Introduction to Intersection Theory in Algebraic Geometry , volume 54 of
BMS Regional Conf.Ser. in Math.
Amer. Math. Soc., Providence, 1984.[GP96] F. J. Gallego and B. P. Purnaprajna. Normal presentation on elliptic ruled surfaces.
J. Algebra , 186(2):597–625, 1996.[Har77] R. Hartshorne.
Algebraic geometry . Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathe-matics, No. 52.[Hil88] David Hilbert. Ueber die Darstellung definiter Formen als Summe von Formenquadraten.
Math. Ann. ,32(3):342–350, 1888.[Hir86] F. Hirzebruch. Singularities of algebraic surfaces and characteristic numbers. In
The Lefschetz centennialconference, Part I (Mexico City, 1984) , volume 58 of
Contemp. Math. , pages 141–155. Amer. Math. Soc.,Providence, RI, 1986.[IKS15] Ilia Itenberg, Viatcheslav Kharlamov, and Eugenii Shustin. Welschinger invariants of real del Pezzo surfacesof degree ≥ Internat. J. Math. , 26(8):1550060, 63, 2015.
EAL ALGEBRAIC CURVES WITH LARGE FINITE NUMBER OF REAL POINTS 19 [KO06] R. Kenyon and A. Okounkov. Planar dimers and Harnack curves.
Duke Math. J. , 131(3):499–524, 2006.[Man17] Fr´ed´eric Mangolte.
Vari´et´es alg´ebriques r´eelles , volume 24 of
Cours Sp´ecialis´es . Soci´et´e Math´ematique deFrance, Paris, 2017.[Mik91] G. Mikhalkin. Congruences for real algebraic curves on an ellipsoid.
Zap. Nauchn. Sem. Leningrad. Otdel.Mat. Inst. Steklov. (LOMI) , 193(Geom. i Topol. 1):90–100, 162, 1991.[Mik00] G. Mikhalkin. Real algebraic curves, the moment map and amoebas.
Ann. of Math. (2) , 151(1):309–326,2000.[Ore03] Stepan Yu. Orevkov. Riemann existence theorem and construction of real algebraic curves.
Ann. Fac. Sci.Toulouse Math. (6) , 12(4):517–531, 2003.[Pet38] I. Petrowsky. On the topology of real plane algebraic curves.
Ann. of Math. (2) , 39(1):189–209, 1938.[Shu99] E. Shustin. Lower deformations of isolated hypersurface singularities.
Algebra i Analiz , 11(5):221–249, 1999.[Shu05] E. Shustin. A tropical approach to enumerative geometry.
Algebra i Analiz , 17(2):170–214, 2005.[Shu06] Eugenii Shustin. The patchworking construction in tropical enumerative geometry. In
Singularities and com-puter algebra , volume 324 of
London Math. Soc. Lecture Note Ser. , pages 273–300. Cambridge Univ. Press,Cambridge, 2006.[ST06] E. Shustin and I. Tyomkin. Patchworking singular algebraic curves. I.
Israel J. Math. , 151:125–144, 2006.[Vir86] O. Ya. Viro. Achievements in the topology of real algebraic varieties in the last six years.
Uspekhi Mat. Nauk ,41(3(249)):45–67, 240, 1986.[Wil78] G. Wilson. Hilbert’s sixteenth problem.
Topology , 17(1):53–73, 1978.
Erwan Brugall´e, Universit´e de Nantes, Laboratoire de Math´ematiques Jean Leray, 2 rue de laHoussini`ere, F-44322 Nantes Cedex 3, France
E-mail address : [email protected] Alex Degtyarev, Bilkent University, Department of Mathematics, 06800 Ankara, Turkey
E-mail address : [email protected] Ilia Itenberg, Institut de Math´ematiques de Jussieu–Paris Rive Gauche, Sorbonne Universit´e, 4place Jussieu, 75252 Paris Cedex 5, France, and D´epartement de Math´ematiques et Applications, EcoleNormale Sup´erieure, 45 rue d’Ulm, 75230 Paris Cedex 5, France
E-mail address : [email protected] Fr´ed´eric Mangolte, Laboratoire angevin de recherche en math´ematiques (LAREMA), Universit´ed’Angers, CNRS, 49045 Angers Cedex 01, France
E-mail address : [email protected] URL ::