A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces
Abstract.
The Welschinger invariants of real rational algebraic surfaces are natural analogues of the genus zero GromovWitten invariants. We establish a tropical formula to calculate the Welschinger invariants of real toric Del Pezzo surfaces for any conjugationinvariant configuration of points. The formula expresses the Welschinger invariants via the total multiplicity of certain tropical curves (nonArchimedean amoebas) passing through generic configurations of points, and then via the total multiplicity of some lattice paths in the convex lattice polygon associated with a given surface. We also present the results of computation of Welschinger invariants, obtained jointly with I. Itenberg and V. Kharlamov.
Key words and phrases:
Real algebraic curves, toric Del Pezzo surfaces, plane tropical curves, enumerative invariants1991 Mathematics Subject Classification:
Primary 14N10, 14P25. Secondary 12J25, 14J26, 14M25Introduction
One of the important questions of real enumerative geometry is: for a real algebraic surface , how many real rational curves in an ample linear system pass through a conjugationinvariant configuration of distinct generic points in ? Similarly to the complex case, where the answer is given by GromovWitten invariants, the recently discovered Welschinger invariants [16, 17] appear to be an ultimate tool to handle the question over the reals. In particular, they led to nontrivial positive lower bounds for the numbers in question, provided that the configuration consisted of only real points [3].
So far no closed or recursive formula is found for the Welschinger invariants, and tropical enumerative geometry [8, 9, 13] provides the only known approach to compute them. In the present paper we express the Welschinger invariants for configurations of real and imaginary conjugate points on real toric Del Pezzo surfaces via the number of certain subdivisions of the corresponding convex lattice polygons.
Welschinger invariants. Let be , or the hyperboloid , or the plane blown up at generic real points, equipped with the standard real structure. Let be a real ample divisor on , and let the nonnegative integers satisfy
(0.1) 
Denote by the set of configurations of distinct points of such that of them are real and the rest form pairs of imaginary conjugate points. The Welschinger number is the sum of weights of all the real rational curves in , passing through a generic configuration^{1}^{1}1The generality means here that all the complex rational curves through the configuration are nodal irreducible, and their number is equal to the corresponding GromovWitten invariant. , where the weight of a real rational curve is if it has an even number of real solitary nodes, and is otherwise. The surfaces as above are the only toric surfaces, whose complex structure determines a symplectic structure which is generic in its deformation class, and thus, by Welschinger’s theorem [16, 17], does not depend on the choice of a generic element (a simple proof of the latter independence in the algebraic setting is found in [4]). The importance of the Welschinger invariant comes from an immediate inequality
(0.2) 
where is the number of real rational curves in passing through a generic configuration , and is the number of complex rational curves in , passing through generic points in . We should notice that even the existence of a positive lower bound for was not reached by any other method, beginning with the case of plane quartics.
A tropical calculation of the Welschinger invariant. Our approach to calculating the Welschinger invariant is quite similar to that in [3], and it heavily relies on the enumerative tropical algebraic geometry developed in [8, 9, 13]. More precisely, we replace the complex field by the field of the complex locally convergent Puiseux series equipped with the standard complex conjugation and with a nonArchimedean valuation
A rational curve over passing through a generic configuration is represented as an equisingular family of real rational curves in over the punctured disc, and its limit at the disc center determines a tropical curve in (called a real rational tropical curve), which passes through the configuration , coordinatewise projection of .
Our first main result is Theorem 3.1 (section 3), which precisely describes the real rational tropical curves, passing through generic configurations of points in , and, for any such real rational tropical curve , determines the contribution to the Welschinger invariant of all the real algebraic curves, projecting to . The proof is based on the techniques of [13].
In the case of configurations, consisting of only real points, this result has been obtained in [9, 13]. We notice that an extension of the tropical formula to the case of configurations of real and imaginary conjugate points, requires some further development of the techniques of tropical enumerative geometry, which we present in this paper. The principal difficulty in the latter case is that the configuration contains fewer points than , since a pair of conjugate points projects to the same point in .
The second main result is Theorem 4.2 (section 4.2), which reduces the count of real rational tropical curves, passing through a generic configuration in , to the count of the total weight of certain lattice paths in the convex lattice polygon corresponding to the divisor . The weight of a lattice path is the sum of Welschinger numbers of certain subdivisions of the given polygon into convex lattice subpolygons, which can be produced from the lattice path in a finite combinatorial algorithm (section 4.2). Here we follow the Mikhalkin’s idea to place the configuration on a straight line.
Applications. The tropical formula turns the computation of Welschinger invariants into a purely combinatorial problem on geometry of lattice paths and lattice subdivisions of convex lattice polygons. We present here some results, obtained in this way jointly with I. Itenberg and V. Kharlamov (see [5] for a detailed presentation and proofs). These results concern the positivity of Welschinger invariants, their monotone behavior with respect to the number of imaginary points in configurations, and the asymptotics with respect to the growing degree of the divisor . We formulate a few natural conjectures.
A. The positivity and monotonicity of the Welschinger invariant. The positivity of for all real toric Del Pezzo surfaces and ample divisors was shown in [3]. For , the invariants can vanish (see, for example, computation of , being a line in the plane, in section 4.3).
Theorem 0.1.
(2) The following inequalities hold:

, for , , , or , with an ample divisor such that ;

, and , for , , and an ample divisor .
Conjecture 0.2.
([5]) For a real unnodal Del Pezzo surface and any ample divisor on , the Welschinger invariants are positive as , and are nonnegative for . Furthermore, they satisfy the monotonicity relation
We notice that the monotonicity and nonnegativity of Welschinger invariants are closely related, since, by [16], Theorem 2.2, the first difference of the function is twice the Welschinger invariant for the surface blown up at one real point.
B. The asymptotics of the Welschinger invariants.
Theorem 0.3.
([5]) The Welschinger invariants of the plane satisfy the relation
(0.3) 
For , or , , it holds that
(0.4) 
This means that the number of real rational curves passing through any generic conjugationinvariant configuration of points in , where is bounded as in the assertion, is asymptotically equal to the number of all complex rational curves in the logarithmic scale.
We propose a natural extension of Theorem 0.3 to all Del Pezzo surfaces and other values of :
Conjecture 0.4.
([5]) Let be an ample divisor on a real unnodal Del Pezzo surface . Then, for any fixed ,
The following statement implies Conjecture 0.4 for the hyperboloid:
Theorem 0.5.
([5]) Let be an ample divisor on of bidegree , , and let a sequence of positive integers satisfy . Then
(0.5) 
Organization of the material. The paper is structured as follows: in section 1 we introduce tropical curves, in section 2 we describe tropical limits of rational curves defined over the field , in section 3 we prove the tropical formula for Welschinger invariants, in section 4 we obtain an explicit combinatorial description for Welschinger invariants via lattice paths and subdivisions of convex lattice polygons.
Acknowledgements. I am grateful to I. Itenberg and V. Kharlamov for very valuable discussions. I thank the referee for a careful reading of the manuscript and pointing out a number of defects in the preliminary version.
1. NonArchimedean amoebas and tropical curves
1.1. Basic definitions and notation
In sections 1 and 3 we assume that is a nondegenerate lattice polygon in , is a toric surface associated with , and is the tautological linear system, generated by the monomials , .
In particular, the surfaces , , , , are naturally associated with the polygons shown in Figure 1:

the triangle , if , ,

the rectangle , if , ,

the trapeze if , ,

the pentagon , if , ,

the hexagon if , .
Observe that
(1.1) 
The amoeba of a curve is defined as the closure of the set . By Kapranov’s theorem (see [2, 6, 7]) the amoeba of a curve , given by the equation
(1.2) 
with as the Newton polygon of , is the corner locus of the convex piecewise linear function
(1.3) 
In particular, is a planar graph with all vertices of valency , consisting of closed segments and rays.
An amoeba with Newton polygon is called reducible, if is the union of two amoebas with Newton polygons such that (Minkowski sum).
Take the convex hull of the set and define the function
(1.4) 
This is a convex piecewise linear function, whose linearity domains form a subdivision ^{2}^{2}2Clearly, does not depend on the choice of a polynomial defining the curve . of into convex lattice polygons . The function is Legendre dual to (see, for instance, [2]), and thus, the subdivision is combinatorially dual to the pair .
We define the tropical curve, corresponding to the algebraic curve , as a balanced graph, supported at (cf. [2, 11]), i.e., this is the nonArchimedean amoeba , whose edges are assigned weights equal to the lattice lengths^{3}^{3}3We define the lattice length of a segment with integral endpoints as . of the dual edges of . The subdivision can be uniquely restored from the tropical curve [11] (we denote a tropical curve and the supporting amoeba by the same symbol, no confusion will arise).
1.2. Rank of a tropical curve
Let be a tropical curve with Newton polygon , given by a tropical polynomial (1.3). On the righthand side of formula (1.3) we remove unnecessary linear functions, take the terms in the remaining linear functions as variables, and factorizing by common shift, obtain the space , where is the set of the vertices of the dual subdivision . The set of tropical curves with Newton polygon , which are combinatorially isotopic to the given curve (or, equivalently, are dual to the same subdivision of ), is defined in by certain linear relations and inequalities, and thus forms a convex polyhedron. Its dimension is called the rank of the tropical curve . Clearly, the rank of a tropical curve is bounded from below by the virtual rank
where is the set of vertices of , and is the valency of the vertex . In terms of ,
where , are the sets of vertices of and , respectively.
Definition 1.1.
Let be distinct points, a tropical curve with Newton polygon . The set of tropical curves such that are vertices of , and imposes a number of linear conditions on the variables , introduced above. We say, that the pair of configurations is in generic position if the aforemention set of curves either is empty, or has codimension in . We say that a configuration is generic, if it is generic for any division into a pair of configurations and any tropical curve with Newton polygon .
Lemma 1.2.
The set of generic configurations of points in is dense in the set of all tuples in . In particular, if a tropical curve with Newton polygon passes through a generic configuration so that are vertices of , then
(1.5) 
Proof.
The requirements, imposed by the pair of configurations on , can be written as linear conditions on the variables in , two for a vertex of , and one for . The failure of generality in this case means just a linear relation to the coordinates of . Since there are only finitely many combinatorial types of tropical curves with Newton polygon , we obtain that the set of generic tuples is the complement of finitely many hyperplanes in . ∎
A maximal straight line interval, contained in , is called an extended edge of . The edges of , forming an extended edge, are dual to a sequence of parallel edges of ordered so that each two successive edges of this sequence belong to one polygon . We say that an extended edge of is dual to each of the edges of in the corresponding sequence.
Having a tropical curve with Newton polygon and a generic configuration of point on it, we call the set of vertices of , coinciding with points of the configuration, and the set of extended edges of , containing points of the configuration in their interior, a basic set of extended edges and vertices of .
2. Tropicalization of real rational algebraic curves
We work over the field and its real subfield .
2.1. Tropical limit
Fix some generic collections of points and in . Let be generic points, satisfying , , and let , , , be generic points in , satisfying , , , . That is
and
(2.1) 
(2.2) 
Let a rational curve , given by a polynomial
(2.3) 
be defined over , and pass through , , and , , .
Changing the parameter , we make all the exponents of in , , integral and the function integralvalued at integral points. Introduce the polyhedron
It defines a toric variety , which is naturally fibred over so that the fibres , , are isomorphic to , and is the union of toric surfaces , , with being the faces of the graph of . By the choice of , , and we shall write that . Then (2.3) defines an analytic surface in a neighborhood of , which, by [13], Lemma 2.3, fibers into equisingular rational curves , and whose closure intersects along the curve that can be identified with . Passing if necessary to a finite cyclic covering ramified along , we can make nonsingular everywhere but may be at finitely many points, corresponding to the vertices of , and, in addition, make the surfaces , , smooth and intersecting transversally in . As in [13], we define the tropical limit (tropicalization) of the curve to be the pair, consisting of the tropical curve , and a collection of real curves , , which are defined by
(2.4) 
respectively, where , .
2.2. Tropical limits of real rational curves
A tropical curve is called nodal if the polygons of the dual subdivision are triangles and parallelograms. Recall that, by [13], Lemma 2.2, if is nodal. Notice also that the weights of the edges of a nodal tropical curve are constant along its extended edges, and thus, one can speak of weights of extended edges of nodal tropical curves.
An irreducible nodal tropical curve is called real rational of type with , satisfying (0.1), and , if

,

the weights of the semiinfinite edges of are or , i.e., the edges of lying on are of length or .
Proposition 2.1.
Under the assumptions of section 2, is a real rational tropical curve of type for some , passing through and in such a way that precisely of the points are trivalent vertices of , whereas and the remaining points among (which we denote ) are not vertices of , and, moreover, lie on edges of of even weight. Furthermore, for ,

if is a parallelogram, then is the product of a monomial and few irreducible binomials,

if is a triangle, then is either a real rational curve crossing at , , or points, at which it is unibranch, or is the union of two imaginary conjugate rational curves such that any component of crosses at precisely three points and is unibranch there.
Proof.
Our argumentation is similar to that in [13], section 3.3, used to establish that the nonArchimedean amoebas of nodal curves in toric surfaces passing through a respective number of generic points, are nodal.
Step 1. Let and be vertices of for some , , and , , , , not be vertices. By (1.5),
(2.5) 
By [13], Lemma 2.2, we have
where
being the number of gons in , and being the number of gons in whose opposite edges are parallel. Substituting the two latter relations into (2.5) and using the Euler formula for the subdivision , one obtains
(2.6) 
Denote by the Euler characteristic of the normalization. Let , , be all the components of , , repeating each component with its multiplicity in , and let be the number of local branches of centered along . Since the rational curves , , degenerate into in a flat family, all the components of are rational in view of the inequality for geometric genera (see [1], Proposition 2.4, or [10]^{4}^{4}4The hypotheses of Proposition 2.4 [1] require that all members of the family are reduced curves, which can be achieved in our situation by the normalization of the whole family .). Denote by the set of components , defined by irreducible binomials, and by the set of remaining components . Let be the union of regular neighborhoods in the threefold of the intersection points of with . Then
(2.7) 
the latter inequality following from [13], Lemma 3.2 and Remark 3.4, where stands for the number of local branches (counting multiplicities) of the curves centered at with running over all the edges of .
Step 2. For an estimation of the righthand side of (2.7) we shall construct some graph .
First, we compactify into by adding an infinite point . The tropical curve is then compactified by closing the semiinfinite edges with the point .
For any , the reduced curve is split into irreducible real components and pairs of imaginary conjugate irreducible components. All such real components or pairs of imaginary conjugate components for all , except for real irreducible components from the set , and the point are taken as the vertices of the graph . If is a common edge, and components or and of contain a common pair of imaginary conjugate points in , then we join and by an edge. If is an edge of , and a component of contains a pair of imaginary conjugate points in , then we join and by an edge.
The constructed graph is then transformed as follows. Let be a part of , consisting of two imaginary conjugate components from the set . The curve intersects with and , where is a pair of parallel edges of . If , , then we remove the vertex and all ending at it edges, and instead join by an edge any two curves and which have been joined with . If and , then we get rid of the vertex of the graph and all adjacent edges, instead connecting with any curve , which has been joined with . In this manner, we get rid of onebyone all the vertices of , corresponding to the curves which consist of two imaginary conjugate components from the set .
We observe that the graph naturally projects to , when sending any vertex of to the vertex of dual to , and sending edges of to respective segments of the compactified extended edges of .
The required graph will be a subgraph of .
First, we define a subgraph of as follows. Each point , , lies on an edge of . Let be the dual edge of . By [13], formula (3.7.17), , where is the truncation of the polynomial , defined by (2.4), on the edge , and are taken from formula (2.2). That is, contains a pair of distinct imaginary conjugate points, and , which belong to curves such that . Hence the edge of , containing , is covered by the projection of some edges of . We choose one such covering edge of , denote it by , and build the graph from the edges , , with their endpoints as vertices of .
Notice that due to the generic position of the points , , the graph is the union of trees, the valency of its vertices that differ from is at most two, and the intersection of the projections of any two of its edges is finite.
A vertex of of valency two corresponds to some curve , . If is the union of two imaginary conjugate rational curves such that each of them crosses at precisely three points, and is unibranch at these intersection points, then we call the vertex of an extendable vertex. Notice that the components of are defined by polynomials with the same Newton triangle, and are determined by , , up to a finite choice in view of [13], Lemma 3.5. Furthermore, due to the generality of the coefficients in (2.2), the intersection is either empty or a pair of imaginary conjugate points for any . Exactly one of these pairs lies on an edge of , whose dual edge is not covered by . However, the latter edge is covered by some edges of with an endpoint . We choose one such edge of and append it to .
Performing this procedure for all extendable vertices of , we obtain a graph . Next we determine the extendable vertices of , using the same definition, and append, if necessary, new edges to , obtaining . Repeating this procedure, we finally end up with some subgraph of .
Step 3. An important observation is that no new edge in can join two extendable vertices of . Indeed, the position of two nonadjacent extendable vertices of is uniquely determined by the position of the corresponding four points among , and thus, due to the generality of the latter points, the straight line through the given extendable vertices is not orthogonal to any of the segments joining integral points in . By a similar reason, no edge from can join two extendable vertices of , . In particular, this means that the number of edges in is equal to the total number of extendable vertices in .
Furthermore, assume that is a valent vertex of , , which is not extendable for any , . Then cannot be the union of two imaginary conjugate rational curves, such that each of them crosses at precisely points and is unibranch there. Indeed, if the situation were as described, the position of any of the intersection points of a component of with would determine this component and the fourth intersection point uniquely up to a finite choice. Thus, the relation on the coordinates of the intersection points of with would imply a relation on the coordinates of the points , , contrary to their general choice. In particular, this implies that the contribution of such a vertex to the sum on the righthand side of (2.7) is , and the equality corresponds to the case of being the union of two imaginary conjugate components such that a component crosses at points and is unibranch there.
We also observe that the points and , which are vertices of by assumption, are not projections of the vertices of . Indeed, the points and cannot be projections of the vertices of in view of a general position of these points with respect to , , which in turn determine the projections of the edges of . Next, the projections of the edges of , and thus, the projections of the extendable vertices of are determined by up to a finite choice, where the number of choices is bounded from above by the total number of possible lattice edges in