The moduli of tropical curves and tropical enumerative geometry

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Moduli space of parametrized rational tropical curves

We now turn our focus on the space of rational parametrized tropical curves. Given a rational abstract tropical curve 􀀀, if we specify the slope of every unbounded end, and the position of a vertex, we can define uniquely a parametrized tropical curve h : 􀀀 ! NR.
Therefore, if N denotes the fixed set of slopes of the unbounded ends, the moduli space M0(,NR) of parametrized rational tropical curves of degree is M0,m ×NR, where the NR factor corresponds to the position of the finite vertex adjacent to the first unbounded end.
Definition 3.2.3. The moduli space M0(,NR) of parametrized rational tropical curves of degree in NR is naturally identified with M0,m × NR. Remark 3.2.4. Notice that we have two types of unbounded ends: the unbounded ends with a non-zero slope, which are true unbounded ends, and the unbounded ends which have a zero slope, which are called marked points. Such ends are sent to a unique point in NR. On this moduli space, for each unbounded end i of 􀀀, we have a well-defined evaluation map that associates to each parametrized curve (􀀀, h) the position of the unbounded end, in the quotient of NR by its direction. Let ni be the slope of h along the unbounded end ei of index i 2 [[1;m]]. The evaluation map is evi : M0(,NR) −! NR/hnii (􀀀, h) 7−! h(p) where p 2 ei.

Realization and correspondence theorem in the real case

In this section we prove a correspondence theorem, by refining the realization theorem of Tyomkin [Tyo17] in the case of real curves. The proof follows the same steps as in [Tyo17] and presents similar calculations. We start by giving a bunch of notations which might seem a little heavy, but are useful to deal with real tropical curves having a non-trivial real structure.

Statement of result and plan for the proof

Using Theorem 4.3.9, we now can relate R,s and N@,trop (s) in the case s = (s1, 0, . . . ). From
now on, we assume that only s1 might be non-zero. Moreover, from now on, the complex points in the configuration are purely imaginary. To do such an assumption, one needs to prove the existence of regular values of the evaluation maps that sends a parametrized curve to the coordinates of its intersection points with the boundary. This transversality condition is needed in the proof of invariance in [Mik17]. The existence of regular purely imaginary values is proven close to the tropical limit along with the correspondence theorem, when we check the invertibility of the Jacobian matrix. As the complex points are now purely imaginary, the quantum index is equal to the log-area, and the refined count only needs to be multiplied by 2s1 in order to take into account that the opposite pair of a purely imaginary pair of conjugated points is the same pair.
Let Pt be a symmetric configuration of points depending on a parameter t, chosen as in section 4.2, but with purely imaginary points. This means that one is given a collection ofseries ±i(t) 2 R((t)) [ iR((t)) corresponding to the coordinates of the points ±pi(t) of Pt on the toric divisors. Let μ be the tropicalization of the point configuration, i.e. for a pair of points ±pi(t), we have μi = val i(t). The correspondence theorem, proven in the previous section, provides for t large enough a correspondence between the curves of S(Pt), which are real parametrized curves of degree , and the parametrized tropical curves (􀀀0, h0) of degree (s) such that mom(􀀀0, h0) = μ. This is done by enhancing (h0, 􀀀0) to a real parametrized tropical curve of degree admitting first order solutions, and showing that one can lift every first order solution to a true solution. By counting the first order solutions, we assign a multiplicity to each curve of mom−1(μ), so that the count of mom−1(μ) with these multiplicities gives the invariant R,s. This multiplicity happens to be proportional to the refined multiplicity of Block-Göttsche, thus leading to the relation stated in Theorem 4.4.1.

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Local resolution at the quadrivalent vertices

In this subsection, we solve the enumerative problem for curves of degree (m1,m2,m3), tracing back to subsubsection 4.1.3.3. This resolution is necessary for the counting of first order sollution at quadrivalent vertices, i.e. that have two incoming adjacent fixed edges, and two exchanged adjacent unbounded edges.
Continuing in the setting of subsubsection 4.1.3.3, we deal with real rational curves in the toric surface C(m1,m2,m3). We choose a real point on CE3 and two purely imaginary conjugated points on CE1, and look for real rational curves of degree (m1,m2,m3), maximally tangent to each toric divisor at the given points. The Menelaus theorem ensures that there exists a unique point on CE2 such that each curve passing through the three chosen points also pass through the point on CE2. Such a curve has a parametrization of the form ‘(t) = 􀀀 a(t − c)m1 , b(t − c)2m2(t2 + 1)m3−m2 2 (C)2, .

Computation of the quantum class for toric type I real curves

Let ‘ : CC 99K N C be a type I real parametrized curve. Let S be a connected component of CC\RC, inducing a complex orientation of RC. The map ‘ induces a morphism ‘ : 1(So) ! 1(N C) = H1(N C, Z) ‘ N.
Definition 6.1.16. A type I real curve ‘ : CC 99K N C with S a connected component of CC\RC is of toric type I if ‘1(So) = {0} H1(N C,Z). Lemma 6.1.17 The intersection points of a toric type I real curve with the toric boundary are real. Therefore, the curve has ropip.

Table of contents :

1 Introduction 
1.1 Historique .
1.2 Cadre et Motivations .
1.3 Résultats .
1.4 Plan du manuscrit .
2 Généralités et notations sur les variétés toriques 
2.1 Variétés toriques .
2.2 Courbes paramétrées dans les variétés toriques
2.3 Structure réelle d’une variété torique
2.4 Amibes et Coamibes .
3 Tropical geometry and tropical curves 
3.1 Tropical curves .
3.2 The moduli of tropical curves and tropical enumerative geometry
3.3 Tropicalization .
4 Computation of some refined invariants in toric surfaces 
4.1 Quantum indices of real rational curves
4.2 Refined curve counting in a toric surface
4.3 Realization and correspondence theorem in the real case
4.4 Statement of result and proof .
5 Recursive formula for tropical refined invariants 
5.1 Definition of the invariants and recursive formula
5.2 Proof of the recursive formula .
5.3 Computations
5.4 Recursive formulas for algebraic invariants
6 Curves in higher dimension 
6.1 Quantum class of type I real curve
6.2 Harnack curves in higher dimension
6.3 Enumerative problem
Bibliographie

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