On Welschinger invariants of symplectic 4-manifolds

We prove the vanishing of many Welschinger invariants of real symplectic $4$-manifolds. In some particular instances, we also determine their sign and show that they are divisible by a large power of 2. Those results are a consequence of several relations among Welschinger invariants obtained by a real version of symplectic sum formula. In particular, this note contains proofs of results announced in [BP13].Comment: 26 pages, 9 figures. v3: many details added, previous sections 2 and 3 have been merge

A Viro Theorem without convexity hypothesis for trigonal curves

A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision. It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a coherent way we prove that no convexity hypothesis is required to patchwork such curves.Comment: 26 pages, 18 figure

Pencils of quadrics and Gromov-Witten-Welschinger invariants of CP3\mathbb C P^3

We establish a formula for the Gromov-Witten-Welschinger invariants of $\mathbb CP^3$ with mixed real and conjugate point constraints. The method is based on a suggestion by J. Koll\'ar that, considering pencils of quadrics, some real and complex enumerative invariants of $\mathbb CP^3$ could be computed in terms of enumerative invariants of $\mathbb CP^1\times\mathbb CP^1$ and of elliptic curves.Comment: 14 pages, 4 figures, minor corrections following referee's suggestion

Pseudoholomorphic simple Harnack curves

We give a new proof of Mikhalkin's Theorem on the topological classification of simple Harnack curves, which in particular extends Mikhalkin's result to real pseudoholomorphic curves.Comment: 10 pages, 7 figures. v3: typos corrected, Theorem 1 precise

Un peu de geometrie tropicale

This basic introduction to tropical geometry is hopefully accessible to a first years student in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking. I tried as much as possible to illustrate each new definition with concrete examples and nice pictures. As the title suggests, this text is in French. A Portuguese (Brazil) version, as well as correction of exercises, can be found at http://people.math.jussieu.fr/~brugalle/largerpubli.htmlComment: 16 pages, 15 figures, 13 exercise

Genus 0 characteristic numbers of the tropical projective plane

Finding the so-called characteristic numbers of the complex projective plane ${\mathbb C}P^2$ is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given $d$ and $g$ one has to find the number of degree $d$ genus $g$ curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is $3d-1+g$ so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when $g=0$. Namely, we show that the tropical problem is well-posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of \CC P^2 in terms of open Hurwitz numbers.Comment: 55 pages, 23 figure

Tropical Open Hurwitz numbers

We give a tropical interpretation of Hurwitz numbers extending the one discovered in \cite{CJM}. In addition we treat a generalization of Hurwitz numbers for surfaces with boundary which we call open Hurwitz numbers.Comment: 10 pages, 6 figure

Recursive formulas for Welschinger invariants of the projective plane

Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we prove a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points.Comment: 18 pages, 2 figure