Surgery of real symplectic fourfolds and Welschinger invariants

A surgery of a real symplectic manifold $X_{\mathbb R}$ along a real Lagrangian sphere $S$ is a modification of the symplectic and real structure on $X_{\mathbb R}$ in a neigborhood of $S$. Genus 0 Welschinger invariants of two real symplectic $4$-manifolds differing by such a surgery have been related in a previous work in collaboration with N. Puignau. In the present paper, we explore some particular situations where these general formulas greatly simplify. As an application, we complete the computation of genus 0 Welschinger invariants of all del~Pezzo surfaces, and of all $\mathbb R$-minimal real conic bundles. As a by-product, we establish the existence of some new relative Welschinger invariants. We also generalize our results to the enumeration of curves of higher genus, and give relations between hypothetical invariants defined in the same vein as a previous work by Shustin.Comment: 28 pages, 2 figures. V2: Major edition (hopefully simplifications) of the first version, references precised. V3: Minor edition

Behavior of Welschinger Invariants under Morse Simplifications

We relate Welschinger invariants of a rational real symplectic 4-manifold before and after a Morse simplification (i.e deletion of a sphere or a handle of the real part of the surface). This relation is a consequence of a real version of Abramovich-Bertram formula which computes Gromov-Witten invariants by means of enumeration of $J$-holomorphic curves with a non-generic almost complex structure $J$. In addition, we give some qualitative consequences of our study, for example the vanishing of Welschinger invariants in some cases.Comment: 5 pages. This text is an extension of the previous version to symplectic setting. It is an announcement and does not contain proof

A bit of tropical geometry

This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking. Each definition is explained with concrete examples and illustrations. To a great exten, this text is an updated of a translation from a french text by the first author. There is also a newly added section highlighting new developments and perspectives on tropical geometry. In addition, the final section provides an extensive list of references on the subject.Comment: 27 pages, 19 figure

Enumeration of curves via floor diagrams

In this note we compute some enumerative invariants of real and complex projective spaces by means of some enriched graphs called floor diagrams.Comment: 5 pages, 3 figure

Brief introduction to tropical geometry

The paper consists of lecture notes for a mini-course given by the authors at the G\"okova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise

Lifting harmonic morphisms II: tropical curves and metrized complexes

In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve. This article is the second in a series of two.Comment: 35 pages, 18 figures. This article used to be the second half of arXiv:1303.4812, and is now its seque