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Erwan Brugalle

Publications and source records attributed to Erwan Brugalle.

15 recordsLinked to original sources
arXiv2019

Floor decompositions of tropical curves : the planar case

In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed proofs in the tropical geometry framework, in the case when the ambient variety is a complex surface, and give some examples of computations using floor diagrams. The focusing on dimension 2 is motivated by the special combinatoric of floor diagrams compared to arbitrary dimension. We treat a general toric surface case in this dimension: the curve is given by an arbitrary lattice polygon and include computation of Welschinger invariants with pairs of conjugate points. See also \cite{FM} for combinatorial treatment of floor diagrams in the projective case.

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arXiv2016

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

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. Kollár 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.

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arXiv2015

The double Gromov-Witten invariants of Hirzebruch surfaces are piecewise polynomial

We define the double Gromov-Witten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1-dimensional counterpart. Furthermore we show that each polynomial piece is either even or odd, and we compute its degree. Our methods combine floor diagrams and Ehrhart theory.

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arXiv2015

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.

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arXiv2015

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].

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arXiv2014

Deformation of tropical Hirzebruch surfaces and enumerative geometry

We illustrate the use of tropical methods by generalizing a formula due to Abramovich and Bertram, extended later by Vakil. Namely, we exhibit relations between enumerative invariants of the Hirzebruch surfaces $Σ_n$ and $Σ_{n+2}$, obtained by deforming the first surface to the latter. Our strategy involves a tropical counterpart of deformations of Hirzebruch surfaces, and tropical enumerative geometry on a tropical surface in three-space.

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arXiv2014

Obstructions to approximating tropical curves in surfaces via intersection theory

We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on the relation between tropical and complex intersection theories which is also established here. We give two applications of the methods developed in this paper. First we classify all locally irreducible approximable 3-valent fan tropical curves in a non-singular fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space contains finitely many approximable tropical lines if it is of degree 3, and contains no approximable tropical lines if it is of degree 4 or more.

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arXiv2013

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.

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arXiv2010

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.

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arXiv2010

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.

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arXiv2009

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.html

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arXiv2007

A Non-Algebraic Patchwork

Itenberg and Shustin's pseudoholomorphic curve patchworking is in principle more flexible than Viro's original algebraic one. It was natural to wonder if the former method allows one to construct non-algebraic objects. In this paper we construct the first examples of patchworked real pseudoholomorphic curves in $Σ_n$ whose position with respect to the pencil of lines cannot be realised by any homologous real algebraic curve.

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arXiv2006

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.

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arXiv2006

Symmetric plane curves of degree 7 : pseudo-holomorphic and algebraic classifications

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real schemes (resp. complex schemes) realizable by symmetric real curves of degree 7 with respect to the type of the curve (resp. M-symmetric real curves of degree 7). In particular, we exhibit two real schemes which are realizable by real symmetric dividing pseudoholomorphic curves of degree 7 on the projective plane but not by algebraic ones.

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