85 citations · 255 across the 22 of their papers we have counts for
18 papers · 1 filter
Welschinger--Witt invariants
Erwan Brugallé, Johannes Rau, Kirsten Wickelgren
Welschinger invariants are signed counts of real rational curves satisfying contraints. Quadratic Gromov--Witten invariants give such counts over general fields of characteristic d…
Real phase structures on tropical manifolds and patchworks in higher codimension
Johannes Rau, Arthur Renaudineau, Kris Shaw
This paper generalises the homeomorphism theorem behind Viro's combinatorial patchworking of hypersurfaces in toric varieties to arbitrary codimension using tropical geometry. We f…
Real semi-stable degenerations, real-oriented blow-ups and straightening corners
Johannes Rau
We study totally real semi-stable degenerations (and more generally, smooth semi-stable degenerations). Our goal is to describe the homeomorphism type of the real locus $\mathbf{R}…
The tropical Poincaré-Hopf theorem
Johannes Rau
We express the beta invariant of a loopless matroid as tropical self-intersection number of the diagonal of its matroid fan (a "local" Poincaré-Hopf theorem). This provides another…
Spines for amoebas of rational curves
Grigory Mikhalkin, Johannes Rau
To every rational complex curve we associate a rational tropical curve so that the amoeba $\mathcal{A}(C) \subset \mathbf{…
The dimension of an amoeba
Jan Draisma, Johannes Rau, Chi Ho Yuen
Answering a question by Nisse and Sottile, we derive a formula for the dimension of the amoeba of an irreducible algebraic variety.
Lower bounds and asymptotics of real double Hurwitz numbers
Johannes Rau
We study the real counterpart of double Hurwitz numbers, called real double Hurwitz numbers here. We establish a lower bound for these numbers with respect to their dependence on t…
Signed counts of real simple rational functions
Boulos El Hilany, Johannes Rau
We study the problem of counting real simple rational functions with prescribed ramification data (i.e. a particular class of oriented real Hurwitz numbers of genus ). We in…
Lefschetz (1,1)-theorem in tropical geometry
Philipp Jell, Johannes Rau, Kristin Shaw
For a tropical manifold of dimension n we show that the tropical homology classes of degree (n-1, n-1) which arise as fundamental classes of tropical cycles are precisely those in…
Rational quintics in the real plane
Ilia Itenberg, Grigory Mikhalkin, Johannes Rau
From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy…
Tropical Real Hurwitz numbers
Hannah Markwig, Johannes Rau
In this paper, we define tropical analogues of real Hurwitz numbers, i.e. numbers of covers of surfaces with compatible involutions satisfying prescribed ramification properties. W…
The combinatorics of real double Hurwitz numbers with real positive branch points
Mathieu Guay-Paquet, Hannah Markwig, Johannes Rau
We investigate the combinatorics of real double Hurwitz numbers with real positive branch points using the symmetric group. Our main focus is twofold. First, we prove correspondenc…
On rational equivalence in tropical geometry
Lars Allermann, Simon Hampe, Johannes Rau
This article discusses the concept of rational equivalence in tropical geometry (and replaces the older and imperfect version arXiv:0811.2860). We give the basic definitions in the…
The diagonal of tropical matroid varieties and cycle intersections
Georges Francois, Johannes Rau
We define an intersection product of tropical cycles on matroid varieties (via cutting out the diagonal) and show that it is well-behaved. In particular, this enables us to interse…
Intersections on tropical moduli spaces
Johannes Rau
This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that…
Tropical rational equivalence on R^r
Lars Allermann, Johannes Rau
We introduce an improved version of rational equivalence in tropical intersection theory which can be seen as a replacement of chapter 8 of our previous article arXiv:0709.3705v2.…
Tropical descendant Gromov-Witten invariants
Hannah Markwig, Johannes Rau
We define tropical Psi-classes on the moduli space of rational tropical curves in R^2 and consider intersection products of Psi-classes and pull-backs of evaluations on this space.…
First Steps in Tropical Intersection Theory
Lars Allermann, Johannes Rau
We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equi…