arXiv subjects

Benoît Bertrand

Publications and source records attributed to Benoît Bertrand.

3 recordsLinked to original sources

arXiv2019

Planar Tropical Cubic Curves of Any Genus, and Higher Dimensional Generalisations

We study the maximal values of Betti numbers of tropical subvarieties of a given dimension and degree in $\mathbb{TP}^n$. We provide a lower estimate for the maximal value of the top Betti number, which naturally depends on the dimension and degree, but also on the codimension. In particular, when the codimension is large enough, this lower estimate is larger than the maximal value of the corresponding Hodge number of complex algebraic projective varieties of the given dimension and degree. In the case of surfaces, we extend our study to all tropical homology groups. As a special case, we prove that there exist planar tropical cubic curves of genus $g$ for any non-negative integer $g$.

math.AG↗

arXiv2017

Haas' theorem revisited

Haas' theorem describes all partchworkings of a given non-singular plane tropical curve $C$ giving rise to a maximal real algebraic curve. The space of such patchworkings is naturally a linear subspace $W_C$ of the $\mathbb{Z}/2\mathbb{Z}$-vector space $\overrightarrow Π_C$ generated by the bounded edges of $C$, and whose origin is the Harnack patchworking. The aim of this note is to provide an interpretation of affine subspaces of $\overrightarrow Π_C $ parallel to $W_C$. To this purpose, we work in the setting of abstract graphs rather than plane tropical curves. We introduce a topological surface $S_Γ$ above a trivalent graph $Γ$, and consider a suitable affine space $Π_Γ$ of real structures on $S_Γ$ compatible with $Γ$. We characterise $W_Γ$ as the vector subspace of $\overrightarrow Π_Γ$ whose associated involutions induce the same action on $H_1(S_Γ,\mathbb{Z}/2\mathbb{Z})$. We then deduce from this statement another proof of Haas' original result.

math.AG↗

arXiv2013

On the total curvature of tropical hypersurfaces

This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical hypersurface defined over the field of real Puiseux series, it has a real part RV which is a polyhedral complex. We define the total curvature of V (resp. RV) by using the total curvature of Amoebas and passing to the limit. We also define the "polyhedral total curvature" of the real part RV of a generic tropical hypersurface. The main results we prove about these notions are the following: - The fact that the total curvature and the polyhedral total curvature coincide for real non-singular tropical hypersurfaces. - A universal inequality between the total curvatures of V and RV and another between the logarithmic curvatures of the real and complex parts of a real algebraic hypersurface. -The fact that this inequality is sharp in the non-singular case.

math.AG↗