Related papers: Tropical curves with a singularity in a fixed poin…

In this paper, second installment in a series of three, we give a correspondence theorem to relate the count of genus $g$ curves in a fixed linear system in an abelian surface to a tropical count. To do this, we relate the linear system…

We show that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case(the double local case). Especially we give the normal form of transversely jointed…

Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this paper is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the…

We introduce the notion of families of n-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical…

We prove a new rigidity criterion for families of polarized Calabi-Yau manifolds. Motivated by known non-rigid examples, we conjecture that a family over a quasi-projective curve is rigid if it admits a smooth compactification whose…

We study branched covers of curves with specified ramification points, under a notion of equivalence derived from linear series. In characteristic 0, no non-constant families of covers with fixed ramification points exist. In positive…

We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients,…

Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and…

We study curve singularities in a smooth surface relative to a smooth boundary curve. We consider the semiuniversal deformations and equisingular deformations of curves with a fixed local intersection number $w$ with the boundary, and prove…

This article discusses a combinatorial extension of tropical intersection theory to spaces given by glueing quotients of partially open convex polyhedral cones by finitely many automorphisms. This extension is done in terms of linear…

We consider toric maximum likelihood estimation over the field of Puiseux series and study critical points of the likelihood function using tropical methods. This problem translates to finding the intersection points of a tropical affine…

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…

A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real…

This paper gives a new perspective on singular canards, which is topological in flavour. One key feature is that our construction does not rely on coordinates; consequently, the conditions for the existence of singular canards that we…

Hassett's moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this article we define a tropical analogue of these moduli spaces…

We show that a complex planar curve homeomorphic to the projective line has at most four singular points. If it has exactly four then it has degree five and is unique up to a projective equivalence.

We use a graph to define a new stability condition for algebraic moduli spaces of rational curves. We characterize when the tropical compactification of the moduli space agrees with the theory of geometric tropicalization. The…

We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or…

We construct qualitatively new examples of superabundant tropical curves which are non-realizable in genus $3$ and $4$. These curves are in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively, and have properties resembling canonical embeddings…

We analyze the wave equation in families of pp-wave geometries developing strong localized scale-invariant singularities in certain limits. For both cases of well-localized pp-waves and the so-called null-cosmologies, we observe an…