Related papers: $\boldsymbol{T}$-algebra homomorphisms between rat…
T-curves are piecewise linear curves which have been used with success since the beginning of the 1990's to construct new real algebraic curves with prescribed topology mainly on the real projective plane. In fact T-curves can be used on…
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…
We study the equivariant cobordism rings for the action of a torus $T$ on smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rational $T$-equivariant cobordism rings of smooth…
Given a tropical linear space $L \subseteq \mathbb{T}^n$ and a matrix $A \in \mathbb{T}^{m \times n}$, the image $AL$ of $L$ under $A$ is typically not a tropical linear space. We introduce a tropical linear space $\mathrm{tropim}_A(L)$,…
Let $X/K$ be a variety over a field, and $A/K$ an abelian variety. A regular homomorphism to $A$ (in codimension $i$) induces, for every smooth geometrically connected pointed $K$-scheme $(T,t_0)$ and every cycle class $Z \in CH^i(T\times…
We give a rigorous definition of tropical fans (the "local building blocks for tropical varieties") and their morphisms. For such a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with…
We study whether a given tropical curve $\Gamma$ in $\mathbb{R}^n$ can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by $\Gamma$. We give an affirmative answer to this…
We construct an analog of the Hodge theory on complex manifolds on tropical curves. We use the analytical approach to the problem, it is based on language of tropical differential forms and methods of $L^2-$cohomologies.
We prove that every connected component of an intersection of tropical hypersurfaces contains a point of their stable intersection unless their stable intersection is empty. This is done by studying algebraic hypersurfaces that tropicalize…
In this paper we generalize correspondence theorems of Mikhalkin and Nishinou-Siebert providing a correspondence between algebraic and parameterized tropical curves. We also give a description of a canonical tropicalization procedure for…
The tropical semiring is a semiring of extended real numbers, where the operations of `max' and `+' replace the usual addition and multiplication, respectively. Difference equations obtained from the ultradiscrete limit of discrete…
Given a family of parameterized algebraic curves over a strictly semistable pair, we show that the simultaneous tropicalization of the curves in the family forms a family of parameterized tropical curves over the skeleton of the strictly…
In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence, one utilizing the algebraic structure of the coordinate semiring of an affine supertropical algebraic set,…
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the…
Let $f \colon R \to B$ be a surjective homomorphism of rings with kernel $I$. Gulliksen (when $I$ is generated by a regular sequence) and later Mehta (in general) showed that for any $B$-modules $M$ and $N$, $\mathrm{Ext}_B^{\ast}(M,N)$ has…
A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The…
We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of…
We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the…
To any algebraic variety X and and closed 2-form \omega on X, we associate the "symplectic action functional" T(\omega) which is a function on the formal loop space LX introduced by the authors in math.AG/0107143. The correspondence \omega…
We define the tropical Tevelev degrees, $\mathsf{Tev}_g^{trop}$, as the degree of a natural finite morphism between certain tropical moduli spaces, in analogy to the algebraic case. We develop an explicit combinatorial construction that…