Related papers: Tropical Fano Schemes

Starting from certain rational varieties blown-up from (P^1)^N, we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we develop an…

In this paper, we give a tropical method for computing Gromov-Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds which admit toric degenerations to toric Fano varieties with singularities…

For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the…

We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical…

Let $X$ be a closed algebraic subset of $\mathbb{A}^{n}(K)$ where $K$ is an algebraically closed field complete with respect to a nontrivial non-Archimedean valuation. We show that there is a surjective continuous map from the Berkovich…

Let $X$ be a smooth complex projective variety with nef $\bigwedge^2 T_X$ and $\dim X \geq 3$. We prove that, up to a finite \'etale cover $\tilde{X} \to X$, the Albanese map $\tilde{X} \to {\rm Alb}(\tilde{X})$ is a locally trivial…

We generalise the notion of Gr\"obner fan to ideals in R[[t]][x_1,...,x_n] for certain classes of coefficient rings R and give a constructive proof that the Gr\"obner fan is a rational polyhedral fan. For this we introduce the notion of…

A projective log variety (X, D) is called "a log Fano manifold" if X is smooth and if D is a reduced simple normal crossing divisor on X with -(K_X+D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this…

We explicitly describe the tropicalization of a cluster variety of finite type C, realizing it as the space of axially symmetric phylogenetic trees. We also find all occurring sign patterns of coordinates, for both the cluster variety and…

Let X be a plane in a torus over an algebraically closed field K, with tropicalization the matroidal fan Sigma. In this paper we present an algorithm which completely solves the question whether a given one-dimensional balanced polyhedral…

The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…

The goal of this paper is to explore the genus and degree of the Fano scheme of linear subspaces on a complete intersection in a complex projective space. Firstly, suppose that the expected dimension of the Fano scheme is one, we prove a…

Let $X$ be a complex smooth Fano variety of dimension $n$. Assume that $X$ admits a birational contraction of an extremal ray. In this paper, we give a classification of such $X$ when the pseudoindex is equal to $\frac{\dim X}{2}$.

In this paper, we study the linear systems $|-mK_X|$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $|-mK_X|$ is non-empty and contains an element with "good singularities" for some natural number $m$…

As a generalization of the Mukai conjecture, we conjecture that the Fano manifolds $X$ which satisfy the property $\rho_X(r_X-1)\geq\dim X-1$ have very special structure, where $\rho_X$ is the Picard number of $X$ and $r_X$ is the index of…

Let $X$ be a complex smooth Fano variety of dimension $n$. In this paper, we give a classification of such $X$ when the pseudoindex is equal to $\dfrac{\dim X+1}{2}$ and the Picard number greater than one.

For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data -- the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant…

We provide an overview of the combinatorial theory of horospherical varieties using coloured fans, a generalization of the combinatorial theory of toric varieties using polyhedral fans.

Classical toric varieties are among the simplest objects in algebraic geometry. They arise in an elementary fashion as varieties parametrized by monomials whose exponents are a finite subset $\mathcal{A}$ of $\mathbb{Z}^n$. They may also be…

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…