Related papers: Global minimal models for endomorphisms of project…

Let $\phi$ be an endomorphism of the projective line defined over a global field $K$. We prove a bound for the cardinality of the set of $K$-rational preperiodic points for $\phi$ in terms of the number of places of bad reduction. The…

We show existence of fundamental domains which minimize a general perimeter functional in a homogeneous metric measure space. In some cases, which include the usual perimeter in the universal cover of a closed Riemannian manifold, and the…

Let $f:(X,B)\to Z$ be a 3-fold extremal dlt flipping contraction defined over an algebraically closed field of characteristic $p>5$, such that the coefficients of $\{B\}$ are in the standard set $\{1-\frac 1n|n\in \mathbb N\}$, then the…

Let K be an algebraic number field. For a degree d rational morphism of projective n-space defined over K let R denote its minimal resultant ideal. For a fixed height function on the moduli space of dynamical systems this paper shows that…

Let $k$ be a global field, let $A$ be a Dedekind domain with $\text{Quot}(A) = k$, and let $K$ be a finitely generated field. Using a unified approach for both elliptic curves and Drinfeld modules $M$ defined over $K$ and having a trivial…

We initiate a study of varieties of minimal degree in weighted projective spaces. We call a weighted projective space $\mathbf{P}(w_0,\dots,w_n)$ divisible if $w_i \mid w_{i+1}$ for all $i$. We provide sharp bounds for when a non-degenerate…

We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study…

This note is an appendix to a preprint by E. Hellmann. We give a complete classification of simple objects of the category of vector spaces D over K = Fpbar((u)) equipped with an endomorphism phi whose image generates D and that are…

Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$, and $F$ be the field of fractions of $A$. Let $\phi$ be a Drinfeld $A$-module of rank $r\geq 2$ over $F$. For all but finitely many primes $\mathfrak{p}\lhd A$, one can…

We study the dynamics of the map endomorphism of N-dimensional projective space defined by f(X)=AX^d, where A is a matrix and d is at least 2. When d>N^2+N+1, we show that the critical height of such a morphism is comparable to its height…

Let $\phi$ be a an endomorphism of degree $d\geq{2}$ of the projective line, defined over a number field $K$. Let $S$ be a finite set of places of $K$, including the archimedean places, such that $\phi$ has good reduction outside of $S$.…

The theory of moduli of morphisms on P^n generalizes the study of rational maps on P^1. This paper proves three results about the space of morphisms on P^n of degree d > 1, and its quotient by the conjugation action of PGL(n+1). First, we…

The first aim of this note is to give a concise, but complete and self-contained, presentation of the fundamental theorems of Mori theory - the nonvanishing, base point free, rationality and cone theorems - using modern methods of…

We prove a result that enables us to calculate the rational homotopy of a wide class of spaces by the theory of minimal models.

In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing…

Given a Riemannian $\mathbb{RP}^3$ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or exist one minimal real projective plane together with two…

Let $f \colon X \to X$ be a surjective endomorphism of a normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure of $X$.…

This article addresses the existence of $\Q$-rational periodic points for morphisms of projective space. In particular, we construct an infinitely family of morphisms on $\P^N$ where each component is a degree 2 homogeneous form in $N+1$…

We construct, for each irrational number $\alpha$, a minimal $C^1$-diffeomorphism of the circle with rotation number $\alpha$ which admits a measur

We present a method for computing projective isomorphisms between rational surfaces that are given in terms of their parametrizations. The main idea is to reduce the computation of such projective isomorphisms to five base cases by…