Energy functions on moduli spaces of flat surfaces with erasing forest

Flat surfaces with erasing forest are obtained by deforming the flat metric structure of translation surfaces, the moduli space of such surfaces is a deformation of the moduli space of translation surfaces. On the moduli space of flat surfaces with erasing forest, one can define some energy function involving the area of the surface, and the total length of the erasing forest. Note that on this space, we have a volume form which is defined by using geodesic triangulations. The aim of this paper is to prove that the integral of the energy functions mentionned above with respect to this volume form is finite. As applications, we will use this result to recover some classical results due to Masur-Veech, and Thurston

Dynamical Anomalous Subvarieties: Structure and Bounded Height Theorems

According to Medvedev and Scanlon, a polynomial $f(x)\in \bar{\mathbb Q}[x]$ of degree $d\geq 2$ is called disintegrated if it is not linearly conjugate to $x^d$ or $\pm C_d(x)$ (where $C_d(x)$ is the Chebyshev polynomial of degree $d$). Let $n\in\mathbb{N}$, let $f_1,\ldots,f_n\in \bar{\mathbb Q}[x]$ be disintegrated polynomials of degrees at least 2, and let $\varphi=f_1\times\ldots\times f_n$ be the corresponding coordinate-wise self-map of $({\mathbb P}^1)^n$. Let $X$ be an irreducible subvariety of $({\mathbb P}^1)^n$ of dimension $r$ defined over $\bar{\mathbb Q}$. We define the \emph{$\varphi$-anomalous} locus of $X$ which is related to the \emph{$\varphi$-periodic} subvarieties of $({\mathbb P}^1)^n$. We prove that the $\varphi$-anomalous locus of $X$ is Zariski closed; this is a dynamical analogue of a theorem of Bombieri, Masser, and Zannier \cite{BMZ07}. We also prove that the points in the intersection of $X$ with the union of all irreducible $\varphi$-periodic subvarieties of $({\mathbb P}^1)^n$ of codimension $r$ have bounded height outside the $\varphi$-anomalous locus of $X$; this is a dynamical analogue of Habegger's theorem \cite{Habegger09} which was previously conjectured in \cite{BMZ07}. The slightly more general self-maps $\varphi=f_1\times\ldots\times f_n$ where each $f_i\in \bar{\mathbb Q}(x)$ is a disintegrated rational map are also treated at the end of the paper.Comment: Minor mistakes corrected, slight reorganizatio

Regularity bounds for complexes and their homology

Let $R$ be a standard graded algebra over a field $k$. We prove an Auslander-Buchsbaum formula for the absolute Castelnuovo-Mumford regularity, extending important cases of previous works of Chardin and R\"omer. For a bounded complex of finitely generated graded $R$-modules $L$, we prove the equality $\text{reg}~ L=\max_{i\in \mathbb Z} \{\text{reg}~ H_i(L)-i\}$ given the condition $\text{depth}~ H_i(L)\ge \dim H_{i+1}(L)-1$ for all $i<\sup L$. As applications, we recover previous bounds on regularity of Tor due to Caviglia, Eisenbud-Huneke-Ulrich, among others. We also obtain strengthened results on regularity bounds for Ext and for the quotient by a linear form of a module.Comment: Final versio