We prove effective upper bounds on the number of effective sections of a Hermitian line bundle over an arithmetic surface. It is an effective version of the arithmetic Hilbert–Samuel formula in the nef case. As a consequence, we obtain effective lower bounds on the Faltings height and on the self-intersection of the canonical bundle in terms of the number of singular points on fibers of the arithmetic surface.

We study problems of maximal symmetry in Banach spaces. This is done by providing an analysis of the structure of small subgroups of the general linear group $\mathrm{GL}\left(X\right)$, where $X$ is a separable reflexive Banach space. In particular, we provide the first known example of a Banach space $X$ without any equivalent maximal norm, or equivalently such that $\mathrm{GL}\left(X\right)$ contains no maximal bounded subgroup. Moreover, this space $X$ may be chosen to be super-reflexive.

Let $S$ be a surface of genus $g$ with $p$ punctures with negative Euler characteristic. We study the diameter of the $ϵ$-thick part of moduli space of $S$ equipped with the Teichmüller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order $log\left(\frac{g+p}{ϵ}\right)$. The same result also holds for the $ϵ$-thick part of the moduli space of metric graphs of rank $n$ equipped with the Lipschitz metric. The proof involves a sorting algorithm that sorts an arbitrarily labeled tree with $n$ labels using simultaneous Whitehead moves, where the number of steps is of order $log\left(n\right)$. As a related combinatorial problem, we also compute, in the appendix of this paper, the asymptotic diameter of the moduli space of pants decompositions on $S$ in the metric of elementary moves.

We study curves of negative self-intersection on algebraic surfaces. In contrast to what occurs in positive characteristics, it turns out that any smooth complex projective surface $X$ with a surjective nonisomorphic endomorphism has bounded negativity (i.e., that $C^2$ is bounded below for prime divisors $C$ on $X$). We prove the same statement for Shimura curves on quaternionic Shimura surfaces of Hilbert modular type. As a byproduct, we obtain that there exist only finitely many smooth Shimura curves on such a surface. We also show that any set of curves of bounded genus on a smooth complex projective surface must have bounded negativity.