Related papers: Arithmetic linear series with base conditions

We study Zariski cancellation problem for noncommutative algebras that are not necessarily domains.

We compute the arithmetic volumes of integral models of unitary Shimura curves. This establishes the base case of an inductive argument to compute the arithmetic volumes of unitary Shimura varieties of higher dimension, to appear in…

In this article we describe cell decompositions of the moduli space of Riemann surfaces and their relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational…

In the previous paper [7], we introduced a notion of pairs of adelic R-Cartier divisors and R-base conditions. The purpose of this paper is to propose an extended notion of adelic R-Cartier divisors that we call an l1-adelic R-Cartier…

Zariski chambers are natural pieces into which the big cone of an algebraic surface decomposes. They have so far been studied both from a geometric and from a combinatorial perspective. In the present paper we complement the picture with a…

This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre-Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition…

We evaluate several arctangent and logarithmic integrals depending on a parameter. This provides a closed form summation of certain series and also gives integral and series representation of some classical constants.

Using multiple Bernoulli series, we give a formula in the spirit of Euler MacLaurin formula. We also give a wall crossing formula and a decomposition formula. The study of these series is motivated by formulae of E.Witten for volumes of…

We continue with our study of the arithmetic geometry of toric varieties. In this text, we study the positivity properties of metrized R-divisors in the toric setting. For a toric metrized R-divisor, we give formulae for its arithmetic…

We consider variational problems with regular H{\"o}lderian weight or boundary singularity, and Dirichlet condition. We prove the boundedness of the volume of the solutions to these equations on analytic domains.

Using a Zariski topology associated to a finite field extensions, we give new proofs and generalize the primitive and normal basis theorems.

In this paper we study volumes of moduli spaces of hyperbolic surfaces with geodesic, cusp and cone boundary components. We compute the volumes in some new cases, in particular when there exists a large cone angle. This allows us to give…

We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.

We present new families of weighted homogeneous and Newton non-degenerate line singularities that satisfy the Zariski multiplicity conjecture.

We consider the Zariski-Lipman Conjecture on free module of derivations for algebraic surfaces. Using the theory of non-complete algebraic surfaces, and some basic results about ruled surfaces, we will prove the conjecture for several…

In this paper, we study a general Syracuse problem. We give some necessary conditions concerning the existence of eventual non trivial cycles. Some properties based on linear logarithmic forms are established. New general conjectures are…

A criterion is given for studying (explicit) Baker type lower bounds of linear forms in numbers $1,\Theta_1,...,\Theta_m\in\mathbb{C}^*$ over the ring $\mathbb{Z}_{\mathbb{I}}$ of an imaginary quadratic field $\mathbb{I}$. This work deals…

In this article, we discuss some recent developments of the Zariski Cancellation Problem in the setting of noncommutative algebras and Poisson algebras.

Using the invariant developed in [6], we differentiate four arrangements with the same combinatorial information but in different deformation classes. From these arrangements, we construct four other arrangements such that there is no…

Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…